Yes. Let x, y, and z be the number of green, brown, and gray chameleons, and consider the remainders left when each of the quantities x-y, x-z, and y-z is divided by 3. These remainders won’t change as the chameleons vary their colors. This means that if two populations are to reach 0 simultaneously, they must differ now by a multiple of 3. In this case two do: The populations of green and brown chameleons differ by 6, so it’s possible for both to disappear without any leftovers:
13, 19, 23
15, 18, 22
17, 17, 21
… and then green and brown chameleons can disappear in pairs until all have turned gray.
From Dick Hess, All-Star Mathlete Puzzles, 2009.
