A stratum · combinatorics on one-sided boards

Leapers on a Möbius Strip

A knight that gallops off the right edge of a half-twisted board doesn't reappear where the doughnut would send it — it returns on the left, upside down. Count the ways such pieces can stand without attacking, and you fall off the edge of the encyclopedia: eight sequences OEIS has never held — and one symmetry the twist quietly destroys.

This is the sequel to Leapers on a Torus. There we wrapped the board into a doughnut — both pairs of edges glued straight — and counted non-attacking knights, camels, zebras and giraffes, one per row and column. Here we glue the left and right edges with a half-twist instead. The board becomes a Möbius band: one continuous side, one continuous edge, and a piece crossing the seam comes back with its rows reflected. Glue the other pair too, with a flip, and you get a Klein bottle. On these one-sided boards the counts are still real, still exactly computable — and, as far as we could find on 2026‑07‑13, catalogued nowhere.

The reason a leaper is the right piece for this is subtle and it is the whole point. On a Möbius band a queen's diagonal has no honest answer — it spirals around the twist forever and where it "attacks" depends on a convention you had to invent. A leaper makes a single fixed jump, so its landing square is fixed by the standard rule for any surface: unfold to the flat cover, read the jump there, fold the target back through the gluing. It does not matter which way round the L the knight goes. That is why the clean twisted chess piece was always a bounded one.

See the twist

Pick a surface and a piece, then hover or tap a square to light up everywhere that piece attacks from there. Watch what happens near a glued edge: on the Möbius band and Klein bottle some targets come back through the seam wrapped, and some come back flipped — landing on the far side, mirrored. Then click squares to drop pieces, one per row and column, and try to seat all 6 without an attack.

The flipped squares are the signature of a one-sided board. On the doughnut a knight leaving the right edge re-enters at the same height on the left; on the Möbius band it re-enters at the mirrored height — row i carries over to row n−1−i, exactly the carry-over Bell & Stevens used to put queens on this board in 2008. That single reflection is enough to change every count below.

Eight sequences off the edge of the map

Counting the non-attacking placements — one leaper per row and per column — for each board and each piece gives the table below. The flat and torus rows are known: the flat knight is OEIS A137774 ("non-attacking empresses", a rook that also leaps like a knight — the rook part is the one-per-row-and-column rule), and the torus row is the previous stratum. The Möbius and Klein rows are the new ones: each returned No results on oeis.org, checked 2026‑07‑13.

The law the twist breaks

On the torus there is a hidden law. Multiply both coordinates of a leaper's move by any number coprime to the board size and the count cannot change — a proved bijection of the modular board onto itself. Because a prime board like n = 7 makes every leaper equivalent under that scaling, all four pieces are forced onto the same count there: 210. The doughnut cannot tell a knight from a giraffe at n = 7.

The scaling law needs both axes to wrap the same way. The Möbius band wraps only one axis (and with a flip); the Klein bottle wraps the two axes differently. So the symmetry has nothing to stand on — and the four leapers, welded together on the torus, split apart. Move the dial to a prime board and watch 210 shatter.

How you know it isn't lying

A new integer sequence is only worth anything if the machine that made it was proven right on numbers it did not get to choose. The attack model here — the fold-through-the-gluing rule — is pinned to four independent published grounds before a single new term is believed: