The king who rewrites the salaries

Question

A city has 66 citizens including the king, and each has a salary of 1 dollar. Salaries are whole numbers of dollars and always add up to 66. The king cannot vote, but he may propose any new salary list as often as he likes. Each citizen votes yes if their own salary would rise, no if it would fall, and does not vote if it stays the same. A proposal passes if there are more yes votes than no votes. The king is greedy and patient. What is the largest salary he can eventually obtain for himself?

Accepted Answer

The king can reach 63 dollars, and no more. A schedule that reaches 63. Track (paid citizens, king). 1. Raise 33 citizens to 2 dollars (33 yes); cut the other 32 citizens and the king to 0 (32 no). Passes 33 to 32. Citizens hold 66, king 0. 2. Of the 33: raise 17 to 3 (17 yes), cut 16 to 0 (16 no). Citizens hold 51, king 15. 3. Of the 17: raise 9 to 4, cut 8 to 0. Citizens 36, king 30. 4. Of the 9: raise 5 to 5, cut 4 to 0. Citizens 25, king 41. 5. Of the 5: raise 3 to 6, cut 2 to 0. Citizens 18, king 48. 6. Of the 3: raise 2 to 7, cut 1 to 0. Citizens 14, king 52. 7. Of the 2 at 7: raise one…

The king who rewrites the salaries

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