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Combinatorial game on changing numbers

Source: St. Petersburg MO 2000, 9th grade, P5

April 22, 2023
combinatoricsInvariantsgames

Problem Statement

The numbers 1,2,…,20001,2,\dots,2000 are written on the board. Two players are playing a game with alternating moves. A move consists of erasing two number a,ba,b and writing aba^b. After some time only one number is left. The first player wins, if the numbers last digit is 22, 77 or 88. If not, the second player wins. Who has a winning strategy?
[I]Proposed by V. Frank
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