# Shogi on three surfaces — experimental rules v0.2

These proposed variants use a full twenty-piece shogi army for each player on a Möbius strip, a Klein bottle, or an orientable surface of genus two, written $T^2\#T^2$. Their rules and starting positions are specified below. Balance has not been established.

Player A moves first. In a game, players alternate one board move or one drop from hand. Ownership and facing are separate: a piece belongs to the same player after crossing a seam, even if its forward direction changes.

## Coordinates and boards

Files are **a–i**, left to right. Ranks increase from the bottom of each diagram. Single-board coordinates are written **a1**, **i11**, and so on. The genus-two board uses two labelled sheets: **I:a1** means file a, rank 1 on sheet I; **II:a1** is a different square. Sheet names I and II are not player names.

| Variant | Cells | Side edges | Top and bottom |
| --- | --- | --- | --- |
| Möbius | 9 × 11 = 99 | A side crossing sends rank *r* to *12 − r* and reverses facing. | Boundaries. |
| Klein | 9 × 18 = 162 | A side crossing sends rank *r* to *19 − r* and reverses facing. | Wrap without reversing facing. |
| Genus two | Two 9 × 9 sheets = 162 | Each sheet wraps; the marked staircase edges instead join the sheets. | Each sheet wraps, with the same staircase exceptions. |

A seam is an adjacency, not a gap or an extra square. The diagrams fix the drop arrows, coordinates, and promotion markings used by these games.

### Möbius and Klein transport

A side crossing reflects the whole movement displacement, including its vertical component, and reverses the piece's facing. A continuing diagonal ray then follows that reflected direction. On the Klein board, a rank beyond the top or bottom is wrapped after reflection. For example, a gold at **i18** facing increasing ranks can move diagonally forward across the Klein corner to **a18**, arriving facing decreasing ranks; that move does not lead to **a2**.

The Möbius middle rank is rank 6. A gold can walk around it over nine moves and return to its starting square with reversed facing. Unpromoted pawns and lances cannot cross a side seam because they do not change files.

The 9 × 18 Klein dimensions reduce unrestricted bishop reach compared with 9 × 14. On an empty 9 × 14 board, a bishop reaches all 62 other cells of its color component in one move. On the chosen board it reaches 31 destinations from 144 starting cells, or 16 from the other 18. Successive bishop moves still connect all 81 cells of either color component. No artificial range limit is added.

### Genus-two identifications

Crossing a genus-two seam translates the movement direction; it never reverses facing. The ordinary outer edges wrap within their own sheet. The following marked edges replace ordinary adjacency. A range such as **I:a4–d4** means one transition for each matching file.

| Northward crossing from | Arrives at |
| --- | --- |
| 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 |

| Eastward crossing from | Arrives at |
| --- | --- |
| I:d5 | II:e4 |
| II:d4 | I:e5 |
| I:e6 | II:f5 |
| II:e5 | I:f6 |

South and west reverse the corresponding north and east transitions. All other adjacent edges remain within their sheet, wrapping at its outer edges.

These identifications form a connected closed surface with 162 square faces, 324 edges, and 160 vertices. Its Euler characteristic is −2, giving genus two. At two vertices, eight square corners meet instead of four; each is a **cone point** with total angle 4π.

A diagonal must stop before a cone point. To determine whether a one-square diagonal is allowed, compare its two cardinal orders, such as north-then-east and east-then-north. If they reach the same square, use that square. If they disagree, the diagonal is forbidden. This rule applies to every piece with a diagonal move. Pieces in the two side-adjacent squares do not block a diagonal. Ordinary turns in the drawn staircase are not automatically cone points: for example, northeast from **I:d4** legally reaches **I:e5**.

A knight follows **forward–sideways–forward** through the adjacent cells to determine its destination. It still jumps intervening pieces. This is the edge-crossing order of its straight center-to-center segment when the charts are unfolded; the segment does not pass through a grid vertex. For example, an A knight at **I:a3** jumps leftward to **I:i5**, not **II:i4**.

## Initial armies

The rows below show player A from left to right. A dot is empty. Each army has one king, rook, and bishop; two golds, silvers, knights, and lances; and nine pawns.

| Row | Möbius and Klein | Genus two |
| --- | --- | --- |
| Back | `L N S G K G S N L` | `L N S G K G S N L` |
| Middle | `. B . . . . . R P` | `. B . . . . . R .` |
| Front | `P P P P P P P P .` | `P P P P P P P P P` |

| Variant | A back / middle / front | B back / middle / front |
| --- | --- | --- |
| Möbius | 1 / 2 / 3 | 11 / 10 / 9 |
| Klein | 4 / 5 / 6 | 15 / 14 / 13 |
| Genus two | I:1 / I:2 / I:3 | II:9 / II:8 / II:7 |

B's army is A's half-turn image: reverse the files and ranks, exchange owners, and reverse facing; on the genus-two board also exchange sheets. In particular, B's rook is on the left and bishop on the right. A initially faces increasing ranks, B decreasing ranks.

The rook-side corner pawn of each Möbius and Klein army guards the middle-row side seam; the bishop-side pawn remains in front to screen its diagonal. The Klein armies also leave space around the vertical wrap. The complete initial arrangements and rules have player-exchange symmetry; that does not remove the distinction between moving first and second.

## Piece movement and captures

Use the following shogi patterns relative to current facing, with the seam and cone conventions above. [Japan Shogi Association: piece movement](https://www.shogi.or.jp/knowledge/shogi/03.php).

| Piece | Movement |
| --- | --- |
| King, K | One square in any direction. |
| Rook, R | Any unobstructed horizontal or vertical distance. |
| Bishop, B | Any unobstructed diagonal distance. |
| Gold, G | One square forward, diagonally forward, sideways, or straight backward. |
| Silver, S | One square forward or along any diagonal. |
| Knight, N | Two forward and one sideways, jumping intervening pieces. |
| Lance, L | Any unobstructed distance forward. |
| Pawn, P | One square forward. |

Promoted pawns, lances, knights, and silvers move as golds. A promoted rook adds one-square diagonals; a promoted bishop adds one-square orthogonal moves. Kings and golds do not promote.

A slider stops at the first occupied square. It can capture an enemy non-king piece there, but cannot pass that piece or enter a friendly occupied square. Captured pieces enter the capturer's hand as their unpromoted base type. These capture and hand conventions follow [official shogi rules, Articles 4–5](https://www.shogi.or.jp/match/taikyoku_rules/taikyoku_rules.pdf).

No move may return to its departure square, even with changed facing. There is no pass and no slider range cap. A ray ends at a boundary, blocker, forbidden cone crossing, or repeated directed state. On the genus-two board, direction is globally preserved, so any repeated cell ends that ray. On the Klein board, another empty cell can be revisited with a different transported direction, allowing the ray to continue.

Distinct clear routes can reach one square. If they yield different final facing, the chosen outcome must be specified. Blocking one route does not necessarily block another.

## Promotion

A promotable board move may promote if its starting or ending square is in the applicable promotion band. Merely passing through it does not suffice. This uses the ordinary start-or-end eligibility convention while replacing the enemy camp with the markings below. [Official rules, Article 5](https://www.shogi.or.jp/match/taikyoku_rules/taikyoku_rules.pdf).

| Variant | Promotion band |
| --- | --- |
| Möbius | A: ranks 9–11. B: ranks 1–3, regardless of current facing. |
| Klein | A: ranks 13–15. B: ranks 4–6, regardless of current facing. |
| Genus two | A: sheet II ranks 7–9. B: sheet I ranks 1–3. |

On the Möbius board, a pawn or lance must promote on its last forward rank. A knight must promote on either of its last two forward ranks. Use the facing after seam transport. **This compulsory promotion overrides camp eligibility:** a knight crossing from i8 to a2 arrives facing decreasing ranks and must promote, even though a2 is outside A's promotion camp. These rules prevent immobility on the bounded strip.

The Klein and genus-two boards have no dead ranks, so promotion is always optional. Drops are never promoted immediately, although a later move beginning in the appropriate band can promote. Once promoted, a piece remains promoted until captured.

## Drops, files, and king safety

A drop takes one unpromoted piece from hand and places it on an empty square. It always uses the owner's marked default arrow: **A toward increasing ranks, B toward decreasing ranks**, on either sheet when applicable. The prior captured facing is discarded, and the player cannot choose the opposite drop facing. On the Möbius and Klein diagrams this is a deliberate convention tied to the markings; subsequent travel can reverse the piece.

Retain **nifu**: a player cannot drop a pawn into a file already containing one of their unpromoted pawns, regardless of its facing. Promoted pawns and opposing pawns do not count. Retain **pawn-drop mate**: a dropped pawn may give check but cannot give immediate checkmate. A pawn moved on the board may give mate. [Japan Shogi Association: illegal moves](https://www.shogi.or.jp/knowledge/shogi/05.php).

| Variant | One file for nifu |
| --- | --- |
| Möbius | All 11 cells with the same file letter. |
| Klein | All 18 cells with the same file letter. |
| Genus two | All 18 cells with the same file letter across both sheets. |

On genus two, these are the nine cycles obtained by repeatedly stepping forward; each cycle visits both sheets. Thus an A pawn on **I:a3** prevents an A pawn drop anywhere in file a on either sheet. Drops bypass the staircase crossings, but not this restriction. On the Möbius board, pawn, lance, and knight drops into their dead ranks are also forbidden.

A legal game move or drop must leave its owner's king unattacked. Kings are never captured. Attacks are calculated from movement geometry and blockers, without first asking whether each attacking piece is pinned. A checked player must evade every attack, including every clear winding route. [Japan Shogi Association: check and mate](https://www.shogi.or.jp/knowledge/shogi/04.php).

## End of a game

The following complete ending conventions are **proposed rules for these variants**:

1. A player with no legal move or drop loses, whether checked or not. Resignation also loses.
2. Count the initial position as its first occurrence. On the fourth occurrence of the same position, adjudicate immediately.
3. Between the first and fourth occurrences, if exactly one player gave check on every one of their moves, that player loses. Otherwise the result is a draw, including the case where both players continuously checked.
4. Players may agree to a draw. There is no entering-king material-point declaration and no automatic draw from merely reaching a promotion camp.

A repeated position has the same variant, piece types, owners, promotion states, occupied squares, both hands, and side to move. Facing also counts when it affects movement. King, rook and bishop facing do not count, whether promoted or not: their movement and fixed-camp eligibility are independent of facing. Their displayed arrows still record transport, so two visually different arrows may represent the same rule state. Piece identifiers and move numbers do not count.

Official shogi uses fourfold repetition and penalizes continuous checking, but normally resolves ordinary repetition by replay. The immediate draw, explicit both-checking case, no-legal-action loss, and omission of entering-king claims above are variant decisions. [Official rules, Articles 8–10](https://www.shogi.or.jp/match/taikyoku_rules/taikyoku_rules.pdf).

## Analysis and editing

A scored game alternates turns and enforces king safety. Analysis controls may allow one person to move either army or disable king-safety filtering; other movement, capture, promotion, and drop restrictions still apply. Such relaxed actions are experiments, not legal moves in the scored game.

The editor permits artificial positions, including positions without kings. Missing kings have no king-safety check to apply, and an edited position need not be reachable from the initial setup. Genus-two facing stays fixed to the owner's direction because all its seams preserve direction. Save or export the setup and any game record needed to reproduce an experiment.