Baby game
Source: 2022 Taiwan Mathematics Olympiad
February 19, 2022
combinatoricsTaiwan
Problem Statement
Two babies A and B are playing a game with bottles of milk. Each bottle has a maximum capacity of ml, and initially each bottle holds ml of milk.
Starting from A, they take turns and do one of the following:
(1) Pick a bottle with at least ml of milk, and drink half of it.
(2) Pick two bottles with less than ml of milk, pour the milk of one bottle into the other one, and toss away the empty bottle.
Whoever cannot do any operations loses the game. Who has a winning strategy?Proposed by Chu-Lan Kao and usjl