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This is the most efficient way to place three congruent squares in an equilateral triangle. What proportion of the triangle is shaded?

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You pick three random points on a circle. What is the probability that the resulting triangle contains the center of the circle?

The points are picked uniformly at random, along the circumference of the circle.

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Can you measure exactly 15 minutes using nothing but an 11-minute hourglass and a 7-minute hourglass?

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What proportion of a square is closer to its centre than its edge?

This problem is surprisingly difficult and requires calculus.

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What 4-digit number, when multiplied by 4, reverses the order of its digits?

*ABCD* × 4 = *DCBA*

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Can you place 18 black and 18 white tiles on a 6×6 board, so that there are no “squares” with their four corners having the same colour?

From Martin Gardner’s *“Sphere Packing, Lewis Carroll, and Reversi”*

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Three geysers A, B and C in a national park erupt every 1, 2 and 3 hours respectively. You just arrived: what is the probability that you will see geyser A erupt first?

Inspired by The Riddler on FiveThirtyEight

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What’s the angle between these two congruent equilateral triangles?

Inspired by Catriona Shearer

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Scientists are studying a micro-organism, starting with a single cell. Every day, each cell either splits in two (with probability *p*), or it dies. What is the probability that the entire organism dies eventually?

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Can you arrange the seven Tetrominoes in a 7×4 rectangle, with no gaps or overlaps?

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You have 10 cans of peas. All peas weigh 1 gram, except for one can with peas that weigh 0.9 grams. How often do you need to use a scale to find this lighter can?

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Can the locomotive L switch the position of the two wagons and end up where it started? Only the locomotive can fit under the bridge.

From Martin Gardner’s *“Sphere Packing, Lewis Carroll, and Reversi”*

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25 frogs are sitting in a 5×5 grid. Every frog jumps into an adjacent square (left, right, up or down). What is the largest number of squares that could become empty?

From the Netherlands Junior Maths Olympiad

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I repeatedly toss a fair coin and record the outcome. What is the probability that the sequence “HHH” occurs before “THH”?

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A castle is surrounded by a 5 meter wide, rectangular moat. Can you cross it using nothing except two planks that are 4.8 meters long?

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Can you plant 7 trees so that there are 6 straight lines containing 3 trees each?

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How many ways are there to distribute 10 identical cookies between five different kids?

Kids don’t need to receive the same number of, or any, cookies.

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What is the least number of integers needed, so that any of these could be true?

Median < Mean < Mode

Median < Mode < Mean

Mode < Median < Mean

Mode < Mean < Median

Mean < Mode < Median

Mean < Median < Mode

The mode has to be well-defined, so you can’t have two different integers both appear the most number of times. For example, the set {1, 2, 2, 3, 3} doesn’t have a well-defined mode, because both 2 and 3 appear twice.

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Rearrange these numbers and symbols to make a true equation:

2 3 4 5 + =

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There are 100 strings in a bag. You randomly pick two ends and tie them together, until there are no free ends left. What is the expected number of loops you will create?

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A semicircle lies inside a square. What proportion of the square is shaded?

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How many ways are there to tile a rectangle of size 2×10 with dominoes?

Dominoes are tiles of size 2×1 and can be placed horizontally or vertically. All dominoes need to be contained within the board, and there can’t be any gaps. Can you find a general answer for a board of size 2×*n*? What about a board of size 3×*n*?

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A circle of radius 1 rolls around the inside of another circle of radius 3. What is the length of the path traced out by a point on the small circle?

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I’m thinking about a large integer.

- It is divisible by 1.
- It is divisible by 2.
- It is divisible by 3.
- …
- It is divisible by 30.

Exactly two consecutive of these statements are wrong. Which ones?

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You have 9 balls, one of which is slightly heavier than the others.

How often do you need to weigh two groups of balls, to find the odd one out?

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Two equilateral triangles are drawn inside a square. What is the area of the smaller triangle?

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A cinema announces a special deal: the first person in the queue to have the same birthday as someone in front of them, will get a free ticket.

Which position in the queue is the best?

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At a party, every guest shook hands with everyone else. There were 66 Handshakes in total. How many guests attended the party?

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Four cities form the vertices of a square. What is the shortest way to connect them with each other using railroad tracks?

The tracks may intersect, and you can add “junctions”. Hint: Two diagonals is not the shortest path!

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This is a *Magic Sum Square*, where the sum of every row, column and diagonal is 15. Can you find a *Magic Product Square*?

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Can you cut this *obtuse* triangle into smaller, *acute* triangles? If so, how many cuts do you need?

Note that a right angle is neither acute nor obtuse!

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A cylindrical hole of length 6cm has been drilled through the center of a solid sphere. What is the volume of the remaining sphere?

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When placing 5 queens on a 5µ5 chess board, what is the maxiumum number of fields you can leave “unattacked” (no queen can reach them within one turn)?

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Can you insert mathematical operators, to make this equation true?

0 0 0 0 0 = 120

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I’ll offer you $4 to play this game:

You have to toss a coin repeatedly, until it lands heads. Then you have to pay me back $1 for every toss.

Do you want to play?

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You have two ropes that burn in exactly 60 minutes – but not neccessarily at a constant rate.

How can you measure 45 minutes?

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A farmer has 300 bananas which he wants to sell at a market 100km away.

His camel can carry 100 bananas at once, and eats one banana per km.

What is the most bananas he can take to the market?

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How many guards do you need for this museum, so that every corner can be watched?

Guards have 360° vision, but they cannot move.

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Here you can see some examples of *Trapezium Numbers*. There is just one number between 1,000 and 2,000 that doesn’t form a trapezium. Which one?

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Three ants are sitting at the corners of a triangle. Each ant picks one direction at random and starts walking. What is the probability that none of the ants collide?

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You break a stick in two different places at random. What is the probability that the resulting three pieces form a triangle?

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You have a large number of 5-cent stamps and 17-cent stamps. What is the largest cent value which you cannot make using a combination of these stamps?

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Which regular polygons can be created using a ring of other regular polygons?

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Can you make **24** using the numbers

3, 3, 8, 8,

and the operations

+ – × ÷ ( )

How many people do you need, so that the probability of two having the same birthday is at least 50%

A bag contains two green marbles and two blue marbles.

I pick two marbles at random and tell you that at least one is blue. What is the probability that the other one is also blue?

A small country contains 10 cities and 5 straight roads. Every road connects 4 different cities. Draw a map of the country!

How many guests do I have to invite to my christmas party, to be sure there will be at least 3 mutual friends, or 3 mutual strangers?

Any two guests are either strangers or friends.

In a dark room there’s a drawer with 10 red socks and 10 blue socks. How many socks do you have to take, to be sure to get a matching pair?

A market stall sells five different kinds of fruit.

I want to buy ten items. How many possible combinations are there?

How can I measure exactly 8 liters of water, using just one 11 liter and one 6 liter bucket?

People from the Town of Truth always tell the truth. People from the City of Lies always lie.

A guide from one of the cities is at the intersection and offers you a single question. What should you ask?

Place the numbers from 1 to 9 in the circles, so that the sum along all 3 sides is the same.

How many triangles are there?

You have to deliver five letters to five different houses, but the rain has erased all addresses. If you just distribute the letters randomly, what is the probability that *everyone gets a wrong letter*?

How many diagonals are there in a 10-gon?

What’s the smallest set of integers a, b, c, d and e that satisfy

a + b = c + d + e AND a^{2} + b^{2} = c^{2} + d^{2} + e^{2}

All shapes have the same *perimeter*. Which one has the largest area?

Is the yellow dot on the *inside* or the *ouside* of this spiral?

Can you split this shape into two equal parts, with a single cut?

What’s next?

Where did the missing square go?

Find all pairs of numbers *a* and *b* that satisfy:

*a* + *b* = *a* × *b* = *a* / *b*.

Continue this sequence:

4, 6, 12, 18, 30, 42, 60, 72, 102, 108, …

What’s the area of the Koch Snowflake, where the largest triangle has side length 1?

Can you cover a 8×8 chessboard, with the two opposite corner tiles removed, entirely with dominoes (no gaps or overlaps)?

Rearrange these seven shapes to form the animals above!