## Medium puzzles

A chess tournament has K levels and 2^K players with skills 1 > 2 > ... >2^K. At each level, random pairs are formed and one person from each pair proceeds to next level. When two opponents play, the one with better skills always wins. What is the probability that players 1 and 2 will meet in the final level?

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A coin is tossed 10 times and the output written as a string. What is the expected number of HH? Note that in HHH, number of HH = 2. (eg: expected number of HH in 2 tosses is 0.25, 3 tosses is 0.5)

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Source: Placement Test
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The stick drops and breaks at a random point distributed uniformly across the length. What is the expected length of the smaller part?

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Source: Quant Interview
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100 prisoners are lined up and assigned a random hat These hats are available in 10 different unique and distinguishable colors. Each prisoner can see the hats in front of him but not behind. Starting with the prisoner in the back of the line and moving forward, they must each, in turn, say only one word which must be the color like "red" or "blue" or "green" etc. If the word matches their hat color they are released, if not, they are killed on the spot. They can hear each others answers, no matter how far they are on the line. What strategy should be used to help release maximum number of prisoners?

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N people team up and decide on a strategy for playing this game. Then they walk into a room. On entry to the room, each person is given a hat on which one of the first N natural numbers is written. There may be duplicate hat numbers. For example, for N=3, the 3 team members may get hats labeled 2, 0, 2. Each person can see the numbers written on the others' hats, but does not know the number written on his own hat. Every person then simultaneously guesses the number of his own hat. What strategy can the team follow to make sure that at least one person on the team guesses his hat number correctly?

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