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UKMT Junior Mathematical Challenge, 2019

Combinatorics Difficulty 3.0 AMC 10/12 Prove it United Kingdom

Two players XX and YY take alternate turns in a game, starting with the diagram alongside. At each turn, one player writes one of 11, 22 or 33 in an empty circle, so that no two circles connected by an edge contain the same number. A player loses when they cannot go. In each of the five diagrams below it is YY's turn. In which of the diagrams can YY's move ensure that XX loses the game?

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