Riven & OneZero
Riven
I was just working on a twist to the knight’s tour—imagine forcing the knight to return to its starting square after visiting every square exactly once. It adds a constraint that feels almost paradoxical; would be fun to see if you can find a solution or a pattern that breaks it down.
OneZero
A closed tour is actually possible on every even‑sized board larger than 4×4, including the usual 8×8. The key is that the knight alternates square colors each move, so you can’t finish on the same color you started from if the board has an odd number of squares. On an even board the colors balance out, and a Hamiltonian cycle exists. A neat construction is to walk around the perimeter, then peel inward in a spiral, always keeping the colour parity balanced. So the “paradox” is only a trick of the board’s parity, not a flaw in the knight’s logic.
Riven
Nice explanation—parity is the only snag. The trick, though, is how you handle the 4×4 case where the Hamiltonian cycle collapses; the spiral strategy falls apart because the interior is too small to maintain the alternation. For boards larger than that, the spiral works, but you still need a careful seam at the transition between layers. It’s the seams that make the proof interesting, not the parity itself.
OneZero
You’re right about the 4×4—its interior is just a 2×2 block, and a knight can’t jump across that without breaking the parity chain. In fact, the only way to close the tour on a 4×4 is to use the two central squares as a bridge, but that forces you to revisit one of them, so a true Hamiltonian cycle is impossible there. The seam problem you mention on larger boards is exactly where the spiral pattern has to splice the outer ring to the inner ring without flipping the colour balance. One trick is to offset the last leg of the outer ring by one square, then start the inner ring a step later; that keeps the alternation intact. It’s a small tweak, but it’s the kind of local adjustment that makes the whole construction fit.