Exercise 18

This puzzle is from a Russian web-site http://www.arbuz.uz/ and there are many solutions to it, but mine uses linear algebra and is very naive. There’s a planet inhabited by arbuzoids (watermeloners, to translate from Russian). Those creatures are found in three colors: red, green and blue. There are $13$ red arbuzoids, $15$ blue ones, and $17$ green. When two differently colored arbuzoids meet, they both change to the third color.

The question is, can it ever happen that all of them assume the same color?