At one point, a remote island's population of chameleons was divided as follows:
- 13 Red chameleons
- 15 Green chameleons
- 17 Blue chameleons
Each time two different colored chameleons would meet, they would change their color to the third one. (
I.E. If Green meets Red, they both change their color to Blue.)
Is it ever possible for all chameleons to become the same color? (Why or why not?)
The problem boils down to this: you have the numbers 13, 15, 17. You can subtract 1 from two of the numbers, and then add 2 to the third. Will you ever end up with two 0s?
To get two 0s, you have to get two numbers to be alike. (Then you can keep subtracting 1 from those two.) To get two numbers alike, they have to first differ by 3. (For example, you could go from 12-15-18 to 14-14-17 or 11-17-17.) One number has to go up by 2, the other down by 1. (They could both go down by 1, but then their difference wouldn't change.)
For them to differ by 3, they have to first differ by 6 or 0. (Try it.) For that to happen, they had to differ by 9 or 3. This is the pattern; for two numbers to be alike, they had to first differ by a multiple of 3. Since the numbers start out differing by 2, 2, and 4, you'll never get them to be alike. The chameleons will always have at least two different colors.
Spoilers.
