Estonian Mathematical Olympiad

Question

Juku and Miku are playing the following game. In the beginning, there is a positive integer on the board. Each turn, a player subtracts from the number on the board a non-zero digit that appears in his or his opponent's ID code, and replaces the number on the board with the result. Players take turns, Juku starts. The player whose move ends up in a negative number on the board loses. Prove that, among any 10 consecutive positive integers, there is a number n such that, if initially the number n is on the board, then Juku can win the game regardless of his opponent's counterplay.

Accepted Answer

Let a, a+1, , a+9 be 10 arbitrary consecutive positive integers. If there exists a number n among a, a+1, , a+8 for which Juku has a winning strategy, then we are done. We will now assume that if any of the numbers a, a+1, , a+8 is on the board, then the active player loses if his opponent plays perfectly. Then, if n = a+9, Juku can start by subtracting any non-zero digit, after which the number on the board is one of a, a+1, , a+8, putting Miku in a losing position, which means that Juku will win. There must be a non-zero digit in either code, as both codes cannot be strings of zeroes, as they must be distinct and have the same length.