Can you paint a 6x6 grid in red, green and blue, such that its every 3x3 sub-grid contains exactly 5 red, 3 green and 1 blue cell?
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The simple answer:
Yes. (I can paint it in such a way)
There are at least $9!/3!5!=504$ different arrangements that work
My method of producing them is to take any 3 x 3 grid that satisfies the condition, and tile it into a 6 x 6 grid.
This works because any 3 x 3 sub-grid of a 6 x 6 always contains the same cells of the original 3 x 3 grid.
Here is an example of a non-tiling (horizontally anyway) that follows the rules.