$Find conditions for all chips to become of the same color in this game$

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$Find conditions for all chips to become of the same color in this game$

There are a white, b black, and c red chips on a table. In one step, you may choose two chips of different colors and replace each one by a chip of the third color. Find conditions for all chips to become of the same color. Suppose you have initially $13$ white $15$ black and $17$ red chips. Can all chips become of the same color? What states can be reached from these numbers?

Putnam Competition Math Competitions Word Problem

Chen K

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Suppose we want all the chips to be white and the end of the process. Then $(a, b, c)$ will be transformed into one of the three triples $(a + 2, b − 1, c − 1)$, $(a − 1, b + 2, c − 1)$, or $(a − 1, b − 1, c + 2)$. In each case, $a − b=I$ mod 3 is an invariant. But $b − c=0$ mod 3 and $a − c=0$ mod 3 must also be invariant. So $I = 0$ mod 3 combined with $a + b + c =0$ mod 3 is the condition for reaching all white chips at the end.

For the given numbers $a=13, b=15, c=17$, since the difference of any two of these numbers is not $0$ mod 3, we can not reach a state where all chips have the same color ($I=0$ mode 3 is not satisfied). Indeed $I=2$ mode 3, and we can reach a final state where all chips have the same color except 2 that are of a different color.

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$Find conditions for all chips to become of the same color in this game$

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