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Princess in a Palace

A princess is living in a palace which has 17 bedrooms, arranged in a line. There is a door between every two neighboring bedrooms and also a hallway which connects them all. Every night the princess moves through the inner doors from one bedroom to another. Every morning for 30 consecutive days you are allowed to go to the hallway and knock on one of the 17 doors. If the princess is inside, you will marry her. What would your strategy be?

Invisible King

The white king has made himself invisible. Where is he?

35 Moves

Design a game that takes less than 35 moves to get to the position below.

Royal Wedding

A prince decides to get married to the prettiest girl in his kingdom. All 100 available ladies go to the palace and show themselves to the prince one by one. He can either decide to marry the girl in front of him or ask her to leave forever and call the next one in line. Can you find a strategy which will give the prince a chance of 25% to get married to the prettiest girl? Can you find the best strategy?

Remark: Assume that the prince can objectively compare every two girls he has seen.

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Two Monks on a Mountain

Two monks are standing on the two sides of a 2-dimensional mountain, at altitude 0. The mountain can have any number of ups and downs, but never drops under altitude 0. Prove that the monks can climb or descend the mountain at the same time on both sides, always staying at the same altitude, until they meet at the same point.

Spiders and a Fly

There are a fly and three spiders on the edges of a cube. If the spiders’ velocities are at least 1/3 of the fly’s velocity, and all insects can travel only along the edges of the cube, show that the spiders can eventually catch the fly.

Integer Dimensions

A large rectangle is partitioned into smaller rectangles, each of which has integer length or integer width. Prove that the large rectangle also has integer length or integer width.

Fair Split

It is well known how to split fairly a cake between two people – one of them cuts, the other one picks. The question is, how can you split fairly a cake between three people?

Easy: “Fairly” means that every person gets at least 1/3 of the cake.

Hard: “Fairly” means that every person has the opportunity to get at least as much cake as any other.