In https: this page, there are all the 107 ways to cover a $5\times 5$ board with pentominos. But, it is possible to prove that there is no way to cover the $5\times 5$ board using only L-pentominos without see all the 107 ways? Which would be a good coloration for the board in order to prove the last question?
2026-02-22 17:16:28.1771780588
Cover a 5x5 board with pentominos
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Up to symmetry there are only two ways to cover the center square with an L-pentomino:
It's easy to see that neither can be completed to a tiling by L-pentominoes. For example, think about how you'd cover the "inside corner" of the first L-pentomino: in the case on the right, this cannot be done at all, and in the case on the left, there are four ways to do it, and all of them leave the bottom right corner in bad shape.