November 2005 Puzzle Solutions

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Solutions:

1. Bob bets 50 on heads and Carol bets 50 on tails. If Alice bets 49 or less, then she can't possibly win because even if she guesses the correct coin orientation, her 100 will equal the 100 of one of Bob or Carol. If Alice bets 50 or more on heads, then she wins on heads and Carol wins on tails. So, Bob and Carol can guarantee themselves a 1/2 chance to win.

2. If Alice bets 48 or less, then Bob and Carol should bet as in the answer to Problem 1. One of them will surely win. But if Alice bets at least 50, Bob and Carol can do no better than before. So, the result stays the same.

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3. Bob will win with probability 3/4. In the penultimate round, he bets nothing. No matter what Alice bets, she cannot exceed 100 units. If Alice loses in the penultimate round, then since she must state her bet first in the last round, Bob just bets as she does in that round and is guaranteed to win. If Alice wins in the penultimate round, then, in the last round, Bob simply bets all his units on the coin orientation opposite to the one Alice chooses. So Bob will win 3/4 of the time.

4. Suppose Alice bets 2 on heads in the penultimate round. If Bob bets nothing or bets on tails and the penultimate flip lands on heads, then Alice wins for sure by using the strategy from the warm-up. If Bob bets all 51 on heads and the result is tails, then again Alice is sure to win on the last flip. If Bob bets anything on heads and the result is heads, then Bob has at most 102 and Alice has 52. In that case, on the last flip, Alice bets everything on the opposite orientation to Bob's and she will win 1/2 the time regardless of what Bob bets. So, Alice wins at least 1/2 the time and she will win more often if Bob plays foolishly.

5. Yes, Alice can do better than 1/2 odds. Alice should bet nothing in the penultimate round. If Bob loses, Alice wins for sure in the final round using the strategy of the second warm-up. If Bob wins, Alice bets everything on the coin face opposite to the one Bob chooses in the final round. Overall, she will win with probability 3/4.

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