Is it possible to paint the cells of a rectangular grid with $𝐾$ different colours such that:
- No two adjacent (horizontally or vertically) cells have the same colour, and
- Every combination of two colours appears exactly once in some two adjacent (horizontally or vertically) cells, and
- The sides of the rectangle are greater than 1.
I've asked this question on Puzzling StackExchange and there a solution was found for 17 colours on a 7x11 grid. Now I am wondering are there any other solutions for other values of $K$?