ID 5138

*Computer Update Problem *

I spot a computer at work which is doing updates.

When I arrive it says '93% complete'.

I immediately start timing the activity and it takes 8 minutes to change to '95% complete'.

Assuming the 'percentage complete' relates directly to time, how much longer will it take to reach '100% complete'?

ID 5139

Eugenia wants to make a simple bridge for her dog. Currently he has to run through a tiny stream in the back garden and then walks mud into the house. Since Eugenia’s dad owns and runs a machine shop, she can easily get a single sheet of steel, aluminum or wood to bridge the stream.

The length suits the size of the stream, the width suits the size of the dog, and the weight will be as much as she can carry.

The strength of a plain sheet is proportional to the relative strength of the material, its width and the cube of its thickness.

Which of the available materials makes the strongest bridge?

ID 5143

Now that I have made a vast fortune from my patented premium dog biscuits, I can afford to build the luxury mansion of my dreams.

I thought my design requirement was very clear: The water for the walk-through shower can be turned ON and OFF from both ends of the room.

The plumber doesn't understand so I have drawn him a plan. I had a few attempts before I got it right!

Which is the correct drawing?

ID 5146

A plastic ruler is much easier to bend in one direction than another according to the proportionality shown, where **b** is the breadth and **d** is the depth of the rectangular cross-section.

The ruler can be approximated as being 1mm thick and 20mm wide.

How much stiffer is it in the hard-to-bend direction?

WARNING: Unless the ruler has some sort of anti-shatter statement written on it, do not try bending it like this without using eye protection. Older designs of rulers are known to shatter and eject bits of plastic into nearby eyes.

ID 5147

I roll two dice, one with the left hand and one with the right.

If the left hand die gives an odd number, the overall score is zero.

If the right hand die gives an even number, I roll it again and again until it is odd.

The score is the sum of the two numbers, except for the previously mentioned case.

There are exactly 6 possible scores: 0, 3, 5, 7, 9, and 11.

What is the probability of a score of 3?

ID 5148

I roll two dice, one with the left hand and one with the right.

If the left hand die gives an odd number, the overall score is zero.

If the right hand die gives an even number, I roll it again and again until it is odd.

The score is the sum of the two numbers, except for the previously mentioned case.

There are exactly 6 possible scores: 0, 3, 5, 7, 9, and 11.

What is the average score?

ID 5152

There are two gods named Orbis and Fidelis, one on your left, the other on your right, but you do not know which is which. Fidelis always answers correctly. Orbis only answers alternate questions correctly; you do not know if his last answer was correct.

You must determine which god is which using the minimum number of YES/NO questions. How many questions do you need to be certain?

To clarify the matter, if Orbis is answering incorrectly your entire question is evaluated correctly and then the answer is reversed. A single question to both gods counts as two questions.

Inspired by G. Boolos. 'The hardest logic puzzle ever', The Harvard Review of Philosophy (6), 1996.

ID 5164

I am allergic to washing powder, but I don’t want my shirt to smell. The dermatologist has told me to reduce the total amount of washing powder residue on my shirt to less than 1 pico-gram. (I think he just made that number up on the spot!)

My dry shirt measures 280g, but after washing and spin-drying it weighs 350g. The washing and rinsing uses 21 L of fresh water for each operation. I wash my shirt with one 30g tablet of washing powder.

How many times do I have to rinse my shirt to reach the required non-allergenic state?

(remember that the density of water is 1g / mL, 1000 mL = 1 L, 1 pico-gram = 10^{-12}g).

ID 5166

Sammy the sea lion has spotted a gorgeous Lady sea lion before anyone else. She is in the sea and he is on the beach. The picture shows the distances in hundreds meters. He needs to get to her as soon as possible to avoid any potential rivals getting to her first.

He can travel three times faster in the sea compared to on the sandy beach over the distances involved.

Which route should he use?

ID 5167

The stand-in mathematics teacher is forced to choose two students to go on a field trip and he can only choose between the two best girls and the two best boys in the class. He hates the idea of a girl being paired up with a boy, but knows that if he first picks a boy it is much more likely that the next pick will be a girl.

He devises this scheme: He labels 4 otherwise identical tokens with the names of the four students. He puts the tokens into a bag and then reaches in and takes two tokens at exactly the same time, one in each hand.

What is the chance of a boy being paired with a girl with this cunning plan?

ID 5168

Given Cartesian axes of infinite extent, find the ratio of the areas below to above the semi-infinite 45° inclined blue lines shown.

ID 5169

Find the ratio A/B where A is the number of elements in the set of all positive integers

and B is the number of elements in the set of all even positive integers.

ID 5170

Evguenia's birthday parties are always the same, and always boring. She gets seated at the table first. There are three other places set with named seating positions. Granddad always arrives next, and since he refuses to wear his glasses he always sits in a random position. Then her Aunt arrives, and if her place is free she takes it. Otherwise she sits in a random position. Then her mother arrives, and if her seat is taken she is grumpy all evening.

What is the probability that her mother gets to sit in her named seat?

ID 5171

A road traffic junction has no provision for pedestrians. The traffic lights are just changing to stop the RED and ORANGE cars. The GREEN and BLUE cars are next to go. Pedestrians A and B will only start to cross once all cars are stationary.

Compare the crossings for A and B.

(NOTE: Some jurisidictions allow drivers to pass red lights when turning, but that is not allowed at this crossing.)

ID 5173

Jane and Gerry work at a call center selling worthless rubbish to unsuspecting customers. They are in competition with each other to be the best seller today. In the morning they work from the *easy customer* list and in the afternoon they are forced to work from the *difficult customer* list.

In the morning Jane sells to 90 out of her 100 calls, whereas Gerry sells to 85 out of his 100 calls. 90% to 85% means Jane wins.

In the afternoon Jane sells to 30 out of her 100 calls, whereas Gerry, who is sad that Jane is winning, only makes 20 calls, and only makes 5 sales. With 30% to 25% sales figures, Jane again wins.

At the end of the day the boss totals the sales, totals the calls, and computes the aggregated percentage successful sales figure for each seller.

Who wins the competition?

ID 5175

The King of a far away land needs to adjust the temperature of the shower all by himself. The inconsiderate servant who normally performs this duty has broken his leg and is therefore unavailable. (The King has failed to realise that kicking the servant caused the servant to trip, breaking his leg because of the fall!)

The King requires a flow rate of 4 liters per minute of 40°C water to be derived from two calibrated hot and cold water taps. The cold water is at 10°C and the hot water is at 50°C. The mixing unit does not change the calibration of the taps or allow heat to be lost from the water.

What settings are required on the taps?

ID 5178

The King of a far off land has just had the ballroom lights in his palace rewired by probably the most illogical and incompetent electrician in the land. Instead of each of the 8 up/down switch positions turning on one of the banks of lights in the ballroom, each switch has to be in exactly the correct up/down state in order to make all the lights come on at once. In every other switch position all the lights are turned off.

Needless to say the incompetence of the electrician enraged the King so much that the electrician was executed in a gruesome way the same day.

Unfortunately, for some inexplicable reason, electricians seem reluctant to come and fix the problem. The Chief Steward needs to devise a plan to turn the lights on and off every evening so the King isn’t made to look foolish in front of his important guests over the coming weeks.

How many switch positions need to be changed every night to meet the requirements?

Changing any switch lever from down to up, for example, counts as one position change.

ID 5179

Modern petrol-electric hybrid cars typically have so many sensors that you can do interesting things like measure, log, and plot the power going to the wheels. This is such a plot, measured on level ground during the course of one day with relatively constant temperatures, no rain, and not especially windy.

Below 30 mph the car was running on electric power only. Above that it was doing its own hybrid thing with the engine running.

What can you conclude from the plot alone?

(NOTE: The power should at least double every time you double the speed since at twice the speed that part of the journey is half as long.)

ID 5180

Criminal gangs have been known to pay vagrants to search through people’s refuse to find useful information like bank statements, credit card statements, receipts and so forth. With such personal information the criminals can then pretend to be the householder and take out loans, buy things or do other bad things having stolen somebody’s identity.

Steve is fairly careful about shredding such documents, but about 5% of the time an important document slips into the refuse unshredded. Fortunately, unless he is being specifically targeted, it is pretty unlikely that somebody will be going through his refuse every week. Let’s put the odds at 1 in 1000 for each weekly collection.

Assuming that if Steve fails to shred a document, and if the criminals are searching his bins at that time, his identity will be stolen, what is the chance of that happening in a 10 year period?

ID 5209

It has been several years since the Apocalypse, but the Zombies still seem to be everywhere. I have been caught out in the open on my own and am now surrounded by 3 hungry Zombies, intent on eating my brains. Fortunately I have 6 rounds (“bullets”) in my gun. Unfortunately the ammo is old and degraded so it only works 80% of time. Also, although my aim is excellent on a shooting range, my shots are inaccurate when I am nervous, for example when I am surrounded by Zombies! It turns out that the closer they get, the more nervous I get, so the chance of my getting a shot to their head is only 63%. (Everybody knows that only a shot to the head will kill a Zombie). I worked it out, there is roughly 50% chance that any particular round will end up killing a Zombie.

There is just enough time to fire off all 6 rounds.

What is my chance of living to fight another day?

ID 5210

The more debt you get into, the more credit card companies profit. If you buy $100 of goods at a shop you owe the credit card company $100, but typically the shop only gets $98. The shop has to inflate its prices to pay the credit card company. Typically the shop is contractually not allowed to give a discount for cash.

The CostCrashers supermarket chain deals only in cash. The CardPayers supermarkets deals both with both cash and credit card transactions, although 75% of people pay by credit card.

All else being equal, how much cheaper could the CostCrashers prices be compared to the CardPayers supermarkets, given the figures stated earlier?

ID 5212

It is difficult to see a 2% price difference between food items in two different food stores, but the difference becomes noticeable over time.

If the weekly family shopping bill is $200, how much money does a 2% reduction make over the course of a year.

ID 5213

Shops have sales all the time to attract your business. Let the buyer beware! Not all sales are as good as others.

Given the same branded goods being sold, and the same quality of after-sales service, which shop offers the best value, given that two weeks ago all had the same price.

ID 5214

Income tax is a strange concept when inflation is taken into account.

At the end of one year with a 1% interest rate and a 2% rate of inflation, what is the tax owed on a $10,000 saved amount, given a 20% income tax rate?

ID 5215

Some forms of loans are more *iniquitous* than others. Find the loan method with the most extortionate rate.

*(Definition: Iniquitous – grossly unfair and morally wrong) *

NOTE: in some regions, Lenders are required to state the Annual Percentage Rate (APR) in order to make comparisons easier.

ID 5216

Jane was a naughty little girl. When she used to play 2-dice games with her late grandfather she always used to cheat. Her grandfather would pretend not to notice that the dice had landed and that she quickly changed one of the dice to her advantage. Specifically, if it was her throw she changed the lowest die to a 6. If it was his throw she changed the highest die to a 1.

Over a long run of throws, how much bigger was Jane’s average score than her grandfather’s?

ID 5217

My Interocitor is broken. Fortunately I have replacements for every single part and each part takes the same time to fit as any other. I also know that the designers were so unbelievably clever that it is inconceivable that one faulty part would cause other parts to fail.

The Interocitor has 60 parts, all of which are different, and I can change from one part to another in 10 seconds without even turning the power off (*hot-swappable parts*). I will know that it is working immediately.

What is the minimum time in which I can guarantee to have fixed it?

An interocitor is a fictional multi-functional device that first appeared in the 1949 story "The Alien Machine".

ID 5218

A certain Professor of Statistics is trying to explain to his grandson that once an event has happened its probability of happening is 1 because it is certain that it happened. The grandson disagrees.

The grandson throws a pair of dice and as they are about to settle puts a bowl over them, preventing the Professor from seeing the result. The grandson then peeks at the dice, knowing the value. He then explains that he could now show these dice to an infinite number of people, other than the professor, and yet the professor still could not declare with certainty what the result was, despite the fact that on average, everybody knew!

What can we say with certainty about this situation?

ID 5221

My neighbor John has invented a perpetual motion machine. It pumps water with no apparent power input and can even pump water up over a 2m fence.

How would you categorise this invention?

ID 5222

What is the next number in the sequence?

**0, 3, 1, 4, 2 **

(NOTE: It is a single sequence and not two sequences interleaved.)

ID 5223

Three observers, not more than 1km apart on flat windless terrain, report the same event quite differently.

They meet up later. Amy says the low frequency sound occurred first, followed by the high frequency sound about 1 second later. John says she is an idiot and claims the high frequency sound happened first, followed by the low frequency sound. Their teacher recorded the event and can prove that both sounds occurred within 0.01 seconds of each other, but being a kind teacher doesn’t actually tell them they are both stupid.

Which possible true statement demonstrates the least incompetence in the observers. . .

(HINT: the speed of sound does not change significantly for the frequencies heard, and there are no temperature gradients or fog to consider.)

ID 5225

"My neighbor a famous Professor of Physics has an excessively old and doddery gardener who knows absolutely nothing about basic science. Nevertheless this disrespectful old gardener insists that the Professor does not need to install an electric power cable all the way down to the ponds to power a fountain pump. The decrepit gardener insists that he has seen such a setup when he was a child and that it required no external power, had no electric motor, and could spray water well above the height of the source pond - but has no idea how. The Professor is still listening because the power cable will cost thousands of dollars to install.

What should he do?"

ID 5227

"Consider which would hurt less if it was accidentally dropped on your foot, a 30kg bag of cacao beans or a 30kg bag of feathers."

ID 5228

"** Endocrinic Igorosis** is a horrible disease with a 100% mortality rate for the infected. Even after 5 years nobody knows how it spreads. The fatality statistics have been steady at 100 people per 100,000 of the population every year. A remarkable cure has been found which is 100% effective at curing the disease, but only if administered before any symptoms are visible. Sadly, if the cure is given to anybody who is not infected then 3% of them will die. The latest test is 100% effective at finding this awful disease, but has a false positive rate of 5%.

How do we save the greatest number of people?"

ID 5229

"You accidentally knock an almost full opened can of soft drink over onto a carpeted floor.

Roughly how long have you got to make the can upright before over half of the contents are spilled?"

ID 5230

"Henrietta is threading the last bead onto a necklace when Tabby the cat brushes past the bowls of undrilled beads, knocking them over.

500 rondelle beads, 400 square rondelle beads, and the teardrop beads all end up on the work bench. Being startled by the falling beads, she drops the last bead onto the bench. There are now 1000 beads all mixed up on the bench and Henrietta need to find the one drilled bead she just dropped amongst all the undrilled beads.

Henrietta can't see the drilled hole without using the magnifier - which is on another bench. The drilled bead is neither a rondelle nor a square rondelle.

What is the chance she will pick the drilled bead on the first attempt?"

ID 5231

In a particular city it is illegal for those aged over 18 to consume children's chocolate drinks. Jack is walking home and drinking a chocolate drink. Sadly he is now 3 months too old to do so. When he sees a cop he therefore runs away and dumps the drink over a fence. The cop arrests him and charges him with arson. **WHAT!** Jack was wearing a green coat and the arsonist was seen wearing a green coat. At the police station the cops claim that only 2,000 people in this city of 100,000 people wear green coats. The odds of randomly finding him were therefore 2,000/100,000 making it 98% certain that he is the guilty party.

Using the evidence presented, what is the correct conclusion?

ID 5234

There are two gods named Mendax and Fidelis, one on your left, the other on your right, but you do not know which is which. Fidelis always answers correctly. Mendax only answers one question in 7 correctly in a repeating cycle. You do not know which part of the cycle Mendax is currently on.

You must determine which god is which using the minimum number of YES/NO questions. How many questions do you need to be certain?

To clarify the matter, if Mendax is answering incorrectly, your entire question is evaluated correctly and then the answer is reversed. A single question to both gods counts as two questions.

Inspired by G. Boolos. 'The hardest logic puzzle ever', The Harvard Review of Philosophy (6), 1996.

ID 5236

There are three goddesses named Mendax, Fidelis and Furtibus; one on your left, one on your right, and one in front. You do not know which is which. Fidelis always answers correctly. Mendax only answers falsely. Furtibus always answers in a way intended to best hide its identity.

You must determine which goddess is which using the minimum number of YES/NO questions. Each question is heard and answered by all three goddesses. That counts as one question.

How many questions do you need to ask in order to be certain?

[You should ideally think up actual questions before answering!]

Inspired by G. Boolos. 'The hardest logic puzzle ever', The Harvard Review of Philosophy (6), 1996.

ID 5238

"Suppose a particular nuclear waste material has a half-life of 100 years.

What could you do to reduce the radioactivity of the material itself to less than 7% of its current value?

(The half-life of a radioactive material is the time it takes, on average, for half of it to change into something else by spontaneous radioactive decay.)"

ID 5245

Dieter's new design of frequency-doubling power converter has an efficiency of 40%, a good figure for this particular type of device.

Kerstin's design is half the price so it is worthy of consideration, despite the fact that for the same power input, Kerstin's design produces 25% less output power than Dieter's.

What is the efficiency of Kerstin's design?

[ NOTE: Efficiency = 100% x (output power) / (input power) ]

ID 5252

* A phrase you will hear on the news or from people speaking is "the vast majority of". As a silly example you might hear something like "The vast majority of people with big noses also have big ears." What is the mathematical definition of the phrase "the vast majority of"?*

* There is a fault with the cruise control on Hank's car such that the speed continuously and linearly increases with time. When he starts off the speed is set to exactly 60 mph. He is driving on a long straight route with the radio on at full blast and he is not paying any attention to his speed. After 3 hours he notices that his speed has now reached 80 mph. How far did he travel in the first 3 hours? *

* There is a fault with the cruise control on Hank's car such that the speed continuously and linearly increases with time. When he starts off the speed is set to exactly 60 mph. He is driving on a long straight route with the radio on at full blast and he is not paying any attention to his speed. After 3 hours he notices that his speed has now reached 80 mph. For how many miles did he drive above the state speed limit of 70 mph? *

*Cletus is absolutely, definitely, the worst student the driving school has ever seen. When asked to drive at a steady speed he constantly accelerates too hard, overshoots the target speed, and then brakes too heavily. He has what the instructors call a "heavy foot". By some miracle he manages to always hit the same top speed of 20 mph before braking in a continuing non-repeating pattern like the one shown. What is his average speed? *

*In order to test the Archimedes Principle, Sasha put a Lead weight inside a football and makes it air-tight with glue. The football now weighs 500g. Sasha fills an outdoor water butt to the brim with water, but can't do the experiment that day because he has forgotten his gloves and the air temperature is 20°F. After a few days the temperature has not risen, but he finds his gloves and does the experiment. How much water is displaced by the football? If you measure the temperature in Celsius, there is a formula: The temperature T in degrees Celsius (°C) is equal to the temperature T in degrees Fahrenheit (°F) minus 32, times 5/9. *

*Sasha is learning about Archimedes at school. His show-off sister is 3 years older than him and sets him a problem: A water butt is filled to the brim with water and then tilted to a 30° angle relative to the horizontal. A football has a lead weight sealed inside, but not fixed in position. The combined weight of the ball and weight is 330g. The ball floats to a depth of 1.2 inches. Knowing the rough size of a football, estimate how much water spills out of the butt when the ball is lowered into the water gently. *

*The unit price of a thingamajig is $3.50. If I buy 10, the unit price drops to $2.00. At what quantity does it become more costly to buy them singly than to just buy 10? (Thingamajig is something whose name you have forgotten or do not know.) *

*The chocolate biscuit factory you are now in charge of has a problem. There are three 8 hour shifts and 11 shift supervisors who each have their own treasured setup of the machines to give an optimum biscuit pass rate. Every time the controls are adjusted the pass rate drops for several hours until the process settles down again. Each supervisor adjusts the controls to their “optimum” settings when their shift starts! Having analysed what they are doing, you have summarized the settings into 11 controls with two positions each. You need to devise a series of experiments to establish the optimum settings in a convincing manner to improve the productivity of your plant. What is the minimum number of experiments necessary to find the optimum settings for each control? *

*When it is 11am in Geneva (Switzerland) it is 5am in New York (USA). The flying time from Geneva to New York is 9 hours. If the plane to New York takes off from Geneva at 9:35am, what time is it in New York when the plane lands? (There are no stops or disruptions, just a normal flight.) *

*The train driver knows there is a Granny on board so he wants to give her a comfortable journey and to make sure she doesn’t spill her tea. When he is braking, which curve should he follow? (HINT: remember Newton’s Second Law, often expressed as Force = Mass x Acceleration) *

*You may have heard the expression "The shortest distance between two points is a straight line." I have drawn a straight line on the map, directly along one of the grid lines of the map. Is this the shortest path for a ship to travel? (Neglecting winds, tides and so on.) (NOTE: this is for a ship, not a submarine!) *

*Bethany throws a hard ball at John when they are in the park and it hurts. When they are weightless in the space station she somehow manages to throw the same ball at the same speed at John. What is the result?*

* Electrical energy can be measured in Joules, but for household use it is typically measured in kilowatt-hours (kWh). If a 100 Watt bulb is switched on for 8 hours every day for a year, roughly how much energy is consumed? J = 1000 x kW x S where J = Joules, kW = kilowatts, and S = seconds *

* Scientists and engineers need to use quantities which can vary over at least 30 orders of magnitude. Using zeros such as 0.000000000023 is impractical. Scientific notation uses a number between 1 and just less than 10, multiplied by a power of 10. What is the scientific form for 0.000123?*

* Scientists and engineers need to use quantities which can vary over at least 30 orders of magnitude. Using zeros such as 12300000000000 is impractical. Scientific notation uses a number between 1 and just less than 10, multiplied by a power of 10. What is the scientific form for 567001?*

* Engineers need to use quantities which can vary over at least 30 orders of magnitude. Using zeros such as 12300000000000 is impractical. Engineering notation uses a number between 1 and just less than 1000 followed by a standard multiplier for thousands, millions, and so forth. Express the frequency of 3 thousand million Hertz (Hz) in engineering notation.*

* Suppose the light bulb in your fridge uses 1 Watt when on and you pay 10 cents = $0.1 per kWh. Estimate the cost due to a faulty door switch, which keeps the light on over a 10 year period even with the fridge door closed. (1 kWh is 1000 Watts for 1 hour)*

* Peter changes his old fashioned single 100W incandescent light bulb to a super efficient 4 LED bulb system which gives the same light output, but only consumes 20W in total. Estimate the electrical energy saving per year if the bulbs are switched on for 8 hours every day of the year. (1 kWh is 1000 Watts for 1 hour)*

* Peter changes his old fashioned single 100W incandescent light bulb to a super efficient 4 LED bulb system which gives the same light output, but only consumes 20W in total. Estimate the saving per year if the bulbs are switched on for 8 hours every day of the year and you pay $0.1 per kWh. (1 kWh is 1000 Watts for 1 hour)*

*A professional design standard requires that free standing equipment must not tip over if subjected to a force equivalent to one fifth of its weight applied at the worst possible point. The latest design has failed the test. What can the design team do to fix the problem?*

*Answer the traditional English-language nursery rhyme in the form of a riddle: As I was going to St Ives I met a man with seven wives Each wife carried seven sacks Each sack carried seven cats Cats, sacks, man, and wives How many were going to St Ives?" "This is a variant of a traditional old rhyming puzzle, often interpreted as a trick question by assuming that the group was travelling *

*My neighbor Timmy has developed a complicated set of rules for choosing girlfriends. He likes: all girls with long hair; all girls who are not tall; tall girls who wear glasses; short-haired girls who don't wear glasses. Can you simplify the rules without changing the requirements?*

*In a large village community there are regularly women giving birth to triplets, but surprisingly never twins. The average number of babies per mother is 1.9 The local nurse visits the next recent mother on her list. Which is the most probable number of babies she will find?*

*Einstein, the hyper-intelligent house cat, is busy walking from one corner of a rectangular room to the diagonally opposite corner in an apparently random manner. His rather stupid human slaves can’t work out what he is doing. The light and dark carpet tiles on the floor look like part of a chess board, with exactly 10 tiles down the length and 6 tiles across the width. Einstein has decided to move from one corner of the room to the diagonally opposite corner, at each tile moving only down the length or across the width of the room. How many different routes are there from one corner to the diagonally opposite corner?
[HINT: You could try an easier problem first.] *

*Anton, the highest IQ house ant on the planet, is taking his regular nocturnal walk from one corner of the chess board (1,1) to the diagonally opposite corner (8,8), moving only right or up the board at each successive square. He remembers that on square (5,3) there is a 'black-hole', so any valid route must exclude this square. From how many different paths can he choose? [HINT: You could try an easier problem first.] *

*Anton, the highest IQ house ant on the planet, is taking his regular nocturnal walk from one corner of the chess board (1,1) to the diagonally opposite corner (8,8), moving only right or up the board at each successive square. He remembers that on square (5,3) there is a tasty sticky residue, so any valid route must involve this square. From how many different paths can he choose? [HINT: You could try an easier problem first.] *

*Martin, the mathematical mole, has dug an extensive network of underground tunnels which he has approximated, in his magnificent mole mind, as a 3D lattice of 30 x 30 x 30 intersections. The distance between intersections is approximately constant. He is currently at intersection (20, 20, 15) and wishes to get to his secret food cache at (25, 25, 20) by one of the many shortest routes available. There are stones blocking the intersections at (21, 22, 10), (22,19,18), (22, 22, 16), (24, 18, 18) and (26, 26, 18). From how many equally short paths can he choose? [HINT: You could try an easier problem first.] *

*Marek, the pan-dimensional super being, has arbitrarily defined his current location as (0,0,0,0,0,0) in 6D hyperspace. He wishes to reach location (3, 0, 2, 0, 4, 3) by one of the many shortest paths available. Despite his immense power, he can only move one hyperstep at a time, each hyperstep consisting of a unit change in exactly one of the coordinate values. Any hyperstep is of equal ‘length’. What is the smallest number of hypersteps required for him to reach his destination? *

*Marek, the pan-dimensional super being, has arbitrarily defined his current location as (0,0,0,0,0,0) in 6D hyperspace. He wishes to reach location (3, 0, 2, 0, 4, 3) by one of the many shortest paths available. Despite his immense power, he can only move one hyperstep at a time, each hyperstep consisting of a unit change in exactly one of the coordinate values. Any hyperstep is of equal ‘length’. From how many of the shortest paths can he choose? [HINT: You could try an easier problem first.] *

*A team of archaeologists is exploring an underground complex on a remote planet. On each level there is a regular grid of North-South corridors intersecting East-West corridors, with ladders at each junction going both up and down to the next levels. Effectively the complex appears to be a regular 3D lattice of tunnels. The previous team has marked the tunnels and made a list of problematic junctions that need to be avoided. The team is currently at junction (3, 2, 5) and needs to get to junction (12, 9, 8) by one of the many shortest available routes. Which of the listed problematic junctions might be in their way? *

*You are pitching your new idea to a panel of Venture Capitalists (VCs) to secure increased funding. Using your advanced mathematical skills, you have dumbed-down the probability of success to something even VCs can understand. You tell them that if they were to throw 10 normal dice and sum the dots on top, the probability of your success is the same as the sum being less than 50. One of the VCs seems very antagonistic, but you must still give the best possible answer, quickly – and using only mental arithmetic. His question is "Can you guarantee that the sum of dots would be less than 50?" *

*Shakuntala Devi was undoubtedly the most brilliant arithmetic mental calculator of all time. In 1977 she mentally calculated the 23 ^{rd} root of a 201 digit number in a mere 50 seconds. She toured the world showing how she could do calculations faster than they could be entered into and solved by the computers of the day. The problem for you is much simpler: Evaluate (without using a calculator) the 6^{th} root of the 25 digit number consisting of 1 followed by all zeros. *

*Shakuntala Devi was undoubtedly the most brilliant arithmetic mental calculator of all time. In 1977 she mentally calculated the 23 ^{rd} root of a 201 digit number in a mere 50 seconds. She toured the world showing how she could do calculations faster than they could be entered into and solved by the computers of the day. The problem for you is much simpler: Evaluate (without using a calculator) the 20^{th} root of the 11 digit number consisting of 1 followed by all zeros. *

*Symmetry is a very big subject, involving much more than geometry alone. Spotting patterns, and breaks in patterns, is a valuable skill. Without worrying about what the function is, or what the programming language is, can you spot the error in this code simply by spotting a break in the symmetry? The error is on line . . . *

* The sinusoidal waveform shown has a peak amplitude of 20 and a period of 2. What is a rough estimate for the mean value of the waveform over the interval shown (from t=0 to t=5)? (There is no need to use Calculus). HINT: Areas below the x-axis should be considered as negative when calculating the mean. *

*The boss of a 10 person company is always the last to arrive, and gets the worst of the 10 car parking places as a result. Being the boss, he decides to allocate parking places to each of his 9 employees (all of whom use a car to get to work), obviously keeping the best parking place for himself. The employees already hate the boss, who only got the job by marrying the owner. Further enraged by the new rule, they collectively decide to ignore it and just park randomly when they arrive. All places are good for the employees. What is the chance that the boss gets to park in the best parking place?*

*Jake, being bored on a rainy Sunday afternoon, throws a pair of dice 500 times and keeps a record of the results. What is the ratio of probabilities between throwing one five and all the rest twos, compared to throwing all threes. *

*An escaped criminal has stolen a spaceship, and has just instantaneously jumped 1 light year away. On each successive jump she will only be able to jump half the distance of the immediately preceding jump due to heat build-up. Jump engines always take 1 hour to recharge. My ship can only jump 1/2 light year, but it has a better cooling system so the jump distances drop-off more slowly. My maximum jump distances follow the sequence: 1/2 light year, 1/3 light year, 1/4 light year, and so on. I can find her with my subspace-tracker. If she is closer than my maximum jump distance I can get close enough to remotely disable her jump-drive and capture her. My jump engines are fully charged. Can I catch her? *

*Despite advice to the contrary from his friends and parents, Timmy has decided on a new strategy to select future girlfriends. He has two “must-see” programs on 5 days of each week. He requires that any future girlfriend must match-up with at least 90% of these programs. Given that there are 20 TV channels available in his area, what is the probability of a match? *

*There are 4 playing cards face up in a line on a table. Each of the cards has a different value. The cards need to be sorted so that the smallest value is on the left. There is only one action you can perform, namely interchanging the position of two cards (swapping them). You cannot move a card to an empty space. What is the minimal number of swaps achievable on the worst possible arrangement of cards? *

*The playing cards shown need to be sorted into increasing order with the lowest card on the left. Each move consists of picking up a card and inserting it anywhere in the line, including at the beginning or at the end. The cards slide sideways to allow a card to fit in between if necessary. Which is the fewest number of moves possible? *

*Christmas is getting near, and a four person company wants to run a Secret Santa scheme. The idea is that all four names are put into a hat and drawn at random. You buy a present for the person whose name you pick. Sadly, for the last three years in a row, at least one person has picked themselves, ruining the draw. Estimate the probability that the draw will fail this year because somebody picks themselves. *

*John refused to learn how to cook, despite the best efforts of his parents and teachers. Now he is at college, any meals he prepares can contain only some combination of boiled eggs, baked beans, and pizza. How many different meals can he prepare?*

*Jasmine has just been learning about the binary number system at school. On her way home she wondered how far she could count using just the four fingers on one hand, if a curled finger represented a binary 0 and an outstretched finger represented a binary 1. To be clear, she was thinking about counting up from zero in whole numbers. How far could she get? *

*The teacher walked into her classroom to find a scene of devastation. There was red paint on the walls, her lunch had been half eaten, and books were thrown around the room. There were only three children in the room: Alex, Betty, and Clive. All three said that Betty ate the lunch. Betty said Alex painted the wall. Clive said that Alex threw the books. Alex said that Clive painted the walls. The Headmaster was called in to resolve the crisis. On his way to the scene he found Wesley hiding in the corridor. Whilst Wesley would not directly implicate anyone, he did admit (under duress) that each of the three had done one of the crimes, and that every statement they made had been untrue. Given that Wesley is telling the truth, who threw the books? *

*A pirate captain takes his pirate crew into the treasure cave shown. The Captain marks the sturdy plank in units of the depth of a pirate, meaning pirates can stand on the marked positions, but not any closer. All of the pirates, including the Captain, are the same weight, and the bag of Gold is one quarter the weight of a pirate. The Captain, being bold and fearless, is going to walk on the unsecured plank over the edge of the steep cliff to position 8 and collect the Gold. He does not worry about falling to his death in the bottomless pit below. How many pirates are needed to stand on the plank to support the Captain? *

*Mark, who has been a digital design engineer for many years, is celebrating his 55 ^{th} birthday. Since the use of 55 candles for a birthday cake seems excessive, he arranges 7 candles in a line and lights the appropriate candles to represent his age in binary, a lit candle representing a “1” state for that bit. How many candles does he light? *

*Jenny is a computer scientist and is shy about her age. On her birthday she encodes her age in binary in a row of 8 candles. Her boyfriend, who is sitting on the opposite side of the cake, is trying to work out her age from the pattern of lit candles. Knowing that her boyfriend is fluent in binary, Jenny encodes the pattern correctly, but does not reveal if a lit candle represents a “1” or a “0” for that bit position. She also does not reveal if she has written the binary number either left to right increasing (standard notation) or the other way around. Given that Jenny is 27 years old, which age cannot reasonably be read by the boyfriend? *

*Jane would ordinarily like to drive at 60 mph on this particular stretch of road. However there is a large truck driving at 30 mph which is slowing her down. She knows that in 1 mile there will be a multi-lane road section (dual carriageway) where she can easily overtake, so she waits behind the truck. Gerry is impatient and overtakes both Jane and the truck to travel at 60 mph. Roughly how much time does Gerry save in this situation? *

*In the game of noughts and crosses (tic-tac-toe) the winner is the player who gets three of their symbols in a straight line, with each player placing their symbol alternately. In this game, *

*John accidentally drops his text book and it falls open at a random position somewhere near the middle of the book. He immediately counts the sum of the two visible page numbers. What is the probability that the sum of the *

**The History of Art in the Dark Ages** is an epically boring subject and too many students pass by simply answering the 100 multiple choice exam questions randomly. This year the marking scheme has been changed so that of the 4 possible answers, the correct answer scores one point, two wrong answers score 0, but the remaining stupid answer scores minus two points.

What is the expected score for a student who guesses randomly?

**The History of Art in the Dark Ages** is an epically boring subject and too many students pass by simply answering the 100 multiple choice exam questions randomly. This year the marking scheme has been changed so that of the 4 possible answers, the correct answer scores one point, two wrong answers score 0, but the remaining stupid answer scores minus two points.

What is the optimum strategy for a student who is good at Mathematics, but not Art History?

*My dog Charlie rushes into my office and bumps with his nose into a book on the floor at the position shown by the blue arrow. Describe what happens.*

*The contrapositive of a logical statement is formed by negating both the test and the result and then changing their order. For example: If [this is my house ] then [the door is black]. Becomes If [the door is not black] then [this is not my house]. Which is a correct contrapositive of If [this is a fish] then [it cannot live out of the water for very long]. *

*Johnny is a very poor communicator and a very fussy eater. If given a plate with any foodstuff that he doesn't like, he rejects the whole plate and sulks. He rejects 'spam, pizza & chips'. He accepts 'sausage, mash & spam'. He accepts 'pie, spam, & beans'. He accepts 'egg, chips, & spam'. He rejects 'spam, beans, & pizza'. Which food item is he actually rejecting? *

*It is demonstrable by direct computer calculation that any even number greater than 5 can be formed as the sum of two odd prime numbers. Find one of the two primes that sums to 36. *

*Analyze the following statement as if it were true: "HyperBrite cleans off 4x as much dirt as the nearest competitor." What can you say with *

*Since the year 2058, 11 year olds have been required to get high exam marks in one of four elective subjects in order to graduate; harder subjects can earn more points. The maximum possible scores are 100 for Set Theory, 200 for Vector Calculus, 300 for Orbital Mechanics, and 500 for Quantum Cryptography. Typically, students of Set Theory get 95 out of 100 questions correct in the exam, whereas the figures for the other subjects are 25 out of 50 for Vector Calculus, 11 out of 30 for Orbital Mechanics, and 21 out of 100 for Quantum Cryptography. Which subject gives a typical student the highest score? *

* Box A contains 3 blue balls and 1 red ball, all of the same size, weight, and texture. Box B contains 1 blue ball and 2 red balls, all indistinguishable from those in box A. I draw one ball from box A at random, examine it carefully, and put it into box B. I shake box B to mix up the balls, then draw a ball from that box at random. What is the probability that the ball is blue? *

*Two identical airplanes (aeroplanes) set out on a vital mission. The lead plane is carrying a secret message which needs to be delivered by hand. Each plane has a full fuel tank and a 1000 mile range. The planes can transfer fuel in mid-air; this process loses no fuel and happens almost instantly. How far can the lead plane get? (Note that a plane with no fuel can still land safely.) *

*Three identical airplanes (aeroplanes) set out on a vital mission. The lead plane is carrying a secret message which needs to be delivered by hand. Each plane has a full fuel tank and a 1200 mile range. The planes can transfer fuel in mid-air; this process loses no fuel and happens almost instantly. How far can the lead plane get? (Note that a plane with no fuel can still land safely.) *

*The image shows two parabolas, f(x)= x*

*What is the ratio of sides of a circumscribed regular hexagon to an inscribed regular hexagon sharing the same circle (as shown in the picture)? *

*The mathematics department at a school has challenged the pupils to a sort of ‘tug of war’ contest. The staff pull in the direction marked M. The boy students pull in the direction marked B, and the girls of course are G. The ropes are connected together by a fairly large strong equilateral triangular plate. The ropes are attached to the plate on smooth posts so the ropes are free to swing around the pivot. The relative strengths of the pulls and the position of the plate are shown at a particular moment in time. What happens next? *

*Johnny had a lesson at school which explained that ropes and cables are strong under tension but useless under compression. Brick and masonry, on the other hand, are strong under compression but weak under tension. Johnny has made the model suspension bridge (shown in the picture) using string and wooden blocks as a school project. Comment on the design. (Hint: consider the force on the top blocks) *

*When only one small uniform solid cylinder of ice floats (with its axis vertical) in a glass of cold fresh water then 8.3% of the length of the cylinder sticks out of the water, the rest being submerged. Estimate the specific gravity of ice. (Reminder: Specific gravity is the ratio of the density of something relative to the density of cold pure water.) HINT: Archimedes.
*

*You have measured the diagonal of your rectangular lawn using a laser rangefinder onto a conveniently positioned garden gnome. You have also measured the angle of the diagonal. What is the length of the lawn?
*

*Susan holds a pencil completely underwater at a 45° angle to the horizontal plane. Pencils have an average density less than that of water. What happens when she carefully releases the pencil?
*

*A drop of paint falls onto a horizontal flat sheet of clean glass. We suppose that at a particular instant the drop forms a perfect sphere in the air. The paint has spread out into a uniform circular disc (disk) of a diameter that is twice as large as the initial sphere diameter. What is the ratio of the disc thickness, *

*We are going to model the growth of a particular type of bacteria as follows: No growth below 3°C. Doubles in quantity every 20 minutes for temperatures between 10°C and 60°C. Dead at 80°C.
Compare the amount of dead bacteria in two similar portions of food, one of which is left to stand at room temperature for 2 hours, then quickly heated to 80°C, whilst the other portion is immediately heated to 80°C. The initial bacteria populations in the two portions are similar at the start of the day. *

*The teacher likes to reward students for being smart. This year the top three students each get to pick 10 cacao beans from a large bag. Each student is blindfolded and wears a glove to do the selection. Beans are removed one at a time, inspected, then replaced, with the bag contents being thoroughly mixed before the next pick. The bag contains hundreds of fresh beans, and an equal number of already dried beans. The student who gets the most dried beans wins a big prize. How does the best student optimise his chance of winning the prize? *

*This incident occurred in deep space. A space ship had been blasting its rockets at full power for several hours, such that the on-board accelerometers recorded an acceleration of 1g, the Earth normal gravitational acceleration of around 10m/s ^{2}. The Doppler Space Radar showed another space ship on a direct collision course so the Captain immediately cut off the engines. At this instant the other ship was 1000km away and a collision would happen in 1 hour if nothing was done. What can you say with certainty about the speed of the other space ship? *

*Terry the termite is taking a walk across the gap between two roses using a conveniently available cotton thread. Terry is quite good at mathematics, but not nearly as smart as his uncle Huygens. Uncle Huygens explained that the thread forms a shape known as a catenary, a curve which looks a bit like a parabola, but is more complicated than that. Terry is smart, but not that smart, so he approximates the curve as a 30° arc of a circle of radius 100mm. What is Terry’s estimate of the (arc) length of this cotton bridge?
*

*You are in a blue car on the side of the road. There is a lot of fast moving traffic going your way and you need to judge when to pull out into the traffic flow. If you want to wait for a really big gap in the traffic you could be stuck there for hours, so you need to pick a safe gap, but not an ideal huge gap. The traffic is going at an estimated speed of 56 mph (25 m/s). You are confident that you can get your car to accelerate from stationary to 25 m/s in 10 seconds (with constant acceleration). Measuring the gap between the rear of your car and the front of the car behind, what is the minimum gap that would not cause an accident (we assume that the other driver does not brake)? *

*Febe, the highly intelligent house fly, is bored buzzing around the kitchen so she hitches a lift when her Hoomins take a trip in their car. Being highly intelligent she can read the speedometer and convert the mph reading into m/s. She quickly becomes bored in the car and decides to fly directly from one side of the car to the other when the car is traveling in a straight line at 4 m/s. Being a mathematics prodigy, she realises that her speed relative to the ground is 5 m/s. How fast is she flying across the car? *

*A 1 ton car is heading due North at 60 mph. A 2 ton truck is heading due East at 30mph. There is sheet ice all over the intersection and the truck cannot stop. The truck smashes into the side of the car and the pair forms one tangled mess of bent metal. (Fortunately the drivers were wearing seatbelts and the air bags did their job. Nobody was injured.) In which direction does the mangled mess travel? (Hint: conserve momentum not energy). *

*A 1 ton car is heading due North at 60 mph. A 2 ton truck is heading due East at 30mph. There is sheet ice all over the intersection and the truck cannot stop. The truck smashes into the side of the car and the pair forms one tangled mess of bent metal. (Fortunately the drivers were wearing seatbelts and the air bags did their job. Nobody was injured.) At what speed does the mangled mess initially travel? (Hint: conserve momentum not energy). *

*For no clearly explained reason I am standing on a set of bathroom scales in a lift within a tall building. When the lift is stationary I weigh 41kg. When the lift moves I see my weight increase to 60.5kg. Describe the motion of the lift: *

*There is a right angled triangle with a hypotenuse of unit length. Denote an angle (other than the right angle) as alpha. Given that the side adjacent to the angle is of length cos(alpha) and the side opposite the angle is sin(alpha), evaluate the sum. *

* An American salesman flies over to London (UK) and for reasons best known to himself carelessly steps out of a first floor window. He breaks his leg. Why?*

* A elderly monk is arranging the annual charitable gift. He will put bank notes in two envelopes such that one envelope has twice the amount in the other. The number of notes will be undetectable within the heavy envelope. It is required that anyone who opens an envelope does not know if they have the high amount or the low amount. Given that the bank notes available to the monk only occur in units of $1, $2, $5, and $10, which statement is acceptable? Inspired by a comment from Jeff Jordan concerning Two Envelope Paradox*

* Jane sees the following text written on the blackboard in the classroom, evidently left over from a previous lesson. X = X + 1 Which statement is true?*

* In a singles tennis tournament of 64 players, the winner of each game goes forward to the next round. Two sisters are both excellent tennis players. Given that each sister wins all their matches until they meet each other, what is the probability that they meet each other at the final?*

* My black credit card has a 16-digit number. Credit cards have the digits 0-9 with equal probability in each digit position. What is the probability that the sum of the first fifteen digits is equal to the sum of the last fifteen digits?*

*In a fictitious far away country income tax is not charged on the first $10,000 of annual income. After that, tax is charged at a 20% rate on the amount beyond the $10,000 allowance. If income exceeds the allowance by more than $30,000, however, earnings beyond that amount are taxed at 40%. Arthur earns $20,000 per year. Barry earns $50,000 per year. How much more income tax (as a ratio) does Barry pay? *

*Jane wants to use her soapbox racer at the park where there is a sharp-edged ramp. Last time she tried it the bottom of the racer struck the corner of the ramp, making a horrible sound; this was extremely uncool. (Notice that the base of the racer always lines up with the axis of the wheels in these designs.) What should Jane do to restore her cool reputation? *

*In factories where food items are packaged, one clever technique for optimally filling bags is to fill 12 nominally equal hoppers with the food, then computer select the 4 which give the closest fit to the required weight. This is better than taking food items such as crisps and putting tiny broken pieces in to make up the required weight. It is also cheaper for the manufacturer to not greatly exceed the minimum weight, and effectively get paid less for each gram of food as a result. How many combinations does the computer have to check to get the optimal selection?
*

*John got back to his car too late and now he is locked into an outdoor car park for the night. There is an escape path, but it involves driving down a fairly steep grassy slope. He has correctly drawn a diagram of the problem, but can’t quite finish it off. The axle of the wheels is roughly in line with the underside of the car. Hitting the underside of the car on the corner of the slope will mean a tow truck will be needed and there may be costly damage done as well. The faint line above the ground is the path that the car axle travels as it rolls along the ground. This is called *

*Mary has a cat, and a rat, and a hat, and a mat. If the cat is on the mat, and rat is in the hat, but the hat is on the mat, where is the rat?*

*There are 100 unique numbered components in a bin. You select 5 components at random. You then sort the components into numerical order. How may different selections can you make (you restore the components to the bin after each selection)? *

*A motorboat is in the middle of a fast flowing river heading directly for some rocks that also happen to be in the middle of the river. Its engine failed 5 minutes ago so the boat is just being pulled along by the flow. Seeing the rocks in plenty of time the skipper pushes the rudder hard over to the left. The river channel is safe either side of the rocks. What happens next? *

*Three ropes join at a ring which is free to move relative to the ground. The forces and directions are shown at one instant. The sketch is not necessarily drawn to scale. What happens next? *

*It is a serious mistake to think that computers can solve any numerical problem almost instantaneously. Whilst addition and subtraction are very fast, multiplication can take twice as long as that, and division can take ten times as long as addition. In this example the values subscripted by N are to be calculated hundreds of thousands of times. The unsubscripted values are constants. Rearrange the equation to minimise the computational time. We are after the form of the equation; the values of the constants vary as necessary to make the result correct. *

*A hospital runs a queuing system for non-urgent surgical procedures. This queue always has 100 patients in it and one patient is operated on each day (7 days a week). If a patient is not available on the day of surgery they lose their place to a brand new patient who would otherwise have gone elsewhere. (This is easier than rescheduling 100 appointments.) Roughly 5% of patients are not available or do not show up for their procedures. What is the average waiting time reported to the managers of the hospital? *

ID 6229

This is part of a real UK tax statement from 2015-2016 showing how the government spends Income Tax and National Insurance contributions.

Which statement is true?

ID 6231

There is a real-life balance beam which is horizontal when unloaded. First we lock the beam in the horizontal position with a brake. Then we hang a 1 ton bag of feathers at a distance of 6 feet to the left of the pivot. Next we hang a 2 ton bag of scrap iron 3 feet to the right of the pivot. Charlie the cat is sitting about 6 feet to the right of the pivot.

What happens when we release the brake?

ID 6244

When representing negative whole numbers in binary for computers it is convenient to use the two’s complement form.

For example we could represent +3 as 0000 0011 in binary.

To get -3 in the two’s complement form we first invert the bits (change a ‘1’ to a ‘0’, and vice versa) and then we add 1.

Which is the correct two’s complement binary representation of -3?

ID 6247

Rather than write the specifications on the side of the packet, manufacturers now use symbols to avoid having to print the specifications in 7 or more different languages.

Sometimes these symbols can be difficult to understand immediately.

Can you decode what this (actual) label tells you about the environmental rating?

ID 6255

We put something into the blue box and something new comes out.

Can you decode the mystery of this 300+ year old mathematics?

ID 6256

We put something into the blue box and something new comes out.

Now we are asking what do we need to put in to get something?

Can you decode the mystery of this 300+ year old mathematics?

ID 6263

The picture shows problem complexity growth curves for computing problems. If N is the number of elements in the problem, then the growth can be proportional to

N^{2}, N.log(N), N, exp(N), N! and so on.

Which type of growth is the worst (fastest)?

ID 6264

If you defined the sine and cosine functions to Pythagoras, 2500 years ago, he would easily have solved this problem.

Surely you won’t have any trouble with it?

*(No cheating by using a calculator or trig tables.)*

ID 6267

If it takes 5 men 5 days to dig 5 trenches in each of 5 countries, how long will it take 10 men to dig 10 trenches in one country?

(We assume of course that the men have similar capacities to dig trenches, the countries make no difference to trench digging, and everything is just simple.)

ID 6271

Calendar dates are written in different ways according to countries and preferences. For example 21 February 2018 could be written 21/02/2018, 02/21/2018, 21 Feb 2018, and so on.

You want to label filenames on a computer so that the date code is easily and automatically sorted into date order by the operating system (regardless of the date the file was actually written).

What is the correct format?

Y is a year number; M is a month number; D is a day number.

Notice that we have to put leading zeros in the each section when necessary, so we have to write 02 for the month and not just 2, otherwise the date-ordered listing is incorrect.

ID 6273

The FastMoney bank decides to only allow passwords with a length of exactly 8 characters. Each position in the password can contain a lower case letter, an upper case letter, a digit, or a special character. For simplicity we will take the sum as 70 possible characters per position.

The bank then randomly asks for the character in 4 unique positions within the password. For example on one day it might ask for characters in positions 3, 6, 2, and 5.

Assuming that you have chosen to not repeat any character within your password, how many unique key sequences are possible for you to correctly logon to your existing account.

ID 6274

Your boss has asked you to produce a report of average sales figures over the last few years.

What rule should you use to calculate the average value?

ID 6275

John is on a fixed income. His monthly budget is spent as follows:

Food 10%

Rent 50%

Entertainment 10%

Savings - anything left over.

Suddenly the cost of food rises by 10%. The rent is increased by 5%.

By how much does his monthly savings decrease?

ID 6282

Mathematicians love using symbols to avoid writing. If you don’t understand what the symbols mean, the resulting expressions are completely meaningless.

Can you crack the code? Can you guess what the upside-down U is supposed to mean?

ID 6283

Words can be tricky. The same word can have different meanings dependant on where it is used in a sentence, and can have different meanings even within the same sentence.

Am I being mean when I ask what the mean of the means means?

ID 6284

A small unmanned rocket program is having budget difficulties so the project director wants at least a 1% cut in the parts cost. The parts are roughly broken down as follows:

$9M Rocket Engines

$900k Avionics

$90k super-structure

$10k miscellaneous items

Which change best meets the project director’s requirements?

ID 6285

John, now aged 18, goes out to work for the first time. Since he is working, and an adult, he can eat whatever he wants. He likes sugary drinks, chocolate, and crisps. He also gets less exercise than he used to at school. As a result of this lifestyle, his food calorie intake averages out at 100 calories more than he needs every day. The human body typically stores excess calories as fat.

Given that a rough estimate of calories per pound of human body fat is 3500, estimate John’s weight increase by the time he is 28, all other factors being equal.

ID 6303

Pretend the round red blobs are tennis balls. Pretend the blue lines are stretchy strings.

Can you move the tennis balls from the pattern on the left to make the pattern on the right?

*NOTE: the strings are special so that whatever you do they never get tangled up with each other.*

ID 6304

Pretend the round red blobs are tennis balls. Pretend the blue lines are stretchy strings.

Can you move the tennis balls from the pattern on the left to make the pattern on the right?

*NOTE: the strings are special so that whatever you do they never get tangled up with each other.*

ID 6305

Pretend the round red blobs are tennis balls. Pretend the blue lines are stretchy strings.

Can you move the tennis balls from the pattern on the left to make the pattern on the right?

*NOTE: the strings are special so that whatever you do they never get tangled up with each other.*

ID 6306

Can you move the tennis balls from the pattern on the left to make the pattern on the right?

*NOTE: the strings are special so that whatever you do they never get tangled up with each other.*

ID 6307

Can you move the tennis balls from the pattern on the left to make the pattern on the right?

*NOTE: the strings are special so that whatever you do they never get tangled up with each other.*

ID 6308

Can you move the tennis balls from the pattern on the left to make the pattern on the right?

*NOTE: the strings are special so that whatever you do they never get tangled up with each other.*

ID 6309

Can you move the tennis balls from the pattern on the left to make the pattern on the right?

*NOTE: the strings are special so that whatever you do they never get tangled up with each other.*

ID 6310

Can you move the tennis balls from the pattern on the left to make the pattern on the right?

*NOTE: the strings are special so that whatever you do they never get tangled up with each other.*

ID 6316

In ancient texts, when writing materials were expensive, it was usual to write without punctuation and without spaces between the words.

In this well known English language proverb, how much space has been saved, assuming that every character and space takes the same amount of room (a fixed-pitch font)?

ID 6320

For reasons best known to himself, John decides to get off the train before it comes to a complete stop. He jumps with both feet together in order to land well, although the height of the step is only a foot (30cm) off the ground. He makes sure to land facing in the direction of motion of the train. The train is moving at about 5 mph.

What is the most likely outcome?

ID 6321

ID 6322

ID 6323

It is often said that "You cannot fit a square peg in a round hole". Obviously that is not true. Anyone with any mathematical background would use a square peg whose diagonal was equal to the diameter of the round hole.

But we have a different question. If you bevel (cut) each corner of the square peg at a 45° angle, what is the largest cross-sectional area of peg that will fit in a circular hole of radius R?

ID 6324

A radian is a very strange unit of angular measure. Protractors are calibrated in degrees, not radians. You will never find radians mentioned in everyday life.

But computer library functions for sin(), cos() and tan() functions all use radians not degrees, so if you need to use them then you need to convert from degrees to radians. There is no need to remember the conversion factor; just work it out from first principles when you need it.

If you draw an arc with a compass, and the arc length is equal to the radius, then the angle of the arc is 1 radian.

Given that **definition** you should be able to calculate how many degrees make 1 radian.

ID 6325

There is a long distance hot-air balloon race. The winner will be the team that gets the furthest from the starting point. Each team consists of one or more team members. Each team member has a standardised weight by carrying ballast to make up to 150kg of load per person.

Gas for the burner is provided with one standard bottle per person. The spherical hot air balloons use the same materials, but the balloons have a constant volume of hot air per person.

There are four teams: which one has the best chance, everything else being equal.

ID 6336

ID 6337

Did you realize that trigonometry is actually used in computer programs? The right half of the image shows a graph with node (vertex) numbers. These labels need to be rotated around the nodes in order to avoid clashing with the blue links (branches). The left half of the image shows the problem of moving the center of a text label box around a fixed point.

Assuming a standard (Cartesian) x-y co-ordinate system, and given that the node is at (Px, Py), what is the position for the top left corner of the label box?

ID 6339

If you try to look at these graphs as 3D shapes you will get very confused.

I have erased one of the blue lines from one of the images. But which line?

ID 6340

Although it doesn't look like it, the distances along any of the blue strings are equal because it is a 3-dimensional object.

What is the shortest distance between blobs 03 and 09?

You can only travel along a blue string.

ID 6341

ID 6342

ID 6343

Being a bored billionaire, Martin decides to have a swimming pool built in the shape of an inverted regular tetrahedron. Of course he wasn’t sure what a tetrahedron was, so the architect explained that it is a triangular based pyramid with all four sides equal, although in this case there is no actual base because that is the surface of the water.

Martin is bored by the fact that the pool has taken two hours to get to a depth of one third the overall height. Filling at the same rate (in gallons per minute) he decides that it will take another 4 hours to fill completely. Everyone avoids his gaze when he makes this statement out loud.

How much longer will it really take?

ID 6344

ID 6345

You wish to establish how close to vertical a wall is. You have a spirit level and you can see that the bubble is not perfectly centred around its calibration marks when placed flat on the wall. Rotating the spirit level 180° about the vertical axis shows that the spirit level itself is essentially perfectly calibrated. In order to get the bubble centred you need to move the bottom of the spirit level about 1mm away from the wall. The spirit level is 30 cm long.

What is the angular deviation of the wall from vertical?

Note: *you do not have a calculator or trig tables to hand.*

ID 6346

Which of these graphs is the sine function?

(*HINT: Look at the inset picture which shows how the sine function relates to a right-angled triangle.*)

ID 6347

Sine waves are fascinating things. The slope of a sine wave is another sine wave, just shifted in time (phase). You can also add two sinusoidal waves, each of which has a different amplitude and a different zero crossing point (phase) and still end up with a sine wave. Furthermore, the addition of these sine waves obeys the rules of vectors (but using phase instead of direction) so you can draw a triangle and calculate the resulting amplitude and phase from that.

In the picture we are adding a 100V sine wave to a 10V sine wave which is phase shifted by 90° relative to the larger voltage.

What is the amplitude of the resultant sine wave?

ID 6348

Susan is walking down the road in a northerly direction at 3 mph. There is a crosswind of 4 mph.

What is her resultant speed?

ID 6349

Jane is a secret agent in hostile territory. She can swim at 1m/s in still water with the secret load she is currently carrying. She has to cross a river speedily, without exhausting herself, so she swims at her normal pace. Sadly the river is flowing at 3 m/s and is 30m wide at the point she needs to cross.

How long does it take her to cross the river?

ID 6350

John is in the wilderness and encounters a fast flowing river. There is only one spot to cross as the bank is very steep, except at this one point. Directly across from this point is another break in the bank, with no other breaks visible. He therefore has to swim directly across the river.

With his back-pack he can swim at 1 m/s in still water. The river is flowing at 0.8m/s. It is 12 m across the river.

How long does it take him to cross the river, swimming with his normal amount of effort?

ID 6351

A modern sailing vessel, powered only by the wind, is at the center of the image (plan view).

In which direction(s) can it travel?

ID 6352

Much of the brilliance of aircraft design comes in optimising the ratio Lift/Drag for a range of flight speeds and conditions. In level flight the weight of the aircraft is equal to the Lift, but the retarding force, the Drag, is smaller by a factor of 10 or more. This means that the engine can produce 10x less thrust than would be needed to lift the aircraft straight up. Another key specification for a design is the Thrust/Weight ratio, a pretty self-evident measure.

Analyse and digest this technical information before picking an answer.

ID 6354

Peter has understood from his school work that an airplane's speed is relative to the wind, and not directly to the ground, so that if the wind is going in the same direction as the airplane, the airplane goes that much faster relative to the ground.

He therefore decides that it is a good idea to take off with the wind going in the same direction as his new model airplane is going to take off.

Comment on this plan.

ID 6355

Perhaps you have seen birds flying, without flapping their wings, and yet they still go up.

Pick a plausible explanation for this observation.

Image source of the short-toed snake eagle is Wikipedia

ID 6360

ID 6361

The heavy yellow weight is hung from the rigid red post. The blue plate is firmly attached to the ground with stakes (shown as thin vertical lines).

What can you say with certainty?

ID 6362

The heavy yellow weight is hung from the rigid red post. The blue plate is firmly attached to the ground with stakes (shown as thin vertical lines).

The joint at A is able to rotate freely.

What can you say with certainty about part B.

ID 6363

The Ancient Olympic Games were held every four years.

They already existed at the time of Homer in 776 BC.

A Roman emperor banned the games in 393 AD.

If a Game was held in year 2 BC, which year of Common Era was the next Game?

The term *anno Domini* is Medieval Latin and often translated as *in the year of our Lord*. BC is *before Christ*. It should be noted that at the time of Homer they were not using the current Gregorian calendar. Historians have effectively relabelled the old dates to make them easier for us to understand. The Gregorian calendar is widely used, but it can be offensive to non-Christians to use the BC/AD terminology. Hence they are being replaced by BCE and CE, the CE meaning Common Era. Since this change only started happening around 2002, the older system is widely used and you should know both.

ID 6364

Hopefully you realise that wood comes from trees. In fact the bulk of a tree is wood, along with a relatively small amount of leaves.

Where does all this wood come from?

(*Pick the best answer, as some of the answers may be partially true.*)

ID 6365

An explorer finds a metal triangle on a planet.

His Artificial-Intelligence camera reports that two of the internal angles in the triangle are 37° and 95°.

What can you say about this triangle?

ID 6367

ID 6368

ID 6369

In a typical carnival game, the player tries to throw a rubber ring over a wooden peg.

We won't consider any "tricks of the trade" which reduce the player's chance of winning.
Consider the ring falling straight down onto the pegs at random.

Call the peg diameter P, with the inner diameter of the ring as 2P. The pegs are on a square grid of 5P side length and large extent (lots of pegs).

The outer diameter of the ring is small and doesn't affect the outcome.

Success is defined as those cases where the ring falls directly over the peg without hitting it first.

What is the probability of random success?

ID 6370

Jane is sitting on a bench in a museum when she sees a round coin drop from a woman's purse onto the tiled floor. She tries to guess the probability that the coin will not land on the boundary between tiles. She neglects the thickness of the boundary between the tiles and estimates that the coin diameter is one fifth the side length of the tiles. She guesses that there is a 20% chance of the coin landing on the border.

What is the correct probability of the coin not landing on a tile boundary?

ID 6372

A particular carnival game involves throwing a soft ball at stacks of bottles, with each knocked over bottle contributing to a prize. The stallholder demonstrates how easy it is, and yet you just can’t seem to do it, despite hitting the bottles in the same place that the stallholder did.

Can you guess why this might be?

ID 6373

Alan, aged 10, has a devious plan to "prove" that he is good at mathematics. He plans to go on to the site ApusClick.com and take on questions for 17 year olds in front of a single adult witness. He will randomly click on answers to two questions only. If he gets both correct he gets his witness to tell everyone what happened, "proving" his brilliance. If he gets any question wrong, he immediately stops, discards that adult, and picks a new witness.

Alan is especially lazy, and can't even be bothered to remember what the correct answer is when he has answered a question previously.

What is the probability of Alan proving his brilliance if he has a pool of 20 adults to use?

(*Please use a calculator if you need to.*)

ID 6374

In computing, a hash is a fixed length output value computed from an arbitrary length input. As an example, it is unsafe to store plain-text passwords anywhere, so it is usual for a computer to hash the password and store that. Any slight change to the input has a large and unpredictable change on the output.

The SHA-256 hashing function produces a 256 bit result from any input. Given that a hexadecimal (hex) character has values between 0 and 15 (0 to F in hex), how may hex characters are needed to print an SHA-256 hash?

ID 6390

Carlo Iznop has an interesting business model for his investment firm. Each client deposits exactly $1000. He gives them 10% return on their investment after 1 year. This profit must be withdrawn. In the first year he had 30 clients. 2 out of 3 withdrew their money at the end of the year. At that time, word spread about this fantastic investment opportunity giving double the normal rate of return, so the total number of investors increased. Over the years his fame spread and the business boomed.

In the fourth year suspicions were raised about the consistently high profits and the police were called.

What is the most probable outcome?

ID 6391

Suppose you want to photograph the cat when it passes through an invisible beam, but you also want to know where it came from. You automatically record the digital camera images to a memory buffer, one after the other, always overwriting the oldest image. The buffer has a certain length, and when you get to the end, you start again at the beginning. Each position in the buffer has a positive index, starting from 0. You write to the current position, then increase the position counter by one. When you increase beyond the end of the buffer you reset the counter to zero.

The position counter is called POS. The buffer length is called BLENGTH.

What is the correct value for the position counter if you want to go back 20 images from the last image?

ID 6392

A thin-rimmed hollow sphere (a black ball) and a homogeneous solid sphere (a white ball) both have the same weight and the same diameter.

They are both dropped simultaneously from a height of 3 feet (around 90cm) onto a glass table below. Both balls have the same material and the same finish on their outer surfaces.

Which has the most kinetic energy at the moment of impact?

ID 6393

Three spheres with equal diameters and equal masses are allowed to freely roll down a smooth inclined ramp.

The green sphere is homogeneous.

The blue sphere is hollow with a heavy rim.

The thin-walled orange sphere is hollow, with a smooth inner surface and filled with a dense, but remarkably inviscid liquid. (*inviscid* means having a very low viscosity - it's "runny")

When released at the same time, in which order do they arrive at the bottom of the ramp?

ID 6394

An eccentric retired professor of Mathematics has created a walled-off area in his garden in the shape of an isosceles triangle. He tethers a goat to the apex of the triangle with a rope of the correct length so that the goat can graze over exactly half of the garden.

Neglecting practical considerations like the size of the goat compared to the garden, how long is the rope?

(*This question comes in two variants: you can either approximate the answer or you can work out the complete answer - an only slightly more involved process.*)

ID 6395

Jane is studying the subject of fluid mechanics from text books. In the first book she reads she estimates that there were 1000 key facts presented. In the next book she reads this also presents 1000 facts, but half of these are duplicates of what she has already read, but more worryingly, 0.1% contradict previous facts.

If every future book she reads has 1000 facts, but new facts halve in number with each successive book, and contradictions occur randomly at a fixed 0.1% rate, what is her projected accumulation of true facts after she has studied an infinite number of such books?

ID 6396

The horizontal Field of View (the size of the scene which fits into the picture) of a particular digital camera is 3m at a distance of 3m.

What is the angular Field of View?

(*Hint: Draw a little sketch.*)

ID 6404

A woman tries to hide her age from an inquisitive boy by telling him that her age is between 31 and 61 inclusive, but that the boy can only have 4 guesses at her age. The guess will be answered truthfully with higher, lower, or correct.

In the worst (unlucky) case, but with intelligent guesses, how far from the correct age will the boy be?

ID 6414

For a small angle d (in radians), the sine of the angle is approximately equal to the angle.

Often the cosine of a small angle is approximated as 1.

Which is the best approximation for the cosine of this small angle?

(*Hint: Pythagoras*)

ID 6440

Many houses in Switzerland use domestic air-sourced heat pumps for central heating.

A heat pump is a marvellous device. You can put 1kW (kilo-watt) of electrical power in and get 3kW of heat out. Engineers dislike calling this 300% efficient, so they use the term Coefficient of Performance (COP) instead. COP=3 means 3kW out for 1kW in.

Different manufacturers produce units with different performances, so under the same conditions one unit can have a COP of 3 and another can have a COP of 4.

Given the same conditions, how much more electricity will the COP=3 system require than the COP=4 system?

ID 6444

Why would anyone design something as apparently stupid as a penny-farthing bicycle?

And why would people buy it?

ID 6445

The beam is balanced by the spheres, all of which have the same diameter.

All of the red spheres have the same density as each other.

All of the blue spheres have the same density as each other.

What can you say with certainty?

ID 6447

A boy throws a very bouncy rubber ball directly down onto a hard concrete floor from a height of 1.5 m.

To what height does the ball bounce?

ID 6448

Jane, the undercover operative, is in a strange far-away city at night. The sky is overcast and moonless so she can’t get a bearing from it. She has no phone or compass to assist her. She knows that the city streets are drawn on a neat rectangular grid, so in order to keep a note of her direction she adds 1 for a left turn, 2 for a 180° turn, and 3 for a right turn, all relative to the direction she is travelling in before each turn.

She starts off looking directly at the clock in the city square. The rest of the city seems featureless to her foreign eyes.

After an hour of following her suspect, with him deliberately making unnecessary turns to throw off anyone following, she has reached a count of 401.

If facing the clock was looking north, in which direction is she now facing?

ID 6449

Simplify the expression shown,* if you dare*!

All letters inside the bracket represent integer variables.

ID 6450

All the balls hanging from the balance have the same weight. The dotted suspension lines represent stretchy string, whereas the solid lines are inelastic cord.

The grey balls are possible positions for balls of equal weight to the red balls.

Which single position for a grey ball balances the beam?

ID 6451

The blue ball weighs somewhere between 1 and 31 times the weight of a red ball, the ratio being an integer. Enough red balls are placed on the right hand side of the balance to make the beam level. The minimum number of red balls is used to balance any particular weight. There is no restriction on the number of balls in any particular position on the balance.

What is the maximum number of red balls required to balance any blue ball weight within the given range?

ID 6454

The blue square is the plan view of an open-topped box with slippery walls. The washer, which is not shown to scale, is thrown into the box at random, and the washer always lands flat on the bottom. The length of the inside edges of the box is 12 inches. The washer’s outer diameter is 1 inch and its inner diameter is 7/16 of an inch.

What is the probability that the washer lands entirely within the bottom left-hand square (red)?

ID 6458

You have found the picture to the right on a random website on the internet. The author claims that it shows an inverted (upside down) red plastic funnel with nothing hidden inside. The author further claims that the top ring seems to almost float in the position shown, as if supported by springs.

What could you reasonably conclude?

ID 6459

You try to work out the problem shown on the right by entering the values into a cheap hand-held calculator.

What answer do you get?

ID 6461

Two ordinary cars with gasoline (petrol) engines are being compared. At their maximum power outputs, one produces twice the power as the other when measured at the road wheels.

What can you say with certainty?

ID 6462

A recruitment agency gets paid for each contractor they place, and they pass on 80% of the fee to the contractor.

But looking at it the other way around, starting from the contractor's fee, how much has the agency marked up the contractor's salary?

ID 6471

Jane, a specialist in many different bird species, is trying to weigh a parrot. First she weighs a large glass bottle, then she weighs the same glass bottle with a parrot inside. The glass bottle is air-tight, so Jane makes sure that the parrot is never in the bottle for more than 3 minutes so it doesn't suffocate.

At first the bird stands on the bottom of the bottle so Jane takes a reading. Then the parrot hovers beautifully in the middle of the bottle for long enough that the weighing scale reaches a steady value. She records the result and releases the parrot.

Compare the two readings.

ID 6472

I put three dice into a cup, shake them up, and roll them out onto a table. A camera looks at the table from above, and computer software correctly finds and displays the die which is closest to the center (centre) of mass of the collection of dice.

What is the probability that the value displayed is 1, 2, or 5?

ID 6473

I have put 3 ordinary 6 sided dice in a cup, and I am shaking them. Before I cast them out onto a table I want you to decide, using the mystical powers of your mind.

If you win you get to pick a prize from the prize table. Otherwise you will get nothing.

ID 6474

In order to disguise their ages, four women will only admit that the sum of the (integer) ages of Anna, Belinda, and Christina is 105. They will also admit that the sum of the (integer) ages of Daisy, Belinda, and Christina is 89.

What can you say with certainty?

ID 6475

Herlewin the Lesser is trying to work out why his workers are costing him so much. He is paying 170 groats/hour for his workforce, consisting of one of each of a plasterer, an electrician, a carpenter, a builder, and an architect.

He knows that electricians earn twice as much as carpenters, that builders and plasterers earn the same, and that the carpenter and builder together earn as much as the architect. He also knows that the builder earns 30 groats/hour.

How much does the carpenter earn?

ID 6479

Mary is half as old as the younger of her two brothers. In 10 years time she will be as old as her younger brother is now.

How old will the younger brother be in 10 years time?

ID 6480

There is a simple repeating sequence of numbers:

1, 2, 7, 11, 12, 17, 21, 22, 27, 31, 32, 37, …

This pattern continues up to 1000.

How many numbers are there in the sequence?

ID 6504

John has been driving for several hours in busy traffic and realises that most of the cars he has seen are different, neglecting color (colour).

What is the most reasonable explanation for this?

ID 6505

John (left) weighs himself, eats 100g of cooked rice, drinks 1 kg of water, then weighs himself again.

Peter (right) weighs himself, eats 100g of super-high-calorie energy bar, drinks 1 kg of water, then weighs himself again.

All of this activity takes place within the space of a few minutes, and no significant activity has not been mentioned.

Who gains the most weight?

ID 6506

You may have heard people use the expression "give it 110% effort" or similar.

What can you say about this?

ID 6507

Jane has always been a bit of an oddball. Rather than buying stocks and selling them later, she likes to sell stocks she doesn't have, hoping to buy them back later.

How does she make a profit?

ID 6513

Jason, whilst out for a drive in his front-wheel-drive car, comes across a slightly larger car stuck in mud. Jason's tow rope is long enough that Jason's car can stay on firm ground.

What is the optimum place for the rather heavy-set (large) passenger of the stuck car.

ID 6514

You have been offered a game of chance by an eccentric multi-billionaire.

He will toss a coin repeatedly until it comes up heads. If heads appears on the first throw he will pay you $2. If it appears on the second throw, you receive $4; if on the third, you receive $8 and so on, doubling each time.

You know that this is the famous St. Petersburg paradox, with an expectation value of infinity, so his requirement for you to pay him $63 to play the game seems fair. And yet you hesitate …

What is the chance of you winning at this game?

ID 6531

This year Samantha has become very fussy about her birthday present. A square box must be wrapped in blue paper. A round box must be wrapped in red paper. An irregular box must be wrapped in green paper.

The probability of her getting a round box is 50%. The probability of her getting a green box is 1/4.

What is the probability that this spoilt child receives a blue present?

ID 6536

A fairly fit runner is in a race with other equally fit runners. The race distance is 1 mile. Unlike marathon runners, these runners do not take on fluids during the run.

At the end of the run, what should we expect of our chosen athlete?

ID 6537

In a strange and far-off land, supermarkets typically sell date-limited food at a discount at a particular time of day. Successful hunter-gatherers can then benefit from a product which was at its full price until only a few minutes beforehand.

The image depicts the price in units of pounds, a decimal currency such that 29p = £0.29.

What discount has been applied?

ID 6603

In a popular type of train, a Diesel engine drives an electric generator, which in turns powers the electric motors that drive the wheels.

Given that every step in this chain loses power, why is such an apparently complicated system used?

*Hint: If you drive or have been frequently driven in a 'stick-shift' (manual gearbox) car you will have a distinct advantage in answering this question.*

ID 6604

The safety label on this charger for power tool battery packs says 'Rest charger 15 minutes between charges'.

Why is there such a requirement?

ID 6605

In a scene from a film, the baddy is sharing the ill gotten gains with his (comically stupid) henchman by dividing the gold pieces into two 'equal' piles.

'That’s one for you, and that’s one for me'.

'That’s two for you (puts an additional one on the henchman’s pile), and two for me (puts an additional two on his own pile)'.

'That’s three for you (adds one to the henchman’s pile), and three for me (puts three more on his own pile)'.

The final round is 'fifty for you, and fifty for me'.

How many fewer pieces of gold does the henchman get?

*Leslie Green suggested the problem for Aplusclick project. This theme has been repeated at least once. An early version was with Bugs Bunny, 'Racketeer Rabbit' (1946).*

ID 6606

In a scene from a film, the baddy is sharing the ill gotten gains with his (comically stupid) henchman by dividing the gold pieces into two 'equal' piles.

'That’s one for you, and that’s one for me'.

'That’s two for you (puts an additional one on the henchman’s pile), and two for me (puts an additional two on his own pile)'.

'That’s three for you (adds one to the henchman’s pile), and three for me (puts three more on his own pile)'.

The final round is 'N for you, and N for me'.

How many times more pieces of gold does the baddy get?

*Leslie Green suggested the problem for Aplusclick project. This theme has been repeated at least once. An early version was with Bugs Bunny, 'Racketeer Rabbit' (1946).*

ID 6609

Brian (aged 12), having been watching a Sci-Fi movie, is wondering what would stop a small airplane flying to the moon if we could somehow get it up into orbit.

Which of these do you think is the biggest problem?

A) There are no petrol stations on the way to the moon

B) Sound does not travel though space

C) The cabin is not air tight so the pilot couldn’t breathe.

D) The engine needs air to burn the fuel.

E) The engine needs air to cool down.

F) The wings need air to generate lift and control direction.

G) The propeller needs air to drive the plane forwards.

ID 6622

Two hundred and thirty two boys and two hundred and twenty nine girls came out of school at different speeds.

After a while, 38 adults rushed out of the same building even faster than the kids.

The question: 'How many students and teachers went home after school?' has already been asked of 10 year olds, to which the correct answer was 499.

Now we ask a deeper question.

How many reasonable assumptions had to be made to get the correct answer?

ID 6653

Can you move the tennis balls from the pattern on the left to make the pattern on the right?

* NOTE: the strings are special so that whatever you do they never get tangled up with each other.*

ID 6654

A convex polygon is a simple polygon (not self-intersecting) in which no line segment between two points on the boundary ever goes outside the polygon.

Which of these shapes is not a convex polygon?

ID 6705

Mary has devised a rope and pulley hoist system to lift her 2 tonne (2000kg) boat out of the water. She has to pull the rope with a force equivalent to a 10kg weight to lift the boat.

How far does she have to pull the rope to lift the boat 1m out of the water?

*For simplicity we assume that the rope and pulley system is lossless.*

ID 6706

A horse trough has a rectangular cross-section and is continuously being filled from a tap. Water is leaking from a hole in the bottom of the trough, the leak-rate being proportional to the square root of the water height in the trough.

The tap is adjusted so that the water rises to just under the top of the trough as a steady-state condition.

The tap is now adjusted to halve the flow rate.

What is the new steady-state height of water in the trough?

ID 6707

A horse trough has a rectangular cross-section and is continuously being filled from a tap. The trough has two similar holes in it: one at the bottom, and one half way up. The leak-rate from a hole is proportional to the square root of the water height.

The tap is adjusted so that the water rises to one fifth the height of the trough as a steady-state condition.

The tap is now adjusted to double the flow rate. What is the new steady-state height of water in the trough?

ID 6715

John finds a box of similar looking books in a box in the attic. The books are labelled with large letters: V, X, II, I, IX, and VI.

How many books can he be confident are missing from this collection?

ID 6716

Here is a number sequence with a definite mathematical rule to move from one number to the next.

3, 7, 16, 43, 124

The tricky thing is that one number (not necessarily at the end) has been deleted from the sequence, but which number?

(*Hint: This is just mathematics. It works in any language, not just English.*)

ID 6717

Here is a number sequence with a definite mathematical rule to move from one number to the next.

2, 5, 23, 47, 95

The tricky thing is that one number (not necessarily at the end) has been deleted from the sequence, but which number?

(*Hint: This is just mathematics. It works in any language, not just English.*)

ID 6718

Here is a number sequence with a definite mathematical rule to move from one number to the next.

3, 7, 11, 17, 27, 43

The tricky thing is that one number (not necessarily at the end) has been deleted from the sequence, but which number?

(*Hint: This is just mathematics. It works in any language, not just English.*)

ID 6719

Sammy the Squirrel hides acorns in groups of 3, 4, or 5 per stash, the actual number being pretty random. He creates 200 such stashes in the autumn. When winter comes he only manages to find 5% of his stashes.

On average, how many acorns does he find from his stashes?

ID 6720

Joe, the ice-cream van man, has observed in his 20 years on the job, that almost all children like ice-cream.
In order to prove this he asks 999 of his child customers if they like ice-cream.

All but two say they like ice-cream.

What is the biggest problem with his assertion that around 99.9% of children like ice-cream?

ID 6721

Here is a number sequence with a definite mathematical rule to move from one digit to the next.

8, 1, 7, 0, 3, 6, . . .

The tricky thing is that one digit (not necessarily at the end) has been deleted from the sequence, but which digit?

(*Hint: This is just mathematics. It works in any language, not just English.*)

ID 6724

Spotting patterns is very important.

Can you spot which number doesn't fit the pattern?

3, 6, 12, 26, 48, 96, . . .

ID 6725

Spotting patterns is very important.

Can you spot which number doesn't fit the pattern?

1, 2, 5, 14, 42, 122, . . .

ID 6726

Spotting patterns is very important.

Can you spot which number doesn't fit the pattern?

231, 248, 266, 282, 316, 333, 350, 367, . . .

ID 6727

This sequence of numbers repeats so they could be written in a circle (after deleting the 1 and 6 at the end, since they just show the repeat point).

There is a definite mathematical rule to go from one number to the next.

1, 6, 4, 3, 0, 5, 3, 1, 6, . . .

Sadly one (and only one) of the numbers has been mis-typed, but which one?

ID 6728

You are required to follow my instructions very carefully:

After every mathematical operation you are to calculate the total.

Think of a number.

Add 3.

Multiply by 3.

Add 6.

Multiply by 2.

Subtract 30.

Divide by 6.

What is the result?

ID 6730

An elite special forces soldier weighing 100 kg (with full kit) is being winched up to a steadily hovering helicopter at a constant speed of 5 m/s.

What is the steady-state load on the winch cable?

NOTE: take the gravitational constant as 10 N/kg or 10 m/s^{2}

ID 6731

An elite special forces soldier weighing 100 kg (with full kit) is being winched up to a steadily hovering helicopter at a constant acceleration of 10 m/s^{2}.

What is the steady-state load on the winch cable?

NOTE: take the gravitational constant as 10 N/kg or 10 m/s^{2}

ID 6732

A village lies in a mountain valley, with valley walls so high that the sun is blocked out for 6 months of the year. People can get very depressed when they live without sunshine for months on end.

At a council meeting it has been suggested that a mirror is installed on the valley wall to reflect sunlight onto the village below.

What would be the most useful response to the suggestion?

ID 6733

Ugg, the primitive human, finds a perfectly circular fountain with a diameter of 1.8m. Of course Ugg doesn't know what a fountain is, or what a diameter is, but he decides to measure the circumference of the fountain, despite not knowing what a circumference actually is. Ugg can only measure using his walking stick and a piece of chalk. The walking stick is remarkably straight, and by sheer chance it just happens to be exactly 1m long.

Ugg is not as clever as you, so he would not think of pressing the stick against the curve and moving the pressure point down the length of the stick to follow the curve exactly.

What is Ugg's count of the number of sticks needed to surround the strange looking historic artefact he has found?

*Unusually, and just for this question, you are encouraged to open another browser tab and search for any information on the Internet which will help you to solve this problem. *

ID 6734

List the following storage technologies in order of their data retention capabilities, the longest storage being listed first:

USB flash drives

paper/books

stone tablets

magnetic tape

ID 6735

In a remote African village it is traditional for the women to set off early in the morning to walk through the desert to fetch water. This is a long and arduous journey. There are three water containers available in adequate quantities, but each woman can only carry one of them.

The measure of water is unique to this tribe and we will just call it a unit.

A large container carries 4 units of water, but its design is so poor that on the trip back, half will be lost by leakage and splashing.

A medium container carries 3 units of water, and has a better design, so only one third is lost on the return journey.

The small container is only slightly smaller than the medium container, carrying 2 and 2/3rds units of water. However, it only leaks one quarter of its original content.

We are told that the container itself weighs half as much as the maximum content it can carry.

Which container should the tribal elders instruct the women to take?

ID 6737

A shop is offering a luxury food item at half price. It is not close to its "sell by" date.

How is this economically viable?

ID 6738

Two tiny motor boats set out from the shore of a lake, each with a full fuel tank and 4 full fuel cans. Each fuel can stores the same amount of fuel as held in the onboard tank. The two boats are tied together until it is time to separate them. The boats use fuel at a steady rate, regardless of the amount of fuel on board. One tank of fuel lasts for 30 minutes.

Each boat can only hold 4 full fuel cans, and a fuel can, once opened, has to be completely emptied into the fuel tank of the boat it is on. Fuel in the onboard tanks cannot be shared.

If fuel cans are optimally transferred, what is the maximum time one boat can continue on its journey?

ID 6765

Three tiny motor boats set out from the shore of a lake, each with a full fuel tank and 3 full fuel cans. Each fuel can stores the same amount of fuel as held in the onboard tank. The three boats are tied together until it is time for one or more to separate. The boats use fuel at a steady rate, regardless of the amount of fuel on board. One tank of fuel lasts for 1 hour.

Each boat can only hold 3 full fuel cans, and a fuel can, once opened, has to be completely emptied into the fuel tank of the boat it is on. Fuel in the onboard tanks cannot be shared.

If fuel cans are optimally transferred, what is the maximum time one boat can continue on its journey?

ID 6766

You are on a critical mission. You reach the shore of a lake where 60 similar tiny motor boats are moored. None of the boats have any fuel on board. There are 52 full cans of fuel available, with each can being the same capacity as the on-board fuel tanks of the boats.

There are plenty of volunteers on hand to help you.

The boats use fuel at a steady rate, regardless of the amount of fuel on board. One tank of fuel lasts for 1 hour. Each boat can only hold 3 spare fuel cans, and a fuel can, once opened, has to be completely emptied into the fuel tank of the boat it is on. Fuel in the onboard tanks cannot be shared.

You need to get away as far as possible from this place by boat. How many boats should start the journey?

ID 6767

You are on a critical mission. You reach the shore of a lake where 55 similar tiny motor boats are moored. None of the boats have any fuel on board. There are 52 full cans of fuel available, with each can being the same capacity as the on-board fuel tanks of the boats.

There are plenty of volunteers on hand to help you.

The boats use fuel at a steady rate, regardless of the amount of fuel on board. One tank of fuel lasts for 1 hour. Each boat can only hold 3 spare fuel cans, and a fuel can, once opened, has to be completely emptied into the fuel tank of the boat it is on. Fuel in the onboard tanks cannot be shared.

You need to get away as far as possible from this place by boat.

What is the maximum number of hours you can travel by boat?

ID 6768

The sine of an angle, A, is identically equal to the cosine of some other angle.

What is the other angle?

ID 6771

I have 8 books with weights of 1001g, 1003g, 1005g, 1007g, 1011g, 1013g, 1017g, and 1019g, where the weights are all accurate to better than ±0.1g.

I have a set of balance scales which will balance provided the imbalance is less than ±0.5g.

I weigh four books at a time, two on each side of the balance.

How many unique sets of 4 books will balance?

(*Swapping books from the left pan to the right pan does not constitute another set.*)

ID 6773

You have been told to suspend the blue object from one of the red mounting points. You have done some calculations and determined that one mounting point is only just inadequate to support that much load. The rope is easily strong enough, with plenty of margin, it is just the mounting point that is problematic.

You boss really really wants the blue object suspended, NOW. He is a very important person and tells you to use two mounts as it will halve the load. The angle of the mounting ropes to the horizontal is 25°.

What is your response? (And remember, if anything goes wrong you were the engineer, and your boss is just a marketing expert.)

Just in case it is relevant: sin(25°) = 0.422; cos(25°)=0.906; tan(25°)=0.466

ID 6774

Desmond the Dragon has been naughty. He has been burning cottages with his fiery breath and eating sheep between meals.

The villagers have decided this must stop, so they have tied him up with a tungsten cable which can hold twice Desmond's weight.

Is this an adequate precaution?

ID 6776

Today there is a tug of war between a human, an orangutan, and a gorilla. You may be unfamiliar with the units being used. We call the force necessary to support a 1kg weight one kilogram-force, with the notation 1kgf. Whilst this is not one of the preferred SI units, it is easy to understand from everyday experience.

The human pulls in the compass direction of 000° with a strength of 100kgf. The orangutan thinks it is very funny and pulls with a force of 120kgf in the compass direction of 240°. The gorilla really can't be bothered, so only pulls with a force of 150kgf in the compass direction of 120°.

In which compass direction does the junction of the ropes move?

ID 6778

I would like to weigh myself, but neither of the two bathroom scales available to me can take my somewhat chunky weight. Also, both are broken in the sense that they have zero offsets, and the zero offset button does not work. I have checked the scales with a 20kg weight, and both read correctly when their offset is taken into account.

The left scale reads +1.1kg when unloaded. The right scale reads -0.6kg when unloaded.

I stand on both scales at the same time, one leg on each, although my weight distribution is not necessarily equal.

The left scale reads 48.1kg and the right scale reads 37.3kg.

What is my actual weight?

ID 6779

Shawna wishes to measure the height of a tree for no clearly explained reason. She has determined that the distance from the ground to her eye level is 1.7m when she is wearing her usual fashionable boots. She uses a 45° set-square to sight-along and she uses a spirit level to make sure the ground is level and the base of the set-square is also level.

She walks back from the tree until the top of the tree aligns with the set square, then she measures the distance from where she is standing to the centre of the tree trunk. The distance she measures is 11.2m.

What is her estimate of the height of the tree?

ID 6780

Having just won the lottery, Jeff has gone from tossing burgers for a living to being a multi-millionaire. Sadly his wealth has increased faster than his EQ (*Emotional Quotient*) can handle. He demands that the bath in his new mansion be ripped out and replaced by one twice as big. When the tradesmen try to make suggestions he just shuts them up angrily, and tells them to do exactly what he asked for, no more and no less, without adding in any complexity.

Jeff gets in the bath whilst filling it, but as it gets nearly full it breaks through the ceiling and plummets onto the marble floor below.

What went wrong?

ID 6781

Lenny throws a baseball to Kenny who is 20m away. The ball arrives in 500ms (0.5s).

Neglecting wind resistance and taking **g** = 10m/s^{2}, to what height above the starting point does the ball reach, given that it is thrown from and received at the same height?

Although you may use different symbols, these formulae may provide some reminders:

**s =ut + (1/2)at ^{2} ; v^{2} - u^{2} = 2as ; v = u + at**

ID 6782

Lenny throws a baseball to Kenny who is 20m away. The ball arrives in 500ms (0.5s).

Neglecting wind resistance and taking **g** = 10m/s^{2}, how fast is the ball thrown, given that it is thrown from and received at the same height?

Although you may use different symbols, these formulae may provide some reminders:

**s =ut + (1/2)at ^{2} ; v^{2} - u^{2} = 2as ; v = u + at**

ID 6783

The boss insists that the new car design must accelerate equally quickly from 20mph to 30mph as it does from 50mph to 60mph.

What is your response?

ID 6784

The year is 2045. Most new cars now have electric drive trains. One particular new innovation is the DriverTron with its *Insanity Mode*. In this mode the car automatically applies a constant maximum power to the wheels at all speeds until it reaches 100mph or until the brakes are applied. Needless to say this mode has been banned in all jurisdictions apart from two States in the USA!

If we neglect wind resistance and bearing loss, what equation represents the velocity **v** of the car in *Insanity Mode* at speeds less than 100mph?

**t** is the time from starting at zero speed, and **k** is a constant.

ID 6785

Rechargeable battery technology is familiar to most people with phones and other portable devices. But larger batteries to power cars and homes are less well understood.

A rechargeable battery can be modelled as an ideal rechargeable battery in series with a parasitic resistance. This resistance dissipates power dependent on the square of the charging current. Now given that electric cars typically have a range of only 200 miles, it is quite tiresome to stop every 150 miles and recharge it for 300 minutes (5 hours). It would be much nicer to charge it in 30 minutes, or even 3 minutes, to make it more like a gasoline (petrol) powered car. But is there a downside to this?

We are going to assume for simplicity that a rechargeable battery is charged by a certain number of ampere-hours. If you double the current (measured in amperes) you halve the charging time.

Consider what happens to the energy loss in the parasitic resistance in the battery if instead of charging in 300 minutes you use a hyper-charger and do it in 3 minutes (for the same battery type).

ID 6787

ID 6788

A particular function takes a value X, multiplies it by 4, adds 16 to the result, then takes the square root of the result.

Can you simplify this function?

ID 6789

A particular function takes a value X, adds 16 to it, then takes the square root of the result.

Can you simplify this function?

ID 6790

A linear transform takes a value and converts it into another value using a rule such as 'add 3 then multiply the result by 3'

Which of these is the same transform?

ID 6791

In mathematical language, a function can be considered as a rule to transform something into something else. The inverse function will then change it back again.

If the function gives one over the input, what is the inverse function?

ID 6792

In mathematical language, a function can be considered as a rule to transform something into something else. The inverse function will then change it back again.

If the function gives the square of the input, what is the inverse function?

ID 6793

A wristwatch has had a chaotic price history, which consists of adding $10, a $10 reduction, doubling in cost, and being at a 50% discount.

If these operations were applied in one order the final price would give a maximum, and in another order would give a minimum.

What is the difference between these two costs?

ID 6810

It was announced on the UK news that birds were flying backwards in a certain location at a certain time.

Suggest the most plausible explanation for this report.

ID 6811

You see a person apparently swimming vigorously in the sea near to a jetty. The strange thing is that he is swimming backwards, by which I mean his head should be getting closer to the jetty, but it is actually getting further away.

What should you reasonably conclude?

ID 6816

A table with four legs is resting on an uneven floor. Even if the floor were level, the table would still have been wobbly as the legs are not exactly equal.

You have available a set of wedges to place under the legs to make the table secure.

What is the minimum number of wedges required to secure the table in the worst case of mismatched legs and floor?

ID 6818

A table with four legs is resting on an uneven floor. Even if the floor were level, the table would still have been wobbly as the legs are not exactly equal.

You have available a set of wedges to place under the legs to make the table secure. The table top is perfectly planar.

What is the minimum number of wedges required to make the table top secure and perfectly horizontal in the worst case of mismatched legs and floor?

ID 6819

A Physics teacher is on a camping trip with his young nephew. Seizing the opportunity to impart some Physics to this youth, the teacher sketches out a graph of how well-stirred water in an open pan might respond to being placed in the middle of a well-established large fire, built using locally sourced wood.

Which graph did he draw?

(*Note: you must be able to justify your answer to get full credit on this question*.)

ID 6820

Suppose you were travelling directly from the Earth to the Moon. You can imagine that as you got further from the Earth, the pull of Earth's gravity would be reducing, whilst the pull from the Moon's gravity would be increasing. At some point these two forces would be equal and opposite.

At what fraction of the distance from the Earth to the Moon would this equilibrium point be?

For the purpose of this question, assume that only the Earth and the Moon exist, and that they are stationary. Take the mass of the Moon to be 1/100th the mass of the Earth.
And we will remind you that the gravitational force on a small object is proportional to the mass of the astronomical body, and inversely proportional to the square of the distance to that body.

ID 6833

6 numbers are required to have an average of 9999. The first five numbers are: 9999, 9998, 9997, 9996, and 9995.

What must the last number be to meet the requirement?

Hint:*Look for a sneaky method, rather than doing long-addition or algebra.*

ID 6834

Compare the security of a 4 digit passcode using two different entry methods:

In the first method all 4 digits are entered and an ENTER button is pressed.

In the second method each digit is ENTER'ed in turn, and the next digit can only be entered once the previous digit has been validated.

Consider only the unlucky (worst) case for each method.

What is the ratio of the number of ENTER key presses of the first method compared to the second method?

ID 6841

Whilst the Earth gets all of its heat from the Sun, roughly how much of the Sun's heat does the Earth get?

*Don't over-complicate the question. Just think about how much of the Sun's radiated power is intercepted by the Earth.*

Sun's diameter = 1.4 million km

Earth's diameter = 13,000 km

Distance to the Sun = 150 million km

ID 6842

A spaceship is going to fly from the Earth to Mars. The project has been sabotaged by politicians so that the flight starts when the planets are in the worst possible orientation for the flight.

What is the distance the spaceship has to travel?

Assume circular orbits with a common center (centre) about the Sun. Assume both orbits are in the same plane. Don't worry about matching orbital velocities.

Radius of the orbit of Mars = 230 million km.

Radius of the orbit of Earth = 150 million km.

ID 6843

A physicist with broken legs is stuck in the middle of a frozen lake. For no clearly stated reason she is sitting on a smooth based sled, with a box full of base balls. We consider that the ice is perfectly flat and has low friction. Her only means of propulsion is to throw baseballs.

What is her optimum strategy to reach the edge of the lake?

ID 6844

A particular electronic dictionary file of the English language contains 80,000 words, 1035 of which are three letter words.

If a three letter word is chosen from random letters, estimate the chance that it will be a valid English word?

ID 6848

I have an unbiased coin, a fair six-sided die, and an ordinary pack of 52 playing cards. I toss the coin, roll the die, and pick a card at random.

What is the chance that I don't get a head, roll a number less than 5, and don't get a heart?

ID 6849

99 spheres with the same diameter (2cm), weight, and surface texture are put into a black bag. These spheres are coloured red, green, blue, white, pink, fushia, magenta, indigo, cyan, azure, gray, jade, plum, ruby, salmon, and viridian in some undefined way. A single silver sphere is added to the bag. It is double the diameter of the rest, but has the same texture.

You are blindfolded and allowed to pick one (and only one) sphere from the bag. If you pick a vaguely red sphere you get $10. If you pick the special silver sphere you get $1000. If you pick anything else you get nothing.

What is the probability of your getting more than $10?

ID 6850

Water is not getting from the PUMPING STATION to the FEED point. A section in the pipeline is blocked, and you are required to find the exact section which is blocked. You know that the pumping station is fine because you spoke to them on the phone.

You have been given an accurate sketched map of the pipeline showing 8 taps. If you open the tap and water comes out you know the pipe is unblocked up to that point. Using an optimum strategy, what is the minimum number of taps you have to open to establish which exact section of the pipe is blocked, assuming you are unlucky in your choices.

ID 6860

Examine the following statements:

1) For a given stored volume, a sphere has less surface area than a cube.

2) For a given width limit, a sphere and a cube have the same ratio of volume to surface area.

3) A cube of a given width will have a lower volume to surface area ratio than a cuboid of the same dimensions (except its length is twice its width).

ID 6984

Grandpa drives a dilapidated old banger which only gets 10mpg (miles per gallon). The City Slicker drives the very latest Eco car which achieves an impressive 100 mpg. The City Slicker is always bragging about how his car choice is so good for the environment.

Grandpa does a 1 mile round trip 5 times a week to visit his pals at the retirement home. The City Slicker drives 50 miles to work each day, again 5 times a week.

Compare the fuel usage of these two life-styles.

ID 7037

You wish to pour juice into a glass from a container with a rectangular cross-section (as shown).

We can define the orientation of the container by saying that one of the 4 marked edges is uppermost.

Which edge should be uppermost to give the best control when pouring from a fairly full container?

ID 7038

The picture shows a fairly standard domestic refrigerator door.

We wish to consider the position of the tall glass bottle, neglecting any other items in the fridge.

The glass bottle is currently in the middle of the door. If we move it to the left it is closer to the door hinge. If we move it to the right it is closer to the open area.

Which is the best position for the tall glass bottle?

* Hint: consider Newton. *

ID 7047

The super-villain of a science-fiction movie shrinks the moon to one hundredth of its original diameter.

If we willingly suspend our disbelief for a moment, and we of course assume that the Law of Conservation of Mass applies, what is the resulting average density of the shrunken moon?

ID 7048

We perform a multiplication

9 x 10 x 10 x 10 x 10 x ... x 10 = ?

The triple dots ( ... ), known as an ellipsis, means there are some tens missing from the equation.

Suppose there are a total of 100 tens in the equation.

How many noughts are there in the result?

ID 7049

ID 7060

Consider the planar figure shown to the right. All angles are 90°. There are 4 dimensions shown, with X being clearly shown by the red line segment.

What is the effect of X on the perimeter of the shape?

ID 7066

The image shows a wooden door with a brush-type draft excluder ready to be cut to fit on the bottom of the door. The door hinge is on the left in this picture.

The instructions that come with the part just say to cut it to size, but where should we cut it?

ID 7067

Drawing A shows a massive black beam easily supporting the load of two heavy red beams by means of the blue tension rods and the green nuts. The blue rods are very strong, while the nuts are the weakest elements of the design. It is not your design, but you have been given computer simulations of this design and it all looks fine, ... in theory.

The contractor hired to build the design hates it. The green nuts have to be spun all the way up the blue threaded tie rods, which is very time consuming and therefore expensive. Also both red bars have to be assembled at the same time and that is difficult. The contractor has come up with design B which uses shorter tie rods of the same design but uses the same nuts. They have calculations to show that the upper red beams will not be damaged by offsetting the lower tie bars slightly.

Will you approve the change?

ID 7068

A large sofa manufacturer seems to be forever having sales. There is the Spring sale, Summer sale, Autumn sale, Winter sale, Xmas sale, New Year's sale, and so on. During these periods they sell particular lines for half price.

According to local legislation (Laws) these items have to have been offered at the higher price for a certain time and must revert to the higher price after the sale.

Assuming they stick to the legal requirements, are you getting an excellent bargain during the sales?

ID 7069

At present you have two heater elements supplied by two twin cables as shown in diagram A. The cables are old and damaged so you have decided to replace them with a single twin cable as shown in B. The system runs on low voltage DC so you feel safe to make the change. The heaters were permanently wired in parallel on the old system, so you have not made a change other than to the wiring.

The old cables used to run quite hot so you make sure to double the cross-sectional area of the conductor in the new cable.

Have you made a satisfactory change?

ID 7070

The manufacturer of a particular brand of paint has specified that the 2.5 L of paint in the tin will cover 32 m^{2}.

Neglecting wastage and spillage, how thick is the paint on the wall, assuming the manufacturer's figures are correct?

Note: 1 mL = 1 cm^{3}

ID 7082

The image shows the cross-section of a simplified electrical water heater. The insulation is so good that we shall consider it to be perfect.

Water flows through the heater at a constant rate. Electrical power is supplied at the constant rate of 1kW.

The system is left to reach equilibrium. How much heat is absorbed by the outgoing water?

ID 7083

The image shows the cross-section of a simplified electrical water heater. The insulation is so good that we shall consider it to be perfect.

The design is based on an existing water heater which works well. However, management have decreed that the old heater is too big. Without changing the internal pipe structure significantly, the length of the heated section has been reduced by a factor of 10.

The water flow rate and electrical input power (2kW) have been kept the same. The characteristics of the insulation and heater element are unchanged compared to the previous design.

What is the most likely outcome?

ID 7093

*Estimate* how much it costs to flush the toilet at home given the following:

(1) we only consider the incremental cost, not the fixed costs

(2) water supply cost is £1.5373 per cubic metre (as measured by the water meter)

(3) sewerage services cost £1.6594 per cubic metre (and are estimated as 90% of the supplied water quantity)

(4) a "standard flush" is 6 litres according to manufacturer's installation instructions.

Remember that 1 L = 1000 cm^{3}

The British pound (GBP, £) is just another decimal currency so £1.00 = 100p

ID 7099

You spin a wheel and it randomly lands on $1, $2, $3, or END. If you land on $1, $2, or $3 you get that money and spin the wheel again. You keep receiving money until you land on END.

What is the probability of receiving $2 or more when playing this game once?

ID 7101

This is a UK mains socket, nominally rated at 230 V AC, 13 A (3 kW)

When is it most likely to catch fire?

ID 7104

Consider an extremely rare event which might randomly happen with a probability, P, of 1 in 1 million million (10^{-12}). This might be the probability of your street getting hit by a meteorite during a particular hour, for example, but we don't pretend for an instant that the probability given is actually correct for such an event. Now consider that this event has 10 million chances to occur (N), so for our example we might say our observation period was 10 million hours.

How should you calculate the chance that the event will occur within the stated interval when using a handheld calculator?

ID 7105

Desmond von Dummkopf is showing off to his (so-called) friends again. He is going to ride his unicycle over the strong concrete structure shown.
(5 m/s is roughly 11 mph.)

Does he have any chance of making the jump?

(*We only consider the jump, not the ability to hold the unicycle upright during and after the landing --- if he should succeed in getting to the other side.*)

g = 10m/s^{2}

ID 7111

The sketch to the right suggests the outline of a wire-frame bird-cage. The wire is thin, with large gaps, so the bird can easily see in all directions. The cage is supported by a single cord (red). The size of the bird is very small compared to the size of the cage.

As an experiment we hang the red cord from an electronic force sensor to measure the weight of the bird plus the cage.

What happens to the measured weight when the bird flies around?

ID 7114

It is easy to show that making beams long in the direction of the load makes them disproportionally stiffer.

We have a simple wooden rod with a square cross-section. Let's suppose it has a 2cm x 2 cm cross-section and is 1m long for the sake of clarity. One piece of this wood is not stiff enough for our purposes, so, knowing the stiffness equation, we stack three pieces up vertically. The three rods together now have a cross-section of 2cm x 6cm with the rods stacked upwards (the direction of the load), not sideways.

How much stiffer is this new assembly compared to the original rod?

WARNING: Do not try bending any beams like this without using eye protection. Certain rulers are known to shatter and eject bits of plastic into nearby eyes.

ID 7121

PCR (Polymerase Chain Reaction) is a technique used to make copies of a particular section of DNA. It is therefore also referred to as DNA amplification.

A PCR machine heats the sample (along with various other ingredients) up to 96°C to split the DNA strands, cools it to around 60°C to allow binding to the separated DNA strands, then heats it up to 72°C to complete the copying process. Because DNA is a double helix, and each side is copied, each PCR cycle roughly doubles the amount of the DNA section present.

If we start with one DNA chain, how much would we expect after 30 PCR cycles?

ID 7123

The black washer is constrained to randomly fall flat onto the outer square, the outer edge of the washer being within the outer square. The diagram is exactly to scale.

What is the probability that we see only red through the hole in the washer?

ID 7131

A computer programmer has characterised his new brick-built house methodically over a period of several years. He knows exactly how much heat to apply to the inside of the house for each and every condition of outside temperature, wind, rain, and humidity, when these are constant for a reasonable period of time (equilibrium conditions).

What happens when he runs his new temperature stabilisation program (assuming the program works as intended, in other words bug-free)?

ID 7132

John lives in a fairly well-insulated fairly old brick-built house in a central European location. In the winter the weather gets quite cold at night compared to the day.

Peter lives in a much newer brick-built house in the same village as John. The locations of the two houses in terms of local climate and wind are similar.

John turns off the heating between the hours of midnight and 6am to save money on his heating bill. Peter claims that John is wasting money on his heating bill because John has to keep heating up the walls every day, rather than leaving them at a more even temperature. Peter has evidence that his scheme of leaving the heating on all night is better since his heating bill is 10% lower that John's; their houses are similar sizes, and kept at the same sort of temperature.

Which scheme saves the most on energy costs?

ID 7133

There is a new and dangerous strain of bacteria in town. You have a total of 10 anti-biotics and supplements available, but you have been told that they are all individually ineffective. You hope that some combination of them may be effective, and you also hope that when testing, the relative amounts of each constituent is not critical.

Fortunately you have a fully automated sample handling facility at your disposal, so you can make as many tests as you want, and the computerized system can tell if the result is a success without your intervention. It is the nature of such tests that the sample needs to be incubated for 24 hours to see if it has been successful.

How long will it take to fully test your idea?

ID 7134

Americium-241 is a radioactive isotope with a half-life of 432.2 years. It is widely used in domestic smoke alarms in the form of Americium-241 dioxide.

Estimate how much of the Americium has decayed at the end of the typical 10 year service life of a smoke alarm.

ID 7141

The picture shows two temperature/humidity meters of the same type, bought at the same time and from the same place. The specification in the manual that comes with the meters says that the temperature resolution is 0.1°C and the refresh rate is 10 seconds. The website selling the meters claims the accuracy is plusmn;1°C.

The meter on the left has been brought into the room and left to stabilise for about 2 minutes. (If it had been put on the right the reading on that meter would still have been low.)

Why are the readings so different? Which answer is the most guaranteed?

(*The picture has been edited to remove branding information, and that is all.*)

ID 7147

The sinking of the *unsinkable* RMS Titanic in 1912 (after it hit an iceberg on its maiden voyage) is a well known engineering disaster story. As with many such disaster stories it was not just one thing that caused the disaster, but a whole catalog of errors.

The sides of the ship were 1 inch thick mild-steel plates, overlapped and hot-riveted together. In this case the iron (or steel) rivets were heated until they were red hot (actually glowing red) then pushed into holes in the steel plates and hammered over. As the rivets cooled, they contracted, making really strong joints.

We suppose that one faulty riveted joint is just about acceptable, but that two right next to each other is bad. Suppose we have 4 rivets in a row, and the probability of a faulty rivet is 1 in 1,000. Estimate the probability of two faulty rivets right next to each other.

ID 7148

A manned space mission to Mars requires a reliable electrical power generation system. For this reason three of the same type of power generators are taken, each of which is sufficient to supply the requirements of the mission. The probability of failure of any one of the power systems during the flight is estimated to be 1 in 100,000.

Estimate the probability of a failed mission in terms of just the electrical power supply.

ID 7149

In the recycling of cars, the body of the car gets shredded by powerful machines and the resulting chunks of material first have the ferrous metal (primarily iron and steel) removed by means of permanent magnets. The next step takes the non-ferrous metals (aluminum, brass, etc) and throws these off the conveyor belt using the force from an alternating magnetic field (an *eddy current separator*). The non-metallic waste passes through the eddy current separator a second time to improve the separation.

If we suppose that 10% of the non-ferrous material remains after the first pass, how much remains after the second pass?

ID 7150

You have 4 electronic weighing scales. You test all 4 with the same 50.000kg reference weight at different times of the day, and on different days, to establish their accuracy.

The results are shown below.

You want to pick the best scale to use in your new coffee bean filling station because you know that you have to supply at least as much coffee as you put on the label, but any amount more than that is just wasted as far as your profit is concerned. The bags are marked as 50kg.

Which scale should you pick?

ID 7152

The image shows an electrical power meter set to read in units of kWh. Given that at the start of the test the last three digits were 9.45, and 30 minutes later the last three digits were 9.50, what was the average power consumption during that period?

(The image shows the last three digits as 9.48 as the picture was taken during the test.)

ID 7153

The manufacturer of a 5000 point linear sensor array for flatbed scanners has a new quality manager. The old quality manager ignored single isolated pixel faults within the array, but rejected an array where two faulty pixels were right next to each other. The new quality manager wishes to improve quality by rejecting dual-fault arrays if the two pixels are within 10 pixels of each other.

Pixel faults are entirely random, and occur with a probability P = 10^{-5} (1 in 100,000).

To be clear about this, the pixels are all in a single straight line. If pixels 1 and 11 are faulty, that is the limiting case of a faulty array.

Considering only two-fault arrays, how much more probable is a failed array under the new scheme?

If you don't know where to start with this problem, try an easier one first.

ID 7154

ID 7160

What is the secret message in the text below?

*Mary Evans eventually took Michael, Ewan, and Tom Maudsley into Downton, not into Glasgow’s heavy traffic.*

ID 7161

The cipher-text shown would be very difficult to decipher by hand if it were not for one crucial mistake. The sender has addressed the recipient by name, and we have guessed that the message is for Betty.

What is the message about?

ID 7165

Jason has just realised that playing *Total Death & Destruction XVIII* every night is not going to help him in adult life. Instead he has resolved to spend 20 minutes a night on A+ click, working through all the problems starting from those for 6 year olds, and working his way up. He has realised, somewhat belatedly, that by practicing problems he will get better at solving problems. For the kiddy problems, the game is to solve the problems as quickly as possible whilst still getting the correct answer.

Given that the A+ click problems are supposed to take one minute or less to solve, how long will it take him to solve the first 6000 problems.

ID 7166

Kevin lives in a remote location and he often experiences power outages during poor weather conditions. He has resolved to build a battery backed-up solar and wind powered electrical system to maintain his power in any conditions. In order to design such a system he has measured his power usage over a typical winter's day.

Which are the essential readings from

1) Average power

2) Mean power

3) Minimum power

4) Maximum power

5) Standard deviation of power

6) Variance of power

ID 7167

ID 7171

Analyse the graph shown.

What is the strongest statement you can be sure of from this data?

Image credit: Quora

ID 7175

You have been told that the weight of cabbages produced by the hydroponics facility has a Gaussian (Normal) distribution with a mean of 1000g and a standard deviation of 100g. Estimate the proportion of cabbages in the range 900g to 1100g.

ID 7176

In Timmy's school there are lots of girls of a similar age to him. He estimates that about half are natural blondes and half are brunettes. There don't seem to be any black haired or ginger haired girls for some reason. He has been told that there are roughly as many girls with blue eyes as there are with brown eyes, and other eye colours are also not present.

Estimate the probability that if he picks a girl from his school at random she will be a natural blonde with brown eyes.

ID 7177

We have been told that the local population of men have an average height of 66 inches, and that their height has a standard deviation of 2 inches.

Estimate what proportion of these men have a height in excess of 63 inches.

ID 7178

You are in charge of quality control in a factory producing tens of thousands of self-sealing stem bolts per day. You measure the length of every bolt and produce a statistics report for the management every month. You have curve-fitted the data to a Gaussian (Normal) distribution and extracted the mean and variance data.

Is it correct to calculate the out-of-tolerance rate using statistical tables based on the measured mean and variance data?

ID 7179

"Mathematics all looks like Greek to me", you say. And much of it is, so you are right!

On the left of the equation we have what looks like the letter U with a tail at the start. It is the lower case Greek letter *mu*. To the right of the equals sign we have a huge symbol that is a bit like a capital E, but different. It is the capital letter sigma, also Greek. What we have here is *sigma notation* which you should think of as summation.

It is inconvenient to write a + b + c + d + ... for hundreds of values. Instead we might call all the values x and use a *subscript* to identify the different values. This is very convenient for experimental results as reading 1 has subscript 1, reading 2 has subscript 2, and so on.

Remember that "sub-" means below like submarine, sub-optimum, sub-standard; the *index* is just below the main symbol.

What is mu as given by this equation?

ID 7180

Since we don't actually speak Greek, putting the Greek letters in the correct order can be problematic.

The intention was to write: alpha, beta, gamma, delta, epsilon, from left to right.

Did we succeed?

ID 7181

ID 7182

Multiplication of 2 x 2 matrices is boring and only for little kids ... except perhaps when they contain imaginary numbers.

i is the square root of minus one.

What is the value of X?

ID 7183

In the field of complex numbers we have a number which consists of a real part and an imaginary part.

The imaginary part is some multiple of i, where i is the square root of minus one.

Given Z = 4 + 3i

and W = -3 - 2i

What is the real part of the product of Z and W?

ID 7184

Which material in this list is the most strongly anisotropic?

School problems often deal with uniform isotropic materials: iso- as a prefix has the sense of sameness. Isotropic materials have properties which are the same, independent of direction. The an- (or sometimes just a-) negates the following items such as anaerobic, anastigmatic.

ID 7190

Neglecting a scaling factor, the image shows a general quadratic equation with two constants, b and c.

Quadratic equations always have two solutions (roots) but they can be the same, and they can involve complex numbers.

What is the exact condition that the two roots are the same?

ID 7194

ID 7195

Given that the cosine of an angle is 3/5, find the sine of that angle without using a calculator or trig tables.

*Hint: use Pythagoras.*

ID 7198

Consider the following statement:

"A hovering helicopter has no vertical or horizontal motion. Given that mechanical power requirement is force x velocity the mechanical power required to do so is zero so the helicopter engine is just idling when hovering."

ID 7199

A certain premium digital meter has a stated DC volts accuracy specification of plusmn;[0.15% of reading + 2 counts].

On the 6 V range it will display 6.000 V, in other words the last digit is changing in steps of 1 mV.

You have correctly set the zero reading and you are measuring a voltage of 50 mV.

What is the specified accuracy of your reading, neglecting any other considerations?

ID 7201

An inexpensive hand-held laser range finder projects a red laser beam onto a wall a few meters away, and returns an accurate distance reading with 1 mm resolution.

Assuming this is a simple time-of-flight measurement of a laser pulse, what is the necessary resolution of the time measurement?

The speed of light is 3 10^{8} m/s (300,000 km/s).

ID 7214

The householder has realised that in due course the rug-rats will spontaneously evolve into large adults, and therefore the household hot water requirements will double. In order to double the hot water storage capacity the householder can either put in an extra identical hot water storage tank, or just put in a single double-capacity tank.

Neglecting everything other than the heat-loss through the insulation, which solution gives a lower cost?

We assume that the hot water storage tank is just scaled up for the double-capacity tank, although the insulation thickness remains the same.

ID 7216

Jennifer wishes to transform her rooms from the existing Dungeon Grey to a bright pastel color. She knows from experience that this large a color change will require 3 coats. Her chosen paint is an expensive type with double the reflected light compared to ordinary paints. It also costs twice as much per tin.

Assuming the paint is applied in the same thickness per layer, what is the cost reduction (as a ratio) by using two layers of a cheaper ordinary reflectivity paint before putting on a top coat of the more expensive high reflectivity paint?

ID 7230

When painting a room with a paint brush, in which direction should a right-handed painter proceed?

*Hint: consider also painting at height and older painters.*

ID 7238

Painting with a roller is faster for large areas, but requires more clean-up of the roller and tray.

Let' suppose a roller covers twice the area in a given time compared to a paint brush, but that cleaning the brush takes 5 minutes whereas cleaning the roller and tray takes 20 minutes.

Beyond what length of painting job does the roller become quicker?

ID 7239

With uncharacteristically mathematical insight, John's dad has realised that there is a cost associated with being an unpaid taxi service for John.

John's dad has costed this as $3 for fuel plus an arbitrary $3 "annoyance cost", since John's trips seem to increasingly occur during live sports events on TV.

If a new bicycle costs $200, after how many of these trips will John's dad feel that he is profiting by just giving John a new bicycle, allowing John to make his own way?

ID 7253

The satelite picture shows Great Britain, also known as Britain, that is a large island in the north Atlantic Ocean off the northwest coast of continental Europe.

What can you say with certainty about the length of its coastline?

ID 7263

The photo shows a dial gauge caliper. (*It has been edited to remove the manufacturer's name.*)

What is the reading?

ID 7266

You are unlikely to have seen these things before, but this is the debris from a domestic repair job. (*The 1 foot ruler is just to show the size of the parts.*)

But can you guess what was being repaired?

ID 7267

ID 7268

A satellite in geostationary orbit is essentially weightless. It needs to move about its axis and to nearby positions at the same orbital radius.

Which is the best (correct) statement?

ID 7269

Cathy is all alone in the spaceship, 1km from the mother-ship. The (hideous) fate of the other crew members has been withheld. Cathy's spaceship is totally out of fuel and is dead in space relative to the mother-ship.

Cathy has one chance. She has created a loose bundle of equipment and waste which she estimates to be 1000kg of mass. If she can eject this mass from the airlock with enough speed she should reach the mother-ship before her air runs out. She estimates the current mass of her spaceship as 100,000kg.

Cathy cannot throw the 1000kg mass, but by pushing it and running on the ridged floor in the air-lock, she is confident she can eject the mass at 5m/s.

What is her estimate of the time to reach the mother-ship?

ID 7271

John, the physics student, wishes to dry his wet rectangular towel on a radiator. He wonders how much has to overhang at the back to make the towel stable.

He is reluctant to put too much of the towel behind the radiator as this is the unknown territory of stale pizza, dead bugs, cobwebs, and other indescribable debris.

He models the situation as shown, where the radiator thickness has been exaggerated. He supposes that the towel only rests against the top of the radiator, shown in red. He further supposes that the length of this red section is negligible compared to the length, L, of the towel. He guesses that the coefficient of static friction for this situation is 0.50

What is the length of the long part of the towel in the limiting case?

ID 7274

By the time you have completed the calculations, and set the thrust magnitude, the 1000 kg lunar lander will be heading directly for the moon's surface at a speed of 50 m/s and just 250 m from the surface.

How much thrust is needed to give a perfect landing (neglecting the weight change due to the fuel expended)?

(*The gravitational acceleration on the surface of the moon is 1.6 m/s ^{2}*).

ID 7283

Your boss has told you to buy a more accurate (and therefore more expensive) thermocouple probe to more accurately measure the room temperature using a thermocouple meter (digital thermometer).

Comment on this requirement.

ID 7289

The weather forecast shows an expected temperature of 5°C, but with the expected 10°C wind-chill, the weather map shows -5°C.

Assuming the weather forecast is spot on, will shallow puddles of water freeze?

[*Remember that water freezes at 0°C.*]

ID 7291

The iPhone has a smooth underside resting on the smooth level surface. It has a mass of 130g. The coefficient of static friction between the phone and the surface is 0.2 exactly. I horizontally pull the phone by the attached white cable with a force of 0.2 N.

What is the magnitude of the frictional force?

1kg = 1000g

ID 7292

The 0 - 60 mph (≈100 km/h) test is frequently used to compare cars. Suppose you took a car and pushed it off a cliff.

Neglecting wind resistance, how long would it take to complete this 0 to 60 mph test?

Take gravity as 10 N/kg and 60 mph as 30 m/s (it is actually 26.82 m/s, an inconvenient value for hand calculations).

ID 7293

In a typical lab setup for measuring frictional force, a block (blue) is pulled by a thin string or line (green) by a weight (yellow). The pulley wheel (red) is typically considered to be free to rotate without friction in the bearing.

What is the effect of some small friction in the pulley wheel bearing?

ID 7305

ID 7306

Elapsed time can get quite messy for computers to deal with.

How would you write 1 hour 23 minutes and 30 seconds as decimal minutes?

ID 7310

This is a single sequence where there is a mathematical rule to go from one position to the next. There are no alphabets involved, this is a purely numerical problem.

1, 8, 4, 0, 7, 3, 10, ...

How many more steps before the 10 is repeated?

ID 7311

A top-fuel dragster, burning nitromethane rather than petrol (gasoline), is able to reach a top speed of 336 mph (150 m/s) in around 5 seconds.

If we suppose that it accelerates at a constant value, what is that constant value?

ID 7312

Your school work has finally paid off and you get to design your own electric dragster. The electric motor is so massively powerful that the torque output to the road wheels has to be limited to a certain fixed value to prevent the wheels from spinning.

What is your prediction of the real-world performance of this new design?

ID 7313

Here we show the side view of a kitchen worktop with a grain of uncooked rice (not to scale) ready to be brushed into the blue bin below.

What is the optimal speed with which to brush the rice?

(*Take the gravitational force as 10 N/kg*)

ID 7314

In elementary Calculus we are often given y as a function of x and have to evaluate dy/dx.

In real life we do not necessarily have y and x.

Suppose we have V = ktimes;p

where k is a constant.

What is dV/dt?

ID 7316

A particle has a velocity which increases in direct proportion to its distance from its origin (the zero reference point for distance).

Describe its acceleration characteristic.

ID 7321

Now that John has to handle his own laundry, he has invented a new scheme. The top pullover on the shelf is the next one he wears. After washing, the cleaned and dried pullovers are placed back on top of the pile.

What is the best name for such a system?

ID 7322

ID 7338

A man wishes to throw a ball such that it stays in the air for as long as possible.

The speed at which he throws the ball is some fixed value. The ground around him is featureless and level as far as the eye can see.

What is his optimum strategy?

(*Neglect wind resistance and spin.*)

ID 7339

It is a sad fact of history that as technology evolves it can get harder for a student to get started with a new subject.

There is a lot of new terminology to learn which did not exist earlier on.

Books from 1940 (and earlier) talk about cycles per second. This is a fairly self-evident unit.

But what is the modern equivalent?

ID 7340

Nerdy McGeekface throws a baseball as hard as he can, but unfortunately hits Beefcake O'Reilly square in the back.

Neglecting any subsequent activity, who feels the most impact?

ID 7341

A man who lives half way up a mountain wishes to exercise his dog by throwing a bean-bag for the dog to fetch. For the sake of simplicity, consider the mountain as conical and relatively featureless in his immediate vicinity.

Which is the optimum direction to throw the bean-bag to maximise the exercise given to the dog?

ID 7342

Share prices are notorious for going up and down, but on a long average they consistently go up in value over time. From year to year you see rises and falls. Suppose the annual changes were of

+10%, +15%, -5%, +2%, +7%, -3%, +10%.

Which is the best sequence to maximise your profit?

ID 7351

30 discs, all of the same size, are placed into an opaque bag. 10 are made of aluminium, 5 are brass, 5 are wooden, 6 are made of ceramic, and 4 are magnets. The bag is shaken to mix all the discs up.

A blindfolded person has to remove a disc from the bag using only a magnet on a string.

(*If more than one disc is attached, a second person selects which disc is the closest to a particular mark on the magnet on the string, and only this disc is removed.*)

What is the chance that the disc removed is a magnet?

ID 7355

The grey block is extremely heavy and rigid, with a perfectly horizontal frictionless top surface. The blue inelastic string over the frictionless pulley transfers force without loss from the small weight to the big weight. The big weight has 3x the mass of the small weight.

Initially, the masses are held in place by some unspecified device, with the string taut.

Describe the motion of the small mass when the masses are released.

*Take the gravitational constant as 10N/kg.*

ID 7361

A toy car is released from stationary on a smooth planar slope which is angled relative to the horizontal, the sine of the angle being 0.2 (an angle of about 11.5°).

Taking the acceleration due to gravity as being 10m/s^{2}, and assuming frictionless wheel bearings and negligible air resistance, describe the motion of the car (which was pointing directly downhill when released).

ID 7362

A toy car is released from stationary on a long smooth planar slope which is angled down relative to the horizontal.

Describe the motion of the car (which was pointing directly downhill when released), assuming the wheel bearings have friction and that wind resistance is not negligible.

ID 7364

Not so long ago, but before petrol engines, and before small steam engines, cargo was transported down canals in barges, pulled by horses or even people.

Why would it have been advantageous to use a long rope?

The picture shows "Barge Haulers on the Volga", an 1873 oil-on-canvas painting by the Russian artist Ilya Repin.

ID 7373

Sheila opens up her piggy bank and counts the coins.

She lives in a strange and far away country, and the coins have values of 1p, 2p, and 5p.

There are 145 coins in total.

There are 5x as many 1p coins as 2p coins.

There are 4x as many 1p coins as 5p coins.

How many 2p coins are there?

ID 7374

John randomly chooses a piece of paper from a collection of green, yellow, blue, violet, and rose sheets. There is an equal probability of choosing any particular color.

At exactly the same time, Jane randomly chooses a pen from a collection of green, yellow, blue, violet, and rose pens. There is an equal probability of choosing any particular color.

Neither John nor Jane know what the other has selected.

Jennifer takes the pen and paper and writes a note.

What is the probability that the writing will be visible, given that writing using a pen of the same color as the paper will not be visible?

ID 7375

Jane randomly chooses a pen from a collection of white, yellow, and cyan pens. There is an equal probability of choosing any particular color.

John randomly chooses a piece of paper from a collection of white, yellow, and cyan sheets. The probability of choosing paper of a particular color is proportional to the number of sheets available. There are equal amounts of yellow and cyan, but twice as much white as yellow.

Jennifer takes the pen and paper and writes a note.

What is the probability that the writing will be visible, given that writing using a pen of the same color as the paper will not be visible?

ID 7376

Patricia randomly chooses a pen from a large collection of white, yellow, and blue pens. The numbers of white and yellow pens are the same. There are twice as many blue pens as white pens.

Peter randomly chooses a piece of paper from a large collection of white, yellow, and blue sheets. There are equal amounts of yellow paper and blue paper, but twice as much white as yellow.

Wendy takes the pen and paper and writes a note.

What is the probability that the writing will not be visible? (This occurs when the pen and paper are the same color.)

ID 7403