On a certain island, there are 10 red chameleons,
15 green chameleons and 20 blue chameleons.
Each time two chameleons of different colors meet,
they both change into the third color.
Nothing happens if two chameleons of the same color meet.
Is it possible for all of the chameleons on the island
to end up being the same color?
The answer is that it is not possible.
Note that when the starting numbers are divided by 3,
the remainders are 1, 0 and 2 (red, green and blue,
respectively).
When two chameleons of differing colors meet, two
of the counts decrease by 1 and the third count
increases by 2.
We can easily check that if the remainders are 0, 1 and 2
before two chameleons meet, then the remainders are still
0, 1 and 2 after they meet.
Let R, G and B be the remainders when the numbers
of red, green and blue chameleons are divided by 3.
Suppose R=0, G=1 and B=2.
If a red chameleon meets a green chameleon, then the
resulting remainders will be R=2, G=0, B=1.
If a red chameleon meets a blue chameleon, the
remainders also become R=2, G=0, B=1.
And if a green chameleon meets a blue chameleon, the
once again become R=2, G=0, B=1.
As a result, the three counts will never be the same,
since they always have different remainders when
divided by 3.