Huh, pretty weird. This grid is a counterexample. 1221 3113 3333 3123 Suppose we can turn this into a symmetric grid. Since there are an odd number of 1s and 2s, the line of symmetry would have to be along a diagonal, and the diagonal would have to contain an odd number of 1s and 2s so that the remaining squares can be paired. Note that for either diagonal, zero of the six pairs of cells that are symmetric about it have matching colors. Also, the distribution of colors on either diagonal does not contain an odd number of 1s and 2s. So to make this grid symmetric along either diagonal, we would have to change at least 1 cell in each pair, plus at least 1 cell on the diagonal. This means we'd need at least 7 cells changed, impossible with only three swaps. (Can it always be done with four?)