Chameleons changing colors

Question

On an island there are chameleons of three colors: 13 red, 15 green, and 17 blue. Whenever two chameleons of different colors meet, they both change into the third color. (For example, a red and a green meeting both become blue.)

Is it possible for all 45 chameleons to eventually become the same color?

Accepted Answer

No, it is impossible. The obstruction is an invariant modulo 3. Let (r, g, b) be the counts of red, green, blue. A meeting of two different colors decreases those two counts by 1 each and increases the third by 2. The three possible meetings do: - red+green → blue: (r, g, b) 	o (r-1, g-1, b+2) - red+blue → green: (r-1, g+2, b-1) - green+blue → red: (r+2, g-1, b-1) Consider the differences modulo 3. Look at g - r \pmod 3 (and by symmetry the other pairwise differences): - red+green: g - r changes by (-1) - (-1) = 0. - red+blue: g - r changes by (+2) - (-1) = +3 \equiv 0 \pmod 3. - green+blue:…

Chameleons changing colors

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