On an island live 13 purple, 15 yellow and 17 maroon chameleons. When two chameleons of different colors meet, they both change into the third color. Is there a sequence of pairwise meetings after which all chameleons have the same color?
Let <p, y, m> denote a population of p purple, y yellow and m maroon chameleons. Can population <13, 15, 17> be transformed into <45, 0, 0> or <0, 45, 0> or <0, 0, 45> through a series of pairwise meetings?
We can define the function:
X(p, y, m) = (0p + 1y + 2m) mod 3
An interesting property of X is that its value does not change after any pairwise meeting because
X(p, y, m) = X(p-1, y-1, m+2) = X(p-1, y+2, m-1) = X(p+2, y-1, m-1)
Now X(13, 15, 17) equals 1. However,
X(45, 0, 0) = X(0, 45, 0) = X(0, 0, 45) = 0**
This means that there is no sequence of pairwise meetings after which all chameleons will have identical color.