A standard game of chess played on a single-sided Möbius strip. The board is an 8×8 grid whose long edges are joined with a half-twist, so the surface has only one side and one edge.

1 · The Twist Identification

When a piece moves past h (rightward) or past a (leftward), it re-enters from the opposite file edge with the rank reflected:

2 · Sliding Pieces

Rooks slide along files (around the strip) and ranks (across the width). File-direction slides wrap with the rank-flip.

Bishops and Queens on diagonals: when a diagonal crosses the file boundary, the rank direction is reversed upon re-entry, so the piece traces a non-Euclidean arc across the fold.

3 · Knights & Kings

Standard offsets, with Möbius wrapping applied to the file component.

4 · Pawns

Move forward toward the back rank in the player's colour. After crossing the Möbius boundary, the pawn reverses its forward direction — it is now on the "other face" of the strip. En-passant, double-step, and promotion behave as standard on whichever face the pawn currently occupies.

5 · Check, Checkmate, Stalemate

Detected by projecting all opponent attack squares (including Möbius-wrapped rays) onto the king's position.

6 · Move Notation

Standard algebraic notation, with a ⟆ prefix on any move that crosses the file boundary.