## The Mansion

Can you find your way through this mansion, by using the information from the security cameras? In order to escape, you must use the correct key and unlock the exit door.

**SOLUTION**

The solution is shown below.

Johann Sturcz is a Hungarian prolific artist, who creates a large variety of work, ranging from traditional paintings through modern art to children mazes.

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Can you find your way through this mansion, by using the information from the security cameras? In order to escape, you must use the correct key and unlock the exit door.

The solution is shown below.

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A hundred prisoners are locked up in a prison. The warden devises the following game: he writes 100 **different** numbers on the foreheads of the prisoners. Then, each of the prisoners inspects the numbers on the foreheads of the others and decides to put either a black or a white hat on his head. Once the prisoners put their hats on, the warden arranges them in a line according to the numbers on their foreheads, starting with the lowest one and ascending to the highest one.

If the hats in the resulting line alternate their colors, then the prisoners will be set free. If not, the prisoners will be executed.

Can the prisoners devise a strategy that will guarantee their freedom?

Once the warden writes the numbers on the prisoners’ foreheads, they form a mental circle, arranging themselves alphabetically in it (or according to any other order they agree on). They include the warden in this mental circle and imagine he has infinity written on his forehead.

Then, each prisoner examines the sequence of 100 numbers written on the foreheads of the others, and computes the number of inversions, i.e. the pairs which are not in their natural order. The prisoners which count an even number of permutations put black hats on, and the prisoners that count an odd number of permutations put white hats on. The infinity symbol is treated as the largest number in the sequence.

For example, if a prisoner sees the sequence **{2, -6, 15.5, ∞, -100, 10}**, then he counts seven inversions, which are the pairs **(2, -6)**, **(2, -100)**, **(-6, -100)**, **(15.5, -100)**, **(15.5, 10)**, **(∞, -100)**, **(∞, 10)**, and puts a white hat on.

Next, we prove that the devised strategy works. We consider two prisoners, **P1** and **P2**, who are adjacent in the final line the warden forms. These two prisoners split the mental circle in two arcs: **A** and **B**.

The number of inversions **P1** counts is:

I_1 = I(A)+I(B)+I(A,B)+I(A, P_1) + I(P_1, B),

where I(X) denotes the number of inversions in a sequence X, and I(X, Y) denotes the number of inversions in a pair of sequences (X, Y):

I(X) = \|x_i, x_j \in X : \quad i < j, \quad x_i > x_j\| \\ I(X, Y) = \|x \in X, y \in Y : \quad x > y\|

Similarly, the number of inversions **P2** counts is:

I_2 = I(A) + I(B) + I(B, A) + I(B, P_2) + I(P_2, A)

We sum I_1 and I_2 to get:

\begin{align*} I_1 + I_2 &= 2I(A) + 2I(B) + I(A, B) + I(B, A) \\ &+ I(A, P_1) + I(P_2, A) + I(P_1, B) + I(B, P_2) \\ &= 2(I(A) + I(B)) + \|A\|\|B\|+\|A\|+\|B\| \end{align*}

Since \|A\| + \|B\| = 99 is an odd number, we see that I_1+I_2 is also odd. Therefore, one among **P1** and **P2** would have counted an even number of inversions, and one would have counted an odd number of inversions. Thus, their hats have alternating colors.

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Two friends, logicians – Ein and Stein – get imprisoned in two distant cells in a castle. Both cells have just one door, and a window with 8 bars in the first cell, and 12 bars in the second cell. The first day both logicians get the same letter from the prison master:

“The total number of bars in the two prison cells in this castle is either 18 or 20. Starting tomorrow, every morning I will go first to Ein and then to Stein, and will ask how many bars the other logician has. If one of you answers correctly, I will immediately let both of you leave the castle. If one of you answers incorrectly, I will execute both of you. Of course, you can always decide not to answer and just stay imprisoned.

I have sent a copy of this letter to you and your friend. There is no point in trying to communicate with him – your cells are far away from each other, and he won’t hear you.”

Will the logicians manage to escape the castle eventually? When will they do it?

Solution coming soon.

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You are lost in the middle of a forest, and you know there is a straight road exactly 1 km away from you, but not in which direction. Can you find a path of distance less than 640 m which will guarantee you to find the road?

Imagine there is a circle with a radius of 100 m around you, and you are at its center **O**. Let the tangent to the circle directly ahead of you be **t**. Then, follow the path:

- Turn left 30 degrees and keep walking until you reach the tangent
**t**at point**A**for a total of**100×2√3/3**meters, which is less than**115.5**meters. - Turn left 120 degrees and keep walking along the tangent to the circle until you reach the circle at point
**B**for a total of**100×√3/3**which is less than**58**meters. - Keep walking around the circle along an arc of 210 degrees until you reach point
**C**for a total of**100×7π/6**which is less than**366.5**meters. - Keep walking straight for
**100**meters until you reach point**D**on the tangent**t**.

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What should follow the sequence of diagrams shown below?

The diagrams represent consecutive monarchs of England:

**King****E**dward the**8**-th →**Ke8****King****G**eorge**6**-th →**Kg6****Queen****E**lizabeth**2**-nd →**Qe2**

Therefore, the next diagram should represent **King** **C**harles the **3**-rd → **Kc3**.

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Divide the circle below in two pieces. Then, put the pieces together to get a circle with a dragon, such that the dragon’s eye is at the center of the new circle.

The solution is shown below.

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Can you spot the five James Bond films?

The films are:

- “Diamonds are Forever” (Diamonds + R + Four + Heifer)
- “Moonraker” (Moon + Rake + R)
- “Octopussy” (Octopus + E)
- “Golden Eye” (Gold + Hen + Eye)
- “The World Is Not Enough” (The World + E’s + Knot + N + Oeuf)

Source:

This rebus is taken from the book “A Collection of Spots”. Inside the book you will find 48 more puzzles.

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Since 2018, **Catriona Shearer**, a UK teacher, has been posting on her Twitter various colorful geometry puzzles. In this mini-course, we cover some of her best problems and provide elegant solutions to them. Use the pagination below to navigate the puzzles, which are ordered by difficulty.

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