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Alice and Charlie play a game, taking it in turns with Alice going first.
On a blackboard is written a three-digit number. If the current number on the blackboard is n, a move consists of choosing a non-zero digit, k, of n and replacing n with n − k on the blackboard. This is repeated until the number 100 is written on the blackboard.
The player who writes the number 100 wins.
(a) If the starting number is 125, Alice can always win. State Alice’s first move and how Alice responds to whatever move Charlie makes at each stage.
(b) Find, with proof, for which starting values Charlie has a winning strategy.
From UKMT - Maclaurin Mathematical Olympiad - 2025 - 2
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