Chameleons go on a date

Question:

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?

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C

A

P

T

A

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N

I

N

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E

R

V

I

E

W

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Solution:

Let <p, y, m> denote a population of p purple, y yellow and m maroon chameleons. CanĀ 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.

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