The only inhabitants of a far away planet are chameleons that come in four color

Question

The only inhabitants of a far away planet are chameleons that come in four colors: blue, green, yellow, and red. Whenever three chameleons of three different colors meet they all change to the fourth color. Start with 2 green, 2 yellow, 6 red, and 7 blue chameleons. (That is, the configuration is (2, 2, 6, 7).) Can you arrange their meetings so that all the chameleons change to the same color at the end. (That is, the configuration at the end is unicolor.)

Can you turn (6, 2, 10, 7) into a unicolor configuration?

Explanation / Answer

We know that whenever at least 3 out of the 4 numbers (g, y, r, b) are equal modulo 4, the configuration (g, y, r, b) can be turned into a unicolor configuration.

1. Now we have 2 green, 2 yellow, 6 red, and 7 blue chameleons.

For green, 2 mod 4 = 2

For Yellow, 2 mod 4 = 2

For Red, 6 mod 4 = 2

=> (2, 2 & 6) modulo 4 are equal.

Hence (2, 2, 6, 7) can be turned into a unicolor configuration

2. Now we have 6 green, 2 yellow, 10 red, and 7 blue chameleons.

For green, 6 mod 4 = 2

For Yellow, 2 mod 4 = 2

For Red, 10 mod 4 = 2

=> (6, 2 & 10) modulo 4 are equal.

Hence (6, 2, 10, 7) can be turned into a unicolor configuration

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