riddle of the day

On this island, every encounter changes the balance of colors. The challenge is to determine which endings can actually be reached.

The Last Color

Master
Pure logic

Riddle statement

An island is home to 4 red chameleons, 7 green chameleons, and 10 blue chameleons.

Whenever two chameleons of different colors meet, they both change to the remaining color.

  1. Can all the chameleons eventually become the same color?
  2. If so, which final colors are possible?
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Solution

Answer: all three colors are possible: red, green, and blue.

The key is to make the two unwanted colors equally numerous. Once their populations are equal, pair them with each other until both disappear into the remaining color.

Write each state as $(R,G,B)$.

To finish with all red chameleons:

$ (4,7,10)\xrightarrow{\,1\ R-B\,}(3,9,9)\xrightarrow{\,9\ G-B\,}(21,0,0). $

To finish with all green chameleons:

$ (4,7,10)\xrightarrow{\,2\ G-B\,}(8,5,8)\xrightarrow{\,8\ R-B\,}(0,21,0). $

To finish with all blue chameleons:

$ (4,7,10)\xrightarrow{\,1\ G-B\,}(6,6,9)\xrightarrow{\,6\ R-G\,}(0,0,21). $

Therefore, the island can end with all its chameleons having any one of the three colors.