There are 7 red, 8 blue and 15 green salamanders on a small island. Whenever two of the same colour meet they mutate, one into each of the other colours. Whenever two of a different colour meets they both mutate into the third colour. Can it ever happen that they are all the same colour? Justify your answer. [Hint: Think about the integers mod 3.]
The Attempt at a Solution
I made my computations by brute force and found the following combinations left
29R and 1G
29G and 1B
29B and 1R
There might be other combinations like it but it just shows that it is not possible to get all 30 to be all R, G nor B because when you get 29 of one colour than it’s impossible to turn the other colour into another because to do that, one of the 29 itself must change which after a bit of combining leads to at best 29 again. My solution is not rigours and kind of intuitive. Is it okay? I didn’t make use of the hint which is not good I think.