ID 922

A fence around a rectangular garden has a perimeter of 14 meters.

Its length is 4 meters.

Find the length of the diagonal of this rectangular garden.

ID 924

The volume of water in a glass of V cubic centimeters (cc), varies directly as its height, H centimeters (cm).

The volume is 100 cc when the height is 5 cm.

Calculate the volume when the height is 11 cm.

Note: Older text books use cubic centimeters. This unit is the same thing as cm^{3} and milli-liters.

1 cc = 1 cm^{3} = 1 mL

ID 925

Tim travels 15 meters north of the flag in the football stadium.

He turns west and travels 8 meters.

He turns south and travels 6 meters and then comes back directly to the flag.

Calculate Tim's travel distance.

ID 927

ID 930

A rectangle has width W, length L, and area A.

If integer W ≤ 9 and L is an even number between 9 and 99 which of the following CANNOT be the area of the rectangle?

ID 931

A wire is wound around a circular rod.

The wire goes exactly 5 times around the rod.

The circumference of the rod is 12 cm and its length is 25 cm.

Identify the length of the wire.

ID 932

ID 934

ID 936

ID 938

A boy stacked colored cubes in a square pyramid like the one shown here.

The top layer had 1 cube, the second layer had 4 cubes, and so on.

If the pyramid were 15 layers high, how many cubes would be in the fifteenth layer?

ID 940

A six-pointed regular star consists of two areas.

What is the ratio of the shaded region in the area of the star?

ID 942

The figure shows a red equilateral triangle inscribed within another equilateral triangle. The side of the bigger triangle measures 10 meters.

We want to obtain the smallest area of the red triangle. What would be the distance x in this case?

ID 947

If the volume of a cube is X cubic meters and the total surface area of the cube is X square meters, then what is the cube's edge length?

ID 948

ID 949

The three circles have fixed centers, and the diameter of a circle is 10% less than that of its 'left neighbor'.

The left circle completes a hundred revolutions per minute.

How many revolutions does the right circle complete?

ID 956

L3 and L4 are two parallel lines in a plane.

If L3 has 3 points equally spaced along its length, and if L4 has 4 points also equally spaced along its length, how many different triangles can be formed by connecting the points on the two parallel lines?

A triangle must be formed by 2 points on one line and 1 point on the other.

ID 957

ID 958

For this rectangular-faced solid (cuboid), which plane(s) contain(s) B and is/are parallel to plane AEH?

ID 960

Two similar pyramids have volumes of 216 m^{3} and 64 m^{3}.

What is the ratio of their surface areas?

ID 961

ID 964

There are 2 identical circles. Circle A remains fixed, while Circle B makes 1 turn around the first one, touching it without slipping.

How many turns has Circle B made around its own axis?

ID 968

G is the area of the green region inside the biggest circle.

R is the total area of red regions of the two smaller circles.

Which statement is correct?

ID 969

The area of the external triangle is equal to 1.

Its sides' midpoints are connected to form a second triangle, and so forth.

What is the sum of the areas of all the triangles in this infinite series?

ID 971

The pattern of shading in one quarter of a square is shown in the diagram.

If this pattern is continued indeﬁnitely, what fraction of the large square will eventually be shaded?

ID 974

What is the maximum number of squares you can make using twelve identical matches?

(The matches must not cross each other.)

ID 976

A gardener has to reach the island in the middle of a pond without getting wet.

The gardener has two planks each X meters long.

What is the smallest length of the planks?

ID 979

ID 980

ID 1004

Twenty matchsticks form five squares (one 3x3 and four 1x1).

How many matchsticks do I need to move to make seven squares?

Find the minimum number.

ID 1041

A ladder leans against a vertical wall. The top of the ladder is 4m above the ground. When the bottom of the ladder is moved 1m closer to the wall, the top of the ladder rests 1m higher than the original position.

How far from the wall was the bottom of the ladder in the initial position?

ID 1043

If the length of the hour and minute hand of a clock are 3cm and 6cm respectively, what is the angle shown on the picture at two o'clock?

ID 1050

All inner lines connect the corners of the big square and the midpoints of the opposite sides.

What fraction of the big square is red?

ID 1054

A square is inscribed in a right triangle.

Find the greatest possible ratio of the area of the square to the area of the triangle.

ID 1058

ID 1076

ID 1083

The sizes of the sealed bottle with water are shown in the figure.

Find the height of the water when the bottle is right side up.

ID 1109

The following pattern is cut and folded to a square-based pyramid.

What size does the base have to be to maximize the surface area of the resulting pyramid?

ID 1190

John is using the spinner shown here to define the movement.

Blue means one step up and green means one step down.

The spinner is moved randomly.

If he starts at point 0 and makes 360 moves, where will he most likely be now?

ID 1192

ID 1264

ID 1280

ID 1340

What is the maximum number of possible points of intersection of N different circles?

The picture shows four circles.

ID 1348

ID 1399

Two lines trisect (divide into three equal parts) each side of the polygon ABCD.

Which polygon has the largest area?

ID 1451

The shaded rhombus is formed by joining vertices of the square to the midpoints of its sides.

What is the area of the shaded rhombus?

ID 1479

The diagram shows 15 billiard balls that fit exactly inside a triangular rack.

The rigid rack prevents the balls from sliding.

What is the largest number of balls that can be removed so that the remaining balls are theoretically unable to move?

ID 1518

ID 1587

Two triangles form seven separate regions.

What is the greatest number of such regions that can be formed by three triangles?

ID 1613

There are six ways to travel from point S to point F on a small cube if only right, forward, and up moves are permitted.

Find the number of different pathways available for a 2x2x2 cube.

ID 1614

ID 1876

A, B, and C are squares with sides of length 1;

D, E, and F are isosceles right triangles;

and G is an equilateral triangle.

The net can be folded to form a shape.

What is the volume of the shape?

ID 1878

I drew three lines from the center of a square that has sides with a length of 1 to form two congruent trapezoids and a pentagon.

All three shapes have the same area.

What is the length of the pentagon's shortest side?

ID 1949

What is the maximum number of pieces that an apple can be divided into with four straight planar cuts?

The pieces do not move.

ID 1974

Each of these five circles is tangent to at least 3 others.

The medium sized circles have a radius 3.

What is the radius of the smallest green circles?

ID 2031

ID 2153

In a city, there were seven bridges.

There was a tradition that a newly married couple walks and crosses over each of the seven bridges only once.

If a couple starts and finishes at the same point, which city plan allows the couple to acomplish this task?

ID 2193

A sphere fits inside a cube. The maximum possible ratio of the volume of the sphere to that of the cube is pi / 6, or about 0.52.

If we put many small spheres inside a cube, then what is the largest possible ratio of the spheres' volume to that of the cube?

ID 2198

I want to cut a wooden cube that is five inches on each side into 125 one-inch cubes.

I can do this by making 4 + 4 + 4 = 12 cuts, keeping the pieces together in the cube shape.

What is the minimum number of cuts needed if rearrangement of the pieces after each cut is allowed?

ID 2238

ID 3509

I need 1 + 4 + 9 + 16 + 25 = 55 cubes to build a pyramid with a height of 5 cubes.

Estimate the number of cubes for a pyramid with a height of 30 cubes.

ID 3552

I want to place N cubes so that each cube touches every other one.

What is the largest possible N?

Inspired by Martin Gardner

ID 3564

ID 3599

The Babylonians used a base 60 number system.

What shape inspired them to decide that a circle has 360 degrees?

ID 3608

Connect N points on the circumference of a circle.

What is the largest number of intersections for the chords?

ID 3674

I take a map of the city where I live and lay it on my table.

There is a "You are here" point on the map, which represents the same point in the city.

The point on the map coincides with its real position.

True or False?

ID 3678

ID 3696

I connected the midpoints of a polygon and constructed a new polygon that was a quadrilateral with opposite sides parallel.

What shape was the initial polygon?

ID 3712

A pyramid and a tetrahedron with edges of the same length are glued together on a triangular face.

How many faces does the resulting solid have?

ID 3718

I need 1 + 9 + 25 = 35 cubes to build a pyramid with a height of 3 cubes.

Estimate the number of cubes for a pyramid with a height of 30 cubes.

ID 3741

Inspired by Boris Kordemsky.

Four Knights problem:

Cut the chessboard into 4 congruent parts, each with a queen on it.

How many sides does each part have?

ID 3806

ID 3909

How many 1x1 squares fit into the large square with the side length 4.8?

Find the maximum possible number.

ID 3916

ID 3917

The picture shows three squares with the side lengths of 10, 8, and 4 cm.

What is the difference between the areas of the green and blue regions?

ID 3920

A trapezoid is formed by cutting off the top part of an isosceles right triangle such that the short base is 8 m.

What is the area of the blue trapezoid?

ID 3924

ID 3941

ID 4023

ID 4064

A circle goes through two adjacent vertices of a square and it is tangent to the bottom side of the square.

Find the diameter of the circle X.

ID 4129

ID 4219

A helicopter takes me from Lausanne, Switzerland to the Swiss capital Bern in 20 minutes. Bern is 36 minutes from Brig.

Which of the following could be the time of a flight from Lausanne to Brig?

ID 4324

What is the probability of breaking a stick into three pieces and forming a triangle?

The pieces must intersect at their tips to form the triangle.

ID 4342

What is the maximum number of trees that can be planted, not closer than 3 meters apart, in a square plot of 10.5 meters x 10.5 meters?

ID 4373

A rectangular yard with an area of 24 m^{2} has sides in the ratio 2:3.

What is the length of the fence around the yard?

ID 4382

ID 4386

ID 4388

ID 4447

The size of an NBA basketball court is about 29 by 16 meters.

How many courts can be planned in an 80 x 67 meter school yard, given that at least 3 meters must separate the courts?

ID 4458

ID 4483

The golden border of the hexagonal brooch with maximum size 60 mm includes a gemstone, whose maximum size is 40 mm.

Which area is the largest?

ID 4542

What is the maximum number of sections into which a pancake may be divided by four straight cuts through it?

(NOTE: The pieces cannot be re-arranged between cuts.)

ID 4560

If I use 30 g of batter to make a crêpe of 30 cm in diameter, how much batter do I need for a square pancake of the same thickness and with a side length of 30 cm?

"Archaeological evidence suggests that pancakes are probably the earliest and most widespread cereal food eaten in prehistoric societies." - Wikipedia

ID 4657

ID 4759

ID 4929

The picture shows a large cube 4 x 4 x 4 that was assembled from one-unit cubes.

It will be painted on all 6 sides.

For what size of large cube will the total number of painted faces of the small cubes be equal to the number of unpainted faces?

ID 5103

What is the largest number of straight lines you can draw through 9 points, so that each line goes exactly through 3 points?

You can move points on the plane as you wish.

ID 5108

John cuts a large piece of cheese into small pieces using straight cuts from a very sharp cheese wire.

He does not move the pieces from the original shape while he cuts the cheese.

How many small pieces of cheese can he get using only five cuts?

ID 5112

Gerry has 24 meters of fence and wants to make a rectangular garden with the largest area.

He uses the house as the fourth side of the garden.

What length should he make the long side of the garden?

ID 5116

ID 5257

Take two sheets of A4 paper (210 x 297 millimeters or 8.3 x 11.7 inch).

Roll one into a short cylinder and the other into a tall cylinder.

Which one holds more air than the other?

ID 5260

Two right-angled triangles have integer side lengths.

Whilst all of the sides have different lengths, the hypotenuses are equal.

What is the smallest length of the hypotenuse?

ID 5265

Estimate the maximum number of smaller 1-inch circles that fit in a larger circle, the diameter of which is three times larger.

ID 5307

ID 5337

My garden fence creates a ten meters by five meters rectangle.

If I reuse all of this fencing to make a new rectangular garden, what is the maximum possible percentage increase of the area?

ID 5374

ID 5389

ID 5405

Three semicircles are constructed on the hypotenuse and legs of a right angle triangle.

Compare green (G) and red (R) areas.

ID 5526

There are 3 circles of equal diameter.

A line is tangent to the third circle, as shown.

Find the length of the line segment AB.

ID 5582

The Bermuda Triangle, also known as the Devil's Triangle, is a loosely defined region, where a number of aircraft and ships are said to have disappeared under mysterious circumstances. The triangle's three vertices are in Miami, Florida peninsula; in San Juan, Puerto Rico; and in the mid-Atlantic island of Bermuda.

The distance from Miami to Bermuda is 1668 km.

The distance from Miami to Puerto Rico is 1663 km.

The distance from Puerto Rico to Bermuda is 1571 km.

Estimate the surface area of the famous triangle.

ID 5638

A farmer has 36-meter of fence to enclose a field. The fence is given as 6-meter straight sections.

He wants to make his field as big as possible.

Estimate the maximum area of his field.

ID 5696

ID 5785

All the circles have the same center. The area of each colored region between the circles is equal to the area of the smaller circle.

We extend the model to 100 circles.

How much larger is the largest circle compared to the small circle?

ID 5880

The picture shows two regular stars with heights of length 1 and 3, which have the same vertical axis of symmetry.

What fraction of the design is blue?

ID 5984

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)?

Author: Leslie Green

ID 5991

The ball is covered with hexagons and pentagons.

The sum of interior angles of a hexagon on a plane is 720°.

What is the sum of the interior angle of the hexagon on the ball's surface?

ID 6007

The ball's design stitches together 20 hexagons with 12 pentagons for a total of 32 panels. The ball made its World Cup debut as Adidas' Telstar in 1970 in Mexico. The ball's pattern of white hexagons with black pentagons made it easily visible on television.

FInd the sum of the internal angles of the panels.

ID 6011

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, **t** to the initial diameter of the drop?

Author: Leslie Green

ID 6080

ID 6105

I cut a net from a square sheet of paper to form a cylinder A.

I cut two nets from the identical sheet to forms 2 small cylinders B.

Compare the volumes of the two sets.

ID 6134

ID 6145

ID 6214

The ground clearance (C) is measured between the flat ground and the lowest point in the vehicle's undercarriage.

The wheel base (B) is measured between the centers of the two wheels.

The Breakover Angle (A) is an angle that a vehicle can drive over without the ground touching the vehicle's undercarriage.

What is the formula for the Breakover Angle?

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.*

Author: Leslie Green

ID 6306

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.*

Author: Leslie Green

ID 6346

Leslie Green asks:

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?

Author: Leslie Green

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?

Author: Leslie Green

ID 6414

Leslie Green asks:

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 6509

A piece of wood is a square with a right-angled isosceles triangle on top.

A carpenter cuts it to form a square tabletop with no pieces left over.

What is the minimum number of saw cuts?

Source: Alex van den Branhof, Jan Guichelaar, Arnout Jaspers Half a Century of Pythagoras Magazine. MAA 2011

ID 6532

ID 6648

How many different convex pentagons with the vertexes in 5 of these 8 points can be formed?

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. For example, a regular hexagon is a convex polygon, while a star is not convex.

ID 6653

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.*

Author: Leslie Green

ID 6668

The spiral of Theodorus (also called Pythagorean spiral) is a spiral composed of contiguous right triangles.

Which triangle has an area of 10?

ID 6675

ID 6687

A regular hexagon with a side length 1 can be decomposed into six regular triangles with the same side length.

Which is the only other regular polygon with unit side lengths which can be decomposed into smaller regular triangles and squares with sides of length 1?

ID 6915

ID 6946

ID 6977

Cut a square paper into acute triangles.

What is the smallest possible number of the triangles?

An acute triangle has all angles smaller than a right angle (90°).

ID 6989

I draw different polygons, and then I construct squares on the outside of each polygon, using the whole of each polygon side.

For example, the picture shows a pentagon with the squares.

I count the sum of the marked angles between the squares.

For what polygon is the sum of the gap angles largest?

ID 7195

*Leslie Green* asks:

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 7273

A cube has a green face, two yellow faces, and three red faces.

How many different such cubes can I make?

Two cubes are different if one cube cannot be rotated to look like the other.

ID 7356

Three lines define seven separate regions.

What is the maximum number of regions divided by six lines in the plane?

ID 7366

A diagonal divides a large square into two triangles.

How much larger is the area of the yellow square compared to the area of the green square?