“Escalators and a Hat”

  At an airport, there are 2 adjacent horizontal escalators (moving walkways), parallel to each other, they are moving in opposite directions, each at a speed of 1 ft/sec.

  2 boys (A and B) are riding on one of the escalators together. When they get to the middle of their escalator, they throw a hat over into the adjacent point of the other escalator (which is also its midpoint), and at the same time they start running in opposite directions (A is running forward, B – backwards). When each boy reaches the end of their escalator, he instantly gets on the other escalator and continues to run.

  Each boy is running at a speed of 6 ft/sec (relative to the escalator).

  Who will get to the hat first?

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“Doubled Chess”

  The Doubled Chess game begins with the same starting position as regular chess, the only difference is that each player makes 2 moves at a time (first, White makes 2 moves, then Black makes 2 moves, and so on). Prove that Black does not have a winning strategy.

  (It means that if you are playing Black, and if White is a super powerful computer which is perfect at Doubled Chess, then you will never win).

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“Rectangular Cake with a Hole”

  3 pirates want to cut a rectangular cake into 3 pieces with a knife. First pirate was in a hurry, so he cut a random rectangular piece out of the cake and left with it. The removed rectangle may have any position and orientation (but the entire removed rectangle lies within the cake). How can the pirates divide the remainder of the cake into 2 equal pieces using a single straight cut?

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“Shooting at a Jumping Cat”

  This is a 2-players game.

  There are 10 equally spaced chairs positioned on a straight line. An invisible Cheshire Cat sits on one of them. Alice is trying to find the Cat using a paintball gun. There is no limit on how many times Alice can shoot. Each time, she shoots at one of the 10 chairs. If the Cat is on that chair, he gets marked with paint and therefore becomes visible, so Alice wins. If the Cat is on another chair, he gets startled and jumps to either one of his 2 adjacent chairs (or, if he is on a corner chair -- to his only adjacent chair). Is there a winning strategy for Alice, or could it so happen that Cat may evade her forever? (Note, Cat never moves except right after Alice shoots).

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“Multiplying Bacteria on an Infinite Chess Board”

  This is a solitaire game.

  Consider an infinite chess board occupying 1 quadrant of a plane (bounded by half-axels x and y). The corner unit square has coordinates (0,0), to its right is the unit square (1,0). The unit square above (0,0) is (0,1), and so on.

  In the beginning, there is only 1 bacterium sitting at unit square (0,0).

  Once a minute, exactly 1 bacterium on the board divides producing 2 offspring, 1 of which moves to the adjacent unit square to the right, the other one – to the adjacent unit square upwards, while all other bacteria on the board remain dormant. So, after 1 minute, there will be 2 bacteria on the board in unit squares (1,0) and (0,1), while (0,0) will become unoccupied.

  A unit square cannot contain more than 1 bacterium, therefore a bacterium is allowed to divide only if both of its adjacent squares (the one on top of it, and the one to its right) are unoccupied.

  Consider a square area A consisting of 9 unit squares, the corners of which are unit squares (0,0), (0,2), (2,2), and (2,0).

  Is it possible to completely clear area A? (Meaning, to play in such a way that at some point all 9 squares in A will become unoccupied?

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“Bounded Set on a Plane”

  Do there exist a bounded set A on the plane, and its strict subset B, such that A and B are congruent?

  A “strict subset” of X is its any subset Y, except the set X itself (so X must not equal Y).

  Set X is “bounded” if there exist a circle which contains the entire set X.

  2 sets X and Y are “congruent”, if they have the same shape and size, meaning there exists a way to flip, and/or slide, and/or turn set X thus transforming it into a set Z, so that  Z=Y.

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“Ants on a Stick”

  There are 20 tiny ants positioned at random points on a 1 meter long stick. From their starting positions, all ants simultaneously start crawling at a constant speed of 1 meter per minute (some are headed from right to left, others – left to right). When 2 ants meet, they both instantly turn around and continue crawling in the opposite direction (at the same speed, 1 m/min). If an ant comes to the end of the stick, he falls off. For all possible starting positions (and directions) of 20 ants, what’s their longest possible survival time (the amount of time from the start until the last ant falls off)?   

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