Competition problem — Combinatorics

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Combinatorics Difficulty 5.7 AIME, harder Prove it Baltic Way

Problem:

Four heaps contain $38$ , $45$ , $61$ , and $70$ matches respectively. Two players take turns choosing any two of the heaps and take some non-zero number of matches from one heap and some non-zero number of matches from the other heap. The player who cannot make a move, loses. Which one of the players has a winning strategy?

Solution

Solution:

The first player wins by making moves so that the opponent must face positions of the form $(a, a, a, b)$ , where $a \leq b$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.