## Generalization That Solves the Problem

I was sent the following problem that was offered at the 2008 Rasor-Bareis-Gordon contest at the Ohio State University:

Chameleons on an island come in three colors. They wander and meet in pairs. When two chameleons of different colors meet, they both change to the third color. Given that the initial amounts of the chameleons of the three colors are 13, 15, and 17, show that it may not happen that, after a while, all of them acquire the same color. |

Three solutions have been posted after the competition.

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**Solution 1.** Let *x* be the number of times colors A and B meet, *y* the number of times colors A and C meet, and *z* the number of times colors B and C meet. If we end up with all of color A, then we have the system of three linear equations *x* - *y* + 2*z* = 45,*x* + 2*y* - *z* = 0,*x* - *y* - *z* = 0.

**Solution 2.** Let *a*, *b*, *c* be the number of chameleons of each color. Note that under any of the three possible changes, the value *b* - *a**a* and *b* both decrease by 1 so *b* - *a**a* decreases by 1 and *b* increases by 2 so *b* - *a**a* increases by 2 and *b* decreases by 1 so *b* - *a**b* - *a*

**Solution 3.** Represent the colors by 0, 1, and 2, for, respectively, the 13, 15, and 17 chameleons. Let *a* and *b* be the distinct colors of two chameleons who meet and let *c* be the third color. Then *a* + *b* = -*c* (mod 3).*c*, and *c* + *c* = 2*c* = -*c* (mod 3);

The same problem has been discussed at the wu:forums under the rubric Political slugfest, where ecoist offered a generalization:

Members of |

Curiously, not only the generalization is as easy as the original problem, but, by pointing to a property of the triple 13, 15, 17 that did not stand out otherwise, the generalization actually spelled its own solution.

Indeed, when *n* - 1 members of different parties defect to some other party, the latter acquires *n* - 1*n* - 1 (mod *n*),*n* the number of members in every of *n* parties changes by -1. Assuming that all memberships were different modulo *n*, to start with, we see that the operation of subtracting -1 modulo *n* does not affect this fact: all memberships remain different modulo *n*.

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Copyright © 1996-2018 Alexander Bogomolny

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