The Queen That Comes Back Upside Down

A chessboard has four edges. Glue the left one to the right with a half-twist and it becomes a Möbius strip — a surface with only one side — and a queen that slides off the right edge slides back in on the left, upside down. Place queens below and watch the attacks wrap through the seam. Then meet three counting sequences that this twist produces, and that the great encyclopedia of integer sequences has never recorded.

The board with one side

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 queen's ray stops at the edge.
• Torus — left↔right and top↔bottom glued straight (the “modular” board). A ray that leaves one edge reappears on the opposite one, level.
• Möbius — only left↔right glued, and with a half-twist: the square just right of (i, n−1) is (n−1−i, 0). Row i “carries over” to row n−1−i. Top and bottom stay open edges. This is exactly the board of Bell & Stevens (2008).
• Klein bottle — left↔right glued straight, top↔bottom glued with a horizontal flip. The non-orientable sibling of the torus.

Because a Möbius queen's horizontal and diagonal lines all wrap onto two rows (its own and the carried-over one), it can never be that a single row holds a full row of safe squares — and the most non-attacking queens the n×n board can ever hold is ⌈n/2⌉. Usually that many do fit. Set the board to n = 3 above: the 3×3 Möbius board is the one small board where they don't — you cannot place even 2 non-attacking queens on it, though ⌈3/2⌉ = 2. Try.

A family the catalogue only half-finished

Counting non-attacking pieces on boards is old, careful work, and the On-Line Encyclopedia of Integer Sequences holds much of it. But it holds the orientable members of this little family and almost none of the twisted ones. Here is the exact state of two natural counts — the number of ways to place two non-attacking queens, and the total number of non-attacking placements of any size (the count the flat board calls A287227):

Every “new” cell was recomputed live in the panel above as you changed the board. Four of them are genuinely absent from OEIS (searched 2026‑07‑06). Three sit on boards whose attack rules are pinned to published ground truth — the engine reproduces the flat, torus, and Möbius queen counts (OEIS A000170, A007705, A137279) exactly before it computes anything new — so these are the ones offered as real additions:

The Klein bottle, where the queen becomes a tyrant

Switch the board above to Klein bottle and place a single queen near a corner. It attacks nearly the entire board. That is real, and it is honest to flag why: on a non-orientable surface a “diagonal” is not innocent. Trace one across the twisted top–bottom seam and it does not close after a short loop the way a torus diagonal does — it spirals, threading most of the squares before it returns. So a Klein-bottle queen's reach depends on exactly how you decide a diagonal continues past the flip, and there is no published convention to anchor it (unlike the Möbius board, which Bell & Stevens fixed). The numbers you see for the Klein board are exact for the gluing defined here — they are a genuine curiosity, not a canonical claim. The kings, whose reach is only one square, don't suffer this ambiguity; extending the family to non-orientable kings is the clean next step, left for whoever comes here next.