The King That Doesn't Spiral

Glue a chessboard's edges into a Klein bottle and a queen becomes impossible to count honestly — her diagonal spirals across the twisted seam with no canonical rule. A king has no such problem: he reaches only the eight squares around him, and that reach is fixed by the gluing on every surface. So the king is the piece that lets you ask, cleanly, what a twist does to a chessboard's arithmetic. Place some below. Then meet two counting sequences the encyclopedia never held — and the surprise that the torus and the Klein bottle are the same board until the very densest packing.

One square, no convention

Numbering rows 0…n−1 top to bottom and columns 0…n−1 left to right, the four boards differ only in what is glued to what:

• Flat — nothing glued. The ordinary board; a king near an edge simply has fewer neighbours.
• Torus — left↔right and top↔bottom glued straight. Every king has all eight neighbours; the board is edgeless.
• Möbius — only left↔right glued, with a half-twist: the square just right of (i, n−1) is (n−1−i, 0). Top and bottom stay open edges.
• Klein bottle — left↔right glued straight, top↔bottom glued with a horizontal flip: the square just below (n−1, j) is (0, n−1−j). The non-orientable sibling of the torus.

The companion study on this board glued the same four ways but placed queens, and had to stop short at the Klein bottle: a queen's diagonal, traced across a non-orientable seam, does not close after a short loop — it spirals, so a Klein-bottle queen attacks most of the board, and how many squares depends on an arbitrary choice of how a diagonal continues past the flip. There is no published convention to anchor it. That study named the fix outright: “the clean way to extend the family to the Klein bottle is with kings.” A king moves one square. His eight-square reach is pinned by the gluing and nothing else — on the torus, the Möbius strip, the Klein bottle, all of it. So the king's counts are canonical, and this is that clean extension.

A family the catalogue only half holds

A “non-attacking king placement” is just a set of squares with no two of them king-adjacent — an independent set in the board's king graph. Counting those on grids is careful, classical work, and the On-Line Encyclopedia of Integer Sequences holds the two orientable boards of this family. It does not hold the two twisted ones:

Every cell was recomputed live in the panel above as you changed the board. The two “new” cells returned no results on OEIS (searched 2026‑07‑12, two term-windows each). And they sit on solid ground: the counter reproduces both catalogued siblings — flat kings (A063443) and toroidal kings (A067958) — exactly, before it computes anything new.

Calibration siblings, reproduced not claimed: flat = A063443 (2, 5, 35, 314, 6427, 202841, …), torus = A067958 (2, 5, 10, 133, 1411, 42938, …).

The twin the torus hides

Set the board to Klein bottle and read the non-attacking pairs number. Now switch to Torus at the same n. It doesn't change. That is not a coincidence: both boards make the king graph 8-regular — every square has all eight neighbours — so both have exactly 4n² attacking pairs, and therefore the same number of non-attacking pairs. Two different surfaces, the same count.

So look at the total placements instead. For n = 1, 2, 3 the torus and the Klein bottle agree there too — 2, 5, 10, identical. The first board on which they finally disagree is n = 4: the torus admits 133 placements, the Klein bottle only 129. And the gap is not spread out — the two surfaces admit exactly the same number of placements of 0, 1, 2, and 3 kings. They differ in one place only: the densest packing, four kings, where the torus fits 12 arrangements and the Klein bottle just 8. The twist erases exactly four of the tightest packings. Here they are — every 4-king packing of the 4×4 torus, with the four the Klein bottle destroys outlined in red:

Push n higher and the two counts keep parting ways — but which surface wins alternates. On every even board the torus admits more placements; on every odd board the Klein bottle does (checked exactly through n = 13). The natural suspect is the flip itself: the Klein seam sends column c to column n−1−c, a map that has a fixed centre column exactly when n is odd. We state the alternation as an observed, exactly-computed pattern, not a theorem — the honest status of a fact found by counting rather than proof.