This post originally appeared on the Brilliant blog on 8/31/2012.
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The following is a Student Solution to a Test Yourself posted on Parity, presented by Farrell W., Philippines.
Test Yourself 5: (*) In the forest, there are chameleons of 3 colors: blue, red and green. When two chameleons of a different color meet, they will change into the third color. For example, if a red and blue chameleon meet each other, they will both turn green. If there are currently 12 blue chameleons, 34 red chameleons and 56 green chameleons, would it be possible for all the chameleons in the forest to turn green?
Key Techniques: Invariance; Modulo Arithmetic
Solution: Let represent the number of blue, green and red chameleons in the forest at time , respectively. Let us look at the value taken modulo 3. When a blue and red chameleon meet, the new value is . When a red and green chameleon meet, the new value is . When a green and blue chameleon meet, the new value is . Hence, this value stays constant throughout, regardless of which chameleons meet each other. We say that the value is an invariant of the problem.
At the start, . If all chameleons are green, then . Since these two values are not equal, it is not possible for all the chameleons to turn green.
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