nostr:npub1tm3v7q8jk0pe5zwrl7w9tav09cttdwsu8hfhgd0x9rchdk60vupq088zk5 I doubt a clean proof is possible, because grids of size 8, 10, and 12 do admit solutions (see pics). The solutions for size 8 and 12 generalize to any multiple of 4. I'm not sure if the solution for size 10 generalizes.

cc nostr:npub1ecj3mfr9lzvx7wh6fmh59vz6eet324mdtdlp9qxzqvwuvpglwnxqv6fchy nostr:npub17z3zrlvtcp3fyc94vdqdk7y8ccjc7yay998z7wsvl9f0t7yqmjes3uhe6e nostr:npub16pn60c6d7458s56quf3e58ahjkwwwmcfm94vxtc09qq7q3srdw4szeruz4 nostr:npub1u83gudwdfwjngz5eqxkrnpvsuydvqwjxwz0majvehqrh62804hfq8yr2s5 wonder if you can find a non-computer proof that size 6 is impossible