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Pile of stones and two persons

Source: Tournament of towns, Senior A-Level paper, Fall 2004

December 25, 2004
combinatorics unsolvedcombinatorics

Problem Statement

Two persons are playing the following game. They have a pile of stones and take turns removing stones from it, with the first player taking the first turn. At each turn, the first player removes either 1 or 10 stones from the pile, and the second player removes either m or n stones. The player who can not make his move loses. It is known that for any number of stones in the pile, the first player can always win (regardless of the second player's moves). What are the possible values of m and n?