Puzzle Calendars

What’s the ratio of the areas of the triangle and the crescent-shaped “lune” formed by two circular arcs?

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A deck of 52 cards is shuffled. On average, how many cards do you expect to remain in exactly the same position as before the shuffle?

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What is the area of the shape enclosed by these six hat mono-tiles?

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How many squares does the diagonal of a 15 × 27 rectangle pass through?

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Here is a 3 × 4 grid of stamps. How many different ways are there to tear off four stamps (still joined together along their sides)?

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Can you cut this right-angled triangle with smaller sides 1 and 2 into five identical triangles, each similar to the big triangle?

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How many numbers between 1 and 9999 consist of at most two different digits?

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How many of these 15 coins arranged in a triangle do you have to move, so that the triangle points downwards?

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One cut divides a cake into 2 pieces.

Two cuts can divide it into 4 pieces.

Three cuts can divide it into 7 pieces.

What is the maximum number of pieces you can make with 10 cuts?

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Five people *A*, *B*, *C*, *D* and *E* are seated randomly around a round table. What is the probability that *A* and *B* are next to each other?

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How many squares can you make using four of these points as vertices?

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Rearrange these 8 prime numbers and 1 into a magic square where every row and column adds up to the same number.

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A semicircle is placed inside a square of size 1. What is the diameter *d* of the semicircle?

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You and your friend both throw a fair die. What is the probability that you get a higher number?

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How many different necklaces can be made using five identical blue beads, and two identical yellow beads?

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Use a single, straight-line to cut this shape into two pieces that can be rearranged into a square.

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What is the sum of all the digits of the numbers from 1 to 100?

1 + 2 + 3 + … + (9 + 9) + (1 + 0 + 0)

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All three circles have radius 1. What is the radius of the larger semicircle?

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What is the least number of moves needed to transfer all seven disks from the first tower to the last, without placing a larger disk on a smaller one?

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What six digits should be placed on the second die, so that the distribution of the sum of both dice is the same as two normal dice?

The first die contains faces numbered 1, 2, 2, 3, 3, 4.

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Place the digits from 1 to 7 into each of these regions, so that every circle has the same sum.

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What is the area of the semicircle, which is placed symmetrically inside a quarter circle?

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Alice said Bob did it.

Bob said Alice did it.

Carol said Alice didn’t do it.

Dan said it was either Alice or Carol.

Only one person is telling the truth. Who did it?

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Can you express 7/8 as a sum of distinct “unit fractions” of the form 1/*x*?

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How many different ways are there to connect four cubes?

*Every cube needs to touch at least one other cube, and faces need to line up. You can ignore rotations or reflections.*

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Can you use eight 8s, together with mathematical symbols like + and –, to make 1000?

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What is the ratio of the perimeters of the circumscribed and inscribed hexagon of a circle?

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Two athletes are running at constant speeds on a circular track. If they head in opposite directions, they meet after one minute. If they head in the same direction, they meet after one hour. What is the ratio of their speeds?

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You have two jugs with volumes 3 and 5 liters, but no markings. Can you use them to get exactly 4 liters of water?

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What’s the next number in this sequence?

441, 961, 691, 522, 652, 982, 423, …

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A shop sells chocolates in boxes of 6, 9 or 20. What is the largest amount of chocolates that is impossible to buy?

*For example, it is impossible to buy 10 chocolates using boxes of 6, 9 or 20.*

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A circle is bounded by two quarter circles in a square. What is its area?

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If 5 balls are placed at random into 5 buckets, what is the probability that exactly one bucket remains empty?

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How many times per day do the minute and hour hands on a clock form a straight line?

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What is the area of the intersection of these two rectangles of size 1 × 2?

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What is the sum of the first six cube numbers? What about the first 10 cube numbers?

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Can you find a 10-digit number, so that:

- the first digit is the number of zeros in that number,
- second digit is the number of 1s,
- …
- tenth digit is the number of 9s?

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Three squares are placed next to each other. What is the sum of the three angles *a*, *b* and *c*?

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You randomly draw two cards from a standard deck of 52 cards.

What is the probability that both cards have either the same suit or the same number/symbol?

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What is the fewest number of weights you need, to be able to balance any weight from 1 to 40 on a 2-sided scale?

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What is the size of the 10th square in this sequence? What is the total area after 10 iterations?

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How many paths through this grid are there from A to B, if you can only move right or down?

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What is the radius of the smallest circle that contains three circles of radius 1?

*What about 4, 5 or more circles?*

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A rook and a bishop are randomly placed on a chessboard. What is the probability that one is attacking the other?

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After 1 and 36, what is the next number that is both a square number and a triangular number?

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Given a random string of *n* left and *n* right brackets, how often do you need to cut it into two parts and swap both sides, to make it “algebraically valid”?

*Puzzle created by James Tanton*

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There are 26 red and 26 black cards in a deck. A “run” is a set of consecutive cards of the same colour. What is the expected number of runs in a standard shuffled deck?

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Seven *different* digits are placed in a row. The products of the first three, middle three and last three digits are all equal. What is the middle digit?

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How many integers between 1 and 1,000,000 have a sum of digits equal to 16?

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Given a line and two points A and B, which point P on the line forms the largest angle APB?

*Puzzle created by James Tanton*

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Frogs can slide horizontally or vertically by one cell, or jump over one frog horizontally or vertically into an empty space. Can the dark and light frogs switch places?

*Puzzle created by James Tanton*

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Can you find a number that starts with 7 and can be divided by 7 by moving the first 7 to the end?

*What about a general approach for any digit?*

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Five people, A, B, C, D and E, are randomly seated around a circular table. What is the probability that A and B are next to each other?

*Can you generalise this to n people?*

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How do you construct a circle inside another circle, that takes up exactly 1/3 of its area?

*Puzzle created by Kiran Bacche*

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Only one of these statements is true. Which one?

A. All of the below

B. None of the below

C. One of the above

D. All of the above

E. None of the above

F. None of the above

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A classic shunting puzzle: can you swap the position of the train cars *A* and *B*? The short segment at the top can fit either car, but not the locomotive *L*.

*The locomotive can drive forwards or backwards, and push or pull the cars from either side. You can also connect the two cars to each other.*

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Two squares of size 20cm are randomly placed inside a larger square of size 1m (edges parallel). What is the probability that the two smaller squares overlap?

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Points A, B and C are each 1/3 along their side of the triangle. What fraction of the triangle is shaded?

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A far-away country uses a currency system with just two coins: worth 13 cents and –9 cents respectively. Is it possible to purchase an item that costs 1,568 cents?

*Puzzle created by James Tanton*

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Can you prove that every integer has a multiple that is a Fibonacci number?

*How about a (non-zero) multiple that can be written using just 0s and 1s, in base 10?*

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What proportion of this square, which contains four quarter-circles, is shaded?

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How many cards, on average, do you have to draw from a standard shuffled deck (52 cards) until you get the first Ace?

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Into how many pieces can you cut a circular pizza with 10 straight cuts from edge to edge?

*Pieces don't have to be wedges, or even have the same size.*

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A group of 20 friends play “Secret Santa”, and draw the name of their target randomly out of a hat.

What is the probability that at least one person draws their own name?

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A pandemic is spreading across an *n* × *n* grid. Each tile gets “infected” if at least two of its four neighbours are infected. What is the minimum number of infected squares required initially, so that the pandemic could spread to eventually cover the the entire grid?

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Six circles with radius 1 are arranged in a regular hexagon. What is the area of the dark, enclosed space?

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I write down the digits from 1 to 9 in a random order. What is the probability that the resulting number is divisible by 11?

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How many squares can you draw with their vertices on a

*How about a more general grid?*

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How often, on average, do you have to roll a die, to see all six sides come up at least once?

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An isosceles triangle is placed inside a square. We draw the incircle of the isosceles triangle, and one of the other triangles on either side. What is the ratio of the radii of these two circles?

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The numbers *a*, *b*, *c*, *d* and *e* are positive integers so that

What is the maximum possible value of any of these integers?

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Can you place one star in every row, column and region? Stars can’t be adjacent, even diagonally.

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This shape consists of three congruent, right-angled, isosceles triangles. Can you divide it into *four* congruent regions?

*“Congruent” means that the four regions need to have the same size and shape.*

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Knights always tell the truth and knaves always lie.

You are approached by two of them, and one says “we are both knaves”. Who are they actually?

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Continue this sequence:

23, 21, 24, 19, 26, 15, 28, 11, 30, 7, ?, ?

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What is the radius of the largest circle that can be drawn on a chess board, so that its circumference lies entirely on black squares?

*Each square has side length 2.*

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How far apart are the centres of these two circles with radius 1, if all three shaded regions have the same area?

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Can you divide this chessboard into four congruent “kingdoms” that each contain one of the kings? (The kingdoms must have exactly the same size and shape.)

*The kings are placed in row 5 and columns 4, 5, 6 and 7, and cannot be moved. The kingdoms also have to be orthogonally connected (they can't consist of multiple disjoint regions).*

*This puzzle has been featured in The Guardian*

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A box contains 100 balls labelled from 1 to 100.

You select 10 balls at random. What is the expected value of the largest number you picked?

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Place the numbers from 1 to 12 into these circles, arranged in a six-sided star, so that the sum along all six lines is the same.

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A white cube is painted black on its outside and then cut into 27 small pieces. The pieces are mixed and randomly reassembled into another cube.

What proportion of the surface area of the new cube is expected to still be black?

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A 3-4-5 triangle lies inside a square. What is the area of the square?

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Place the digits from 1 to 8 in these boxes, so that consecutive digits are not adjacent (even diagonally).

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How often, on average, do you have to flip a (fair) coin, until you get 10 consecutive heads?

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Two trains with 2 and 3 carriages are travelling in opposite directions on the same track. There is a siding, but it only has space for one carriage or locomotive. How can the trains pass each other?

*Locomotives can drive forwards and backwards, and attach to carriages on either side.*

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In this triangle, the length of all three sides and its height are four consecutive integers. What is the area of the triangle?

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Two 3-digit numbers sum to a third number with 3 digits.

The digits of all three numbers are permutations of each other. What are these numbers?

*None of the digits are 0s.*

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What is the average distance between two points picked randomly on the circumference of a circle?

*Note: This problem requires calculus.*

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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*

*This puzzle has been featured in The Guardian*

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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?

*This puzzle has been featured in The Guardian*

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?

*This puzzle has been featured in The Guardian*

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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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How many triangles are there?

*This puzzle has been featured in The Guardian*

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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?

*This puzzle has been featured in The Guardian*

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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)?

*This puzzle has been featured in The Guardian*

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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?

*This puzzle has been featured in The Guardian*

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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!