# Colour in a 5 by 5 grid, with evolution rules, so that the starting grid is the same as the output grid

Colour in a 5 by 5 grid using the colours Red, Blue, Green and Yellow.

Each cell is considered to have 4 neighbours (to the left, right, above and below.) The grid wraps, so e.g. the top right is a neighbour of the bottom right and top left.

Produce a new grid as follows: if a cell is surrounded by 4 of the same colour, it turns (or remains) Red; if surrounded by 3 of the same colour and one different, it turns Blue; if surrounded by 2 of the same colour and any two other colours (e.g. G, G, R, Y or B, B, R, R) it turns Green; and if surrounded by 4 different colours, it turns Yellow.

Question: other than a grid that's coloured all red, is it possible to colour a grid so that the output, i.e. the grid that gets produced after one iteration, is identical to the starting grid? (You don't have to use all of the colours.) And, if not, can you prove that this is impossible?

I don't know the answer to this question, but after having spent some time trying to solve, I suspect it is impossible. (Hopefully I didn't miss something really obvious...) I'm hoping someone can supply a proof!

• I'm curious how this arose. Is it a pure puzzle, or did it grow out of some interesting research? May 24 at 14:40
• @Scott Sauyet I made this puzzle: reddit.com/r/PictureGame/comments/tfzsux/… for r/picturegame on Reddit and then this one is another variant in the same genre. However I couldn't post it on Picture Game as I didn't have a solution! So it's a pure puzzle. May 24 at 15:28
• I guess there would $4^5^2 = 1125899906842624$ possible grids. May 24 at 16:24