Puzzle 1. We are given a biased coin that comes up heads with probability p. Using this coin, construct an unbiased coin.

Solution. The solution is the “von Neumann trick”: flip the biased coin twice, interpreting HT as heads and TH as tails. If the flips come up HH or TT, then try again. Under this scheme, “heads” and “tails” are equiprobable, each occurring with probability p(1 - p) in any given trial. Conditioned on either HT or TH occurring, it follows that the simulated coin is unbiased.

Puzzle 2. n people sit in a circle. Each person wears either a red hat or a blue hat, chosen independently and uniformly at random. Each person can see the hats of all the other people, but not his/her own hat. Based ...

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