A little kingdom contains 66 people, a king and 65 citizens. Each of them, including the king, has a salary of one gold piece. When democracy comes, the king is denied a vote, but he has the power to suggest changes, in particular regarding the redistribution of salaries. The salaries must total 66, and each salary must be a whole number of gold pieces. The citizens will vote on each suggestion, which will pass if more citizens vote for it than against it. Each voter will reliably support a measure if it will increase his salary, oppose it if it will decrease his salary, and otherwise abstain from voting.
The king is greedy. What’s the highest salary he can arrange for himself?
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Impressively, he can get 63 gold pieces. He starts by proposing that just over half of the 65 citizens (33 voters) have their salaries doubled to 2 gold pieces, at the expense of the other 33 voters, including himself. Then he does the same thing again, proposing that just over half (17) of the 33 salaried voters receive an increase to 3 or 4 gold pieces, while the remaining 16 in that group are reduced to zero. By continuing in this manner he can reduce the number of salaried voters to 9, 5, 3, and finally 2, with each receiving 33 gold pieces. Then the king can propose that three of the unsalaried citizens receive a salary of 1 gold piece if these two large salaries are now reassigned to himself.
From Peter Winkler’s excellent Mathematical Puzzles, 2021.
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