*This year’s Advent Puzzle Calendar will start on 1 December 2019. New puzzles will be added at 9am GMT every day until Christmas. The solution will be revealed on the following day.*

What is the area of the highlighted area, created by four quarter-circles?

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

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Can you place 18 black and 18 whitetiles on a 6x6 board, so that thereare no “squares” with their fourcorners having the same colour?

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

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

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

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

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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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25 frogs are sitting in a 5x5 grid. Every frog jumps into an adjacent square (left, right, up or down). What is the largest number of squares that could become empty?

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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

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Rearrange these numbers and symbols to make a true equation:

2 3 4 5 + =

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What is the average distance between two points picked at random in a square of side length 1?

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How many ways are there to tile a rectangle of size 2×10with 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 theinside of another circle of radius 3.What is the length of the path tracedout 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 thesestatements are wrong. Which ones?

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

How often do you need to weightwo groups of balls, to find theodd 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 specialdeal: the first person in thequeue to have the same birthdayas someone in front of them,will get a free ticket.

Which position in thequeue is the best?

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At a party, every guestshook hands with everyoneelse. There were 66 Handshakesin total. How many guestsattended the party?

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Four cities form the vertices of asquare. What is the shortest way toconnect them with each otherusing 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* triangleinto 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 hasbeen drilled through the center ofa solid sphere. What is the volumeof the remaining sphere?

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When placing 5 queens on a 5µ5chess board, what is the maxiumumnumber of fields you can leave“unattacked” (no queen can reachthem 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 haveto pay me back $1 for every toss.

Do you want to play?

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You have two ropes that burn inexactly 60 minutes – but notneccessarily at a constant rate.

How can you measure 45 minutes?

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

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

What is the most bananas he cantake to the market?

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How many guards do you need forthis museum, so that every cornercan 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 onenumber between 1,000 and 2,000that doesn’t form a trapezium.Which one?

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Three ants are sitting at the cornersof a triangle. Each ant picks onedirection at random and startswalking. What is the probabilitythat none of the ants collide?

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You break a stick in two differentplaces at random. What is theprobability that the resultingthree 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 youcannot make using a combinationof these stamps?

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Which regular polygons can be created using a ring of otherregular 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 greenmarbles 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 10cities and 5 straight roads. Everyroad connects 4 different cities.Draw a map of the courtry!

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 twoequal 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 thearea 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!