# Chess on three surfaces — experimental rules 0.1

## Design intent and fixed choices

These variants preserve the familiar chess pieces, alternating moves, captures, king safety, checkmate, stalemate, promotion choice, and en passant. Each side has exactly K, Q, two R, two B, two N and eight P. White moves first.

A surface board is not an ordinary board with a cosmetic wrap. Directions must transport through seams, some diagonals fail at cone points, and a closed board has no intrinsic last rank. Those decisions are explicit below. Initial deployment is adapted to stop the rear and side attacks that an ordinary back rank would expose. These rules define coherent experimental games; opening checks and symmetry do not establish balance or enjoyment.

## Coordinates, square complexes and playable cells

Files increase to the right, ranks upward. Cells, not mesh vertices, are occupied by pieces. Möbius uses a–j, ranks1–9 (90 cells). Klein uses a–i, ranks1–18 (162). Genus two uses two labelled charts I and II, each a–i/ranks1–9 (162 total). Charts are not players or separate boards.

Möbius has square faces arranged in a 10×9 rectangle. Identify corresponding left/right edges with rank reversal: stepping east from j,r lands at a,10−r and stepping west from a,r lands at j,10−r. The top/bottom edges are unglued. The derivative of that identification is (dx,dy)↦(dx,−dy). In particular, the remainder of a multi-edge route and a pawn's facing both reverse vertically.

Klein uses a 9×18 rectangle with the same side rule r↦19−r, then top/bottom identified by translation r mod18. A corner step first uses the same full-displacement reflection and then rank wrapping. The two identification derivatives are diag(1,−1) and identity.

Genus two uses the accepted Shogi square complex, reproduced completely here. Each chart otherwise wraps on all four outer edges. The listed north/east transitions replace the ordinary transitions. Their south/west inverses replace the corresponding reverse transitions. All derivatives are identity.

| North from | To |
|---|---|
| I:a4–d4 | II:a4–d4 |
| II:a3–d3 | I:a5–d5 |
| I:e5 | II:e5 |
| II:e4 | I:e6 |
| I:f6–i6 | II:f6–i6 |
| II:f5–i5 | I:f7–i7 |

| East from | To |
|---|---|
| I:d5 | II:e4 |
| II:d4 | I:e5 |
| I:e6 | II:f5 |
| II:e5 | I:f6 |

These face-edge identifications, with paired square-edge endpoints identified consistently with the indicated derivative, specify the cell complex, not merely an adjacency graph.

### Geometry

The square complexes have the following mesh counts and vertex links. Here V, E and F refer to mesh vertices, edges and square faces; playing cells are the faces.

| Board | Square faces F | Mesh edges E | Mesh vertices V | Links | Boundary | Orientation, χ |
|---|---:|---:|---:|---|---|---|
| M10×9 |90|190|100|80 circles,20 intervals|one component,20 edges|nonorientable,0|
| K9×18 |162|324|162|162 circles|none|nonorientable,0|
| G two9×9 |162|324|160|158 four-corner circles,two eight-corner circles|none|orientable,−2|

The two eight-corner G vertices have cone angle4π; their links remain circles, so they are surface points, not pinched vertices. Euler characteristic alone was not used as a manifold certificate.

M10×9 and K9×18 admit the alternating cell coloring parity(file index + rank index). The respective seam conditions are W+H odd and additionally H even on K. Their bishops preserve that coloring. Each side starts with one bishop on each color. G has odd wrapping cycles and no global alternating checkerboard; drawing alternating shades within charts must not claim a conserved bishop color. On an empty G board, bishops can connect the whole board over successive moves.

## Exact movement geometry

A cardinal step crosses the selected cell edge and transports all remaining directions by its derivative. A diagonal unit step with local directions u horizontal and v vertical is allowed precisely when u-then-v and v-then-u both exist and reach the same cell with the same transport. If so that cell/transport is its result. If not, the diagonal is forbidden; it cannot be used by a king, bishop, queen, or pawn. Neighboring side cells used for this calculation do not block the diagonal. On G the forbidden cases are exactly the16 approaches to its two cone vertices. An ordinary bend in the drawn staircase is not automatically forbidden: I:d4 northeast reaches I:e5.

A knight has all eight usual jump choices: choose any cardinal long direction L and either perpendicular cardinal short direction S, then follow the edge word L,S,L, transporting the remaining directions at each crossing. This is the edge order of the unfolded straight knight segment. All intermediate occupancy is ignored, but any missing boundary step makes that route unavailable. The word L,L,S is not interchangeable on G. A jump is never permitted to finish on its departure cell.

Rooks repeat any of the four cardinal steps. Bishops repeat any of the four diagonal steps. Queens have both sets of rays. After each step, continue in the transported direction. A ray stops at a boundary, forbidden diagonal, first occupied cell, its departure cell, or repeated directed state. The departure cell remains occupied during ray tracing, so it blocks a returning ray even if direction has changed. An encountered friendly piece cannot be landed on. An enemy piece can be captured there and ends that route. No path is continued through an occupied square; alternative winding routes are checked independently.

A king moves one cardinal or allowed diagonal step. A non-pawn does not retain a facing field because its next-move set is invariant under vertical reflection. Pawns retain a facing f=+1 (increasing ranks) or−1 (decreasing ranks), independent of owner. Opposite routes to the same destination can give different pawn facings; those are different outcomes and must be selectable. Routes with identical final state are one legal action, even if several clear rays realize it. Friendly-occupied destinations are forbidden. A king is attacked but never captured.

### Pawns and their initial double step

A pawn advances one cardinal step in its facing direction only into an empty cell. It captures one allowed forward diagonal step only when that destination contains an enemy non-king, except en passant below. Its facing is transported by the chosen route, including after a diagonal capture through a reflected seam.

Every initial pawn has an unused-double-step flag true. A pawn may instead make two consecutive forward cardinal steps on its first move, provided both are empty and the final cell differs from departure. The flag becomes false on every pawn move, including a capture; it is never restored by returning to an initial square. In ordinary play a true flag therefore also identifies a still-unmoved pawn at its original home square. An analysis setup may explicitly grant a flag, but must label that provenance rather than claim historical reachability. A two-step move evaluates promotion only on its destination. There are no extra pawn-file restrictions.

### En passant on any of the three boards

A double step that ends as an unpromoted pawn records `(owner, source, middle, destination, finalFacing)` for precisely the opponent's next turn. Every other move clears the old record. If a double step promotes, it creates no record. There can be at most one vulnerable pawn and one trace.

On that next turn an opposing pawn may move by one of its allowed forward diagonal routes to the empty recorded middle cell and remove the recorded pawn at its recorded destination, provided that pawn is still there. This is the definition even when the capture crosses a reflected seam or the two pawns now have unusual local facings. The capturer's facing follows its own capture route. The captured pawn is removed from its destination, not from the landing cell. There is no requirement inferred from a rectangular drawing that the pieces share a displayed rank. King safety is checked after both removals and placement of the capturer; en passant that exposes one's king is illegal. The capturing pawn promotes if its landing cell requires it.

For example after Black's a9–a7 double step on M, a White pawn on j1 facing increasing ranks can capture en passant j1–a8, remove the Black pawn on a7, and arrive facing decreasing ranks. The rule uses the recorded route, not a guessed neighboring-square formula.

### Mandatory promotion

A pawn arriving on its promotion target is immediately replaced, as part of that move, by its owner's Q, R, B or N of the player's choice. Captured stock does not restrict the choice. There is no choice to remain a pawn, and the new piece acts immediately in check detection. A move with an omitted required choice is incomplete, not implicitly a queen.

| Board | White target | Black target |
|---|---|---|
| M | last forward rank:9 when arriving f=+1,1 when f=−1 | same facing-relative rule |
| K | rank15 | rank4 |
| G | II:rank9 | I:rank1 |

K/G targets are owner-marked enemy back-rank lines, deliberately replacing a nonexistent intrinsic far edge. A pawn can cycle through other ranks without promoting; arrival at its target is compulsory regardless of facing. Crossing a marked rank during a two-step move does not count unless it is the destination. Every initial pawn has an unobstructed forward-only path to a target when other pieces are removed: M7–8 single steps,K8–11,G6–9. That is geometric access, not a guarantee a pawn can force promotion through defenders.

## Initial positions and rationale

Black is the exact player-exchange image specified below. All initial pawns face increasing ranks for White and decreasing ranks for Black. Initial king/rook identities matter only for M castling. Inventory is16 pieces per side on each board.

M retains the ordinary back-rank relative ordering with one advanced knight and two flank guards:

```text
rank2: . P N P P P . P P .
rank1: P R . B Q K B N R P
files: a b c d e f g h i j
```

Black has that same file ordering on ranks8/9. Pawns at a1/j1 stop direct side-join rook captures; front pawns and the knight on c2 shield the rooks, bishops, and queen. Moving just the knight off c1 supplies the missing front screen while retaining8 pawns. Bishops d1/g1 occupy opposite color classes. Merely adding two flank guards while leaving the ordinary rook files open failed the opening audit and is not the starting position.

The K compact camp is:

```text
rank7: . . . . P . . . .
rank6: . P . P P P . P .
rank5: . . B R Q R . B .
rank4: . P . N K N . P .
files: a b c d e f g h i
```

Black is the rank-reflected copy on ranks12–15. The king and knights screen the major pieces from rear wrapping; seven of the pawns form front/flank screens, and the eighth leads on e7. The bishops c5/h5 are on opposite color classes. The h5 bishop has open development routes initially; every initial route stops before reaching an enemy. The9×18 choice avoids the nearly universal single-move bishop reach of some shorter quotients: on this empty board bishop destinations number31 from144 cells or16 from18 cells, within two81-cell color components. Its area162 and deployment are provisional design choices, not a claimed optimum.

G uses the same compact idea, with the second bishop at g2 rather than h2 because there is no global color restriction:

```text
I:rank4: . . . . P . . . .
I:rank3: . P . P P P . P .
I:rank2: . . B R Q R B . .
I:rank1: . P . N K N . P .
files:   a b c d e f g h i
```

Black occupies the half-turn image on II:rank6–9. The centered king and queen have distinct roles and remain on e under the half-turn. All five sliders begin locally screened. A conventional RNBQKBNR row would exchange queen and king files under this G symmetry; that rejected assumption is not used. The asymmetric staircase remains visible and movement through it is part of development.

### Castling: Möbius only

K/G have no castling. Their compact camps have no ordinary shared king/rook home rank, so grafting a rook teleport onto them would add an unrelated special route. The deliberate omission is limited to those two variants.

M has the named original king at f1 (White)/f9 (Black), original western rook b1/b9, and original eastern rook i1/i9. Castling uses only the short drawn home-rank segment, never a winding alternative. East: king f→h, rook i→g. West: king f→d, rook b→e. Both pieces move in one turn.

All cells strictly between the named king and rook must be empty. The king and selected rook must remain on their original squares with their respective rights. Moving the king permanently loses both rights. Moving the named rook or capturing it permanently loses that side's right. Returning to a home cell, substituting another rook, or promoting a new rook cannot restore it.

The source king must not be attacked. Separately test the king on its one-step transit cell and its two-step destination with the source vacated and the rook still at its home; neither may be attacked. Then move both pieces simultaneously and test the actual final king position again. All winding attacks count. The extra explicit final check handles a rook move uncovering another route; the explicit pre-rook landing check ensures the new rook position is not used to shelter a landing square that was attacked during the king's traversal. This is the selected surface castling convention. Ordinary moves never offer castling as a geometrically long king move.

## Legal games, state, repetition and endings

Scored play has exactly one king of each color; they are never removed. A legal action obeys its piece/special rules and leaves its own king unattacked. Attacks use geometric capture rules and blockers without first filtering a potential attacker for its own king safety. Thus pinned pieces still attack. Pawn advances and castling are not attacks. King attacks use one-step moves, not castling. These avoid recursive definitions of check.

The state contains variant/rules version; every occupied cell's owner/type and pawn-facing/unused-double-step flag; side to move; M's four persistent castling-right flags; the optional last-double-step trace; halfmove counter; ply count; the repetition record; and terminal result if any. Non-pawn facing and physical piece IDs are absent from the repetition key. The initial position counts once.

The repetition key consists of variant/rules version, the full placement including relevant pawn fields, side to move, castling rights, and en-passant rights only if the next player has at least one legal en-passant capture. The retained EP key gives the middle and vulnerable destination; distinct capture outcomes are not conflated. An expired record, or an EP record whose only apparent captures expose the capturer's king, is equivalent to none. Source IDs, move count, halfmove clock and earlier repetition counts are not part of a position key. Prior history still affects the game because occurrence counts decide the ending.

After every legal move, clear/rebuild EP, update rights, switch turn, reset the halfmove counter for any capture or pawn move (including promotion), otherwise increment it, then add the resulting key to history. Determine the result in this order:

1. No legal action and the player to move is checked: that player loses by checkmate.
2. No legal action and not checked: draw by stalemate.
3. Only the two kings remain: draw.
4. The current key has now occurred at least5 times: automatic draw.
5. At least150 consecutive halfmoves without a pawn move or capture: automatic draw.
6. Otherwise continue. Players can agree a draw or resign in any ongoing game; resignation loses. No pass or clock/timeout rule is included in this untimed first implementation.

The fivefold/150-halfmove choices retain the automatic thresholds of FIDE Article9.6. Claimable threefold/50-move draws are omitted. General FIDE dead-position adjudication is deliberately not imported: apart from the proved bare-kings case, positions that cannot mate continue to an agreed, repetition or move-count draw. No orthodox insufficient-material lookup is asserted on these surfaces. Bare kings is safe because kings' adjacency attacks are reciprocal, so no legal king move can give a check to the other king. Checkmate takes precedence over an otherwise simultaneous automatic draw. Continuous checking has no special loss rule in chess.

Termination follows independently of the150-halfmove rule. There are finitely many placements of at most32 pieces, types, colors, pawn facings/unused rights, castling rights, EP traces and turns. Promotion replaces rather than adds a piece. Any indefinitely continued legal game would revisit a key infinitely often; the fifth occurrence instead ends it. Pawn cycling therefore cannot defeat termination even though pawn moves reset the counter.

## Player exchange including state/history

M/K use Φ(s,x,y)=(s,x,H−1−y); G uses Φ(s,x,y)=(1−s,8−x,8−y). Exchange White/Black, reverse every pawn facing, preserve pawn unused flags, exchange side to move and winner, and transform every EP trace cell by the same Φ. On M the east/west labels remain east/west and their owner rights exchange. K/G rights are always absent. Halfmove/ply counters remain unchanged. Transform every past position and move with the same map in its original temporal order; never reverse or independently canonicalize history.

Each map is involutive. It commutes with all cardinal edges and their derivatives (with local direction map (dx,−dy) on M/K and(−dx,−dy) on G), hence with the two-order diagonal test, all eight L,S,L knight words, blockers and rays. Pawn facing reversal maps forward routes to forward routes. All promotion targets and initial placements are exchanged. M's explicit castle source/path/final tests also map cell for cell. Captures, king safety, rights updates, EP duration, repetition equality and ending precedence consequently commute with Φ. This is an equal-treatment certificate before choosing who moves first, not a proof of equal first/second winning chances.

## Development examples

These are legal illustrative lines, not opening recommendations or minimax results. Long coordinate notation avoids ambiguity from seam routes.

* M:1.e2–e4 e8–e6 2.Bd1–g4 Bd9–g6 3.Nc2–d4 Nc8–d6. The pawn advances release both bishops. The bishops then bear on the opposing central pawns along g4–f5–e6 and g6–f5–e4; the knights add central pressure.
* K:1.e7–e9 e12–e10 2.Nd4–c6 Nd15–c13 3.Bc5–d4 Bc14–d15. The leading pawns contest the central file; rear knights vacate diagonal exits for the screened bishops. Choosing the rear bishop routes shows that the compact camp can develop in both directions on a closed board.
* G:1.I:e4–I:e5 II:e6–II:e5 2.NI:d1–I:c3 NII:f9–II:g7 3.BI:c2–I:d1 BII:g8–II:f9. The leading pawns meet across the I:e5/II:e5 seam and block one another's direct advance. Knights then clear the rear bishop exits. A different legal first move is the double step I:e4–II:e5 through intermediate I:e5; it is a distinct seam-crossing thrust and is recorded separately in the fixtures.

## Baseline source and chosen departures

Primary baseline read directly: [FIDE Laws of Chess effective1 January2023](https://handbook.fide.com/chapter/E012023), especially Articles3,5,9. The borrowed baseline is ordinary piece inventory and movement identity, capture blocking, pinned-piece attacks, mandatory promotion choice, en passant, king safety, checkmate/stalemate, castling's lost rights, and automatic fivefold/75-move draws. The surface geometry, deployments, pawn targets/facings, trace-based EP, exact surface castling safety, K/G castling omission, bare-kings-only immediate material draw, omitted claimable draws and untimed play are explicitly this proposal, not FIDE regulations for these boards.


## Saved games and edited setups

JSON game records preserve the initial position or explicitly labelled edited setup, every action, special rights, repetition counts, and the final result. Loading replays the complete record and rejects inconsistent final data. There is no arbitrary move-count import limit. Very long records take proportionally longer to validate. Setup mode begins a new history and does not claim that an edited position occurred in a game.

The application provides untimed local play for two people or an optional [local practice opponent](./opponents.md).