Chess on a Folded Magnetic Board

Back in 2021, we got a submission for a game the author called Folding Magnetic Chess. But he hadn't invented this game, and he didn't know if it was officially known by any name. The description he gave for the game was fairly bare bones and did not go into detail. In 2025, we got a submission by Peter L for a game he called Throw Pillow Chess. This game turned out to be the same game, though described in much more detail. Peter also discovered that an earlier game called Pillow Chess was the same game. Grant Cairns described the game in an article called Pillow Chess that appeared in Mathematics Magazine, Vol. 75, No. 3, June 2002, but it focused more on the mathematics of the game than it did on the mechanics of play. Since it is not practical to play the game on a pillow, but it can be played with a folded magnetic Chess set, Folding Magnetic Chess seems to be the best of these three names. However, that name has the unfortunate side effect of suggesting it is a variant of Magnetic Chess, which is a completely unrelated game based on the idea of magnetism. To avoid this association, I suggest calling the game Chess on a Folded Magnetic Board.

Setup

This game is played on a board that is folded between the two middle files, called d and e in algebraic notation. My own magnetic board folds between the 4th and 5th ranks. While it can be rotated 90 degrees to play this game, the light and dark squares are reversed upon doing this.

When folded, the a and h files will connect in the same way the d and e files do, as they do in Cylindrical Chess. Additionally, the a and h files will loop vertically with each other, and so will the b and g files, the c and f files, and the d and e files. Mathematically speaking, each rank will loop into itself, and each file will loop with another file. This can be seen in this photo to which I have added rank and file markers:

Spherical Chess does the same kind of thing, and both Grant Cairns and Peter L. have conceived of this game as a kind of Spherical Chess. The main difference concerns which file each file loops with. In Spherical Chess, ranks and files are like lines of latitude and longitude, and like lines of latitude, the files all meet up at the poles. But in this game, ranks and files are like perpendicular lines of longitude, and there are no poles.

While the game can be played on a board like shown in the photographs above, it would be more convenient to play it on a regular Chess board with the understanding that the board is supposed to have the 3D topography of a folded Chess board. In his article on Pillow Chess, Grant Cairns used an extended diagram of the sort below, though for a problem rather than the opening setup. This extended diagram of the opening setup shows how the edge spaces on a regular Chess board connect with other edge spaces that are brought together by being folded together.

The central board is the actual board, and the other boards are just reflections of the same board, showing how the spaces on the board connect to each other. A counterintuitive detail shown in this diagram is that each corner square on a folded magnetic board is diagonally adjacent to itself. These include not only the four regular corners (a1, a8, h1, and h8) but also the four middle corners (d1, d8, e1, and e8). Without using an expanded diagram, Peter L. defined a diagonal step as a move to a square that may be reached through two orthogonal steps that are perpendicular to each other. Even on the physical folded magnetic board, this would result in each corner square being diagonally adjacent to itself. This is because a corner square is doubly adjacent to another corner square, once horizontally and once vertically. For example, a1 and h1 are horizontally adjacent due to the rank they are on looping back into itself, and they are vertically adjacent due to the file each is on looping into the other. This means that each corner square is two perpendicular orthogonal moves away from itself.

The main consequence for Bishop moves is that a move to one of these corners could reflect back onto the same path that led there, effectively making it a dead end for diagonal moves, just as corner squares are in Chess. The only other consequence is for Knight moves, and this mainly just results in an equivalency between different ways of describing the Knight's move. Further details on these points will be given in the piece descriptions.

Pieces

In general, each piece moves as it does in FIDE chess. However, the board shape can lead to unexpected results, so each piece will be reviewed below.

Pawn

White pawns promote upon reaching the 8th rank, and black pawns on reaching the first. The only quirk of the board they have to worry about is how the left edge is glued to the right.

Rook

The Rook moves any number of spaces orthogonally. It can reach space on either side of the Knight, because the rank it is on loops back on itself. Thanks to the c file looping with the f file, it can also move to the f file, but it may not go past the pieces standing in its way.

The rook can reach 22 squares on an empty board, regardless of its position. Any square it can reach it can reach in at least two ways.

Bishop

The expanded diagram makes it easier to visualize how Bishop moves wrap around the board.

Diagonal steps may be understood to comprise an orthogonal step followed by an orthogonal step in a perpendicular direction. A diagonal line comprises a zigzag of orthogonal steps. (This is one possible definition of "diagonal" on the FIDE board, but it isn't the only viable FIDE definition. However, not all definitions that are equivalent on the FIDE board are equivalent on the folded magnetic board, hence the specification.) Note in particular that "corner squares": a1, a8, d1, d8, e1, e8, h1, and h8 are diagonally adjacent to themselves, so that a bishop traveling c3-b2-a1 slingshots around the 3D corner of the board to continue traveling a1-b2-c3. This behavior is irrelevant for bishops, since it doesn't allow any new squares to be reached (but see Notes for further discussion). 

The main upshots here are that

1) a1 and h1 are NOT diagonally adjacent, although at first glance they might appear so, since they are connected by a "corner" of the pillow.

2) when a bishop travels off the top or bottom of the board, it always returns rotated 180 degrees with respect to the board.

The bishop is quite powerful. On an empty board, it can reach 19 squares if it's on a diagonal containing a corner (a1, d1, e1, h1, a8, d8, e8, or h8), or 23 squares elsewhere. Any square it can reach that isn't on a corner diagonal can be reached in two or, in some cases, four ways.

Knight

This diagram is expanded to only the first two ranks or files, because the Knight's move does not extend any further. So that you can tell them apart, each Knight's legal moves are shown in a different color.

The Knight's move can be thought of as an orthogonal step followed by an outward diagonal step, a diagonal step followed by an outward orthogonal step, two orthogonal steps in one direction followed by another in a perpendicular direction, or one orthogonal step in one direction followed by two more in a perpendicular direction. All of these definitions are equivalent on the folded magnetic board. The Knight can reach 8 squares from the 3rd, 4th, 5th, and 6th files, and can reach only 7 from the 1st, 2nd, 7th, and 8th. Here, the black knight's moves are all unaffected by the strange board except its two forwardmost: the knight travels up through e8 to d8, then moves one perpendicularly to either c8 (overlapping with its left-two-up-one move) or e8. When the white Knight travels two squares left, it ends up on h1, then turns one square up to h2 or down to a1. When it travels two right to d1, it can move one up to d2 (a normal move) or one down to e1. When it travels two squares down, to g1 and then g2, it can then move perpendicularly to f2 or h2 (overlapping with its left-two-up-one move).

Queen

The Queen moves as a Bishop or a Rook. The Queen, which is on the d file, can also move along the e file thanks to them looping into each other. Unlike the Bishop diagram, this one shows diagonal movement directly toward corner squares. If the Queen goes to a1 or h8 along the a1-h8 diagonal, any continuation of its move will just cover spaces it already passed over, making the continuation redundant. In this way, these corner squares are terminal squares. But there is also a path by which the Queen goes to a7, then to h8, then to b8, then can move along a diagonal to h2, then goes to g1 and back to the Queen. In this instance, movement continued through the corner square h8, because that movement was not going directly toward the corner.

The Queen can reach 36 or 37 squares on an empty board depending on if it's on a diagonal containing a corner.

King

The King can move one space in any direction. In this diagram, each King's legal moves are shown with different colors. Black's King has the full eight legal moves. The three additional moves it wouldn't have at this position in Chess are due to the f, g, and h files looping with the c, b, and a files. White's King has only six legal moves. This is because it is on what would be a corner space when the board is folded. Since this space is diagonally adjacent to itself, one of the diagonal moves from it leads directly to it, but the King cannot move where it is already is. Another move is lost due to two moves being redundant. The King may move to d1 through either the usual horizontal move or through a vertical move allowed from the e file looping with the d file.

Rules

Peter L. added the following rules to Throw Pillow Chess:

• A null move is not legal. This means a piece may not take advantage of the looping topography to move back to the space it started from.
• There is no castling.

Making null moves illegal makes sense, because Chess has no provision for letting a player pass his move, and a null move is essentially a pass move. His rationale for no castling is that the Rooks are already connected. However, Pillow Chess and Folding Magnetic Chess do not mention any changes to the rule of castling, and there are two other options. One is to keep castling as it is in Chess, because that is the rule in Chess, and this game is supposed to be an adaptation of Chess to a different topography. The other option is to use the rule found in some versions of Spherical Chess, which is that a King may castle with a Rook by moving toward it in either direction so long as all the other conditions for castling hold. So, for example, if a1 through d1 are all empty, the King could castle with the h1 Rook by moving to c1. Likewise, if f1 through h1 were all empty, the King could castle with the a1 Rook by moving to g1. In each case, the Rook would move to the space the King passed over.

Notes

Fergus Duniho's Notes

This page was written mainly by Fergus Duniho, who also took the photos. But it also incorporates sections that Peter L. wrote for Throw Pillow Chess, and the piece diagrams are all based on the diagrams he originally made. To better fit the magnetic set theme, I changed the piece set from Alfaerie to Magnetic.

Peter L.'s Notes

All riders get a significant benefit from the spherical board. Not only can they reach a lot more squares, but they are twice as hard to block. Knights could probably be given at least an additional one-step orthogonal move (NW in funny notation) to compensate. But, that is a step away from FIDE rules.

Many topological variants, including cylindrical and toroidal chess, have issues with forcing mate in endgames due to a lack of edges to trap the king against. This is far less of an issue in Throw Pillow chess. While there are no "edges" as such, a king on the first rank, even with six or nine possible squares to move to, finds that all those squares are on the first and second rank. This, combined with the rook's control of two "separate" files, means king + rook can force a bare king to the side of the board to checkmate. A bishop pair can easily force mate, without even bothering to chase the king to the board edge. The possibility of these forced mates is the main reason to forbid null moves, in my opinion, even if they seem like a fun quirk of the board.

An alternative definition of "diagonal step" is to consider a diagonal step to consist of a step from one square to another square that shares a corner, but is not orthogonally adjacent, and a diagonal line to consist of a series of diagonal steps where the corners crossed by each step do not share an edge. This alternative is equivalent to the definition given above on the FIDE board, and, indeed, it is also equivalent for FIDE pieces on the Throw Pillow board. However, it would mean that corners are not considered diagonally adjacent to themselves (and would mean that true corners of the board that forced bishops to stop would exist, but without any edges that could stop rooks!). With the official Throw Pillow definition of diagonal, a vao on b1 could capture a piece on a1 by traveling southwest, hopping the piece, then coming back northeast to land on a1 and capture it. Using this alternative definition, that move would not be allowed.

On a mathematical note, it isn't surprising that spherical boards are so tricky to design. The fundamental issue is this: chess boards are flat. Spheres aren't. This sounds silly and reductive, but is in fact a difficult hurdle to overcome. Of the variants covered on the Spherical Chess page, the one with the most consistent bishop movement is Nadvorney's Spherical [sic] Chess, in which the board is actually a Klein bottle. It is no coincidence that the Klein bottle (and the torus, another popular board shape for chess variant designers) are, in fact, flat surfaces. By this I mean that their Euler characteristics are 0. This is the same as for an (unbounded) square grid: if one were to count up the vertices, edges, and faces of a square grid, one would count, per grid cell, one face (the square), four edges (the sides), and four vertices (the corners). Each edge is double-counted since it separates two cells, and each vertex is quadruple-counted since it is shared by four cells. So, the Euler characteristic per cell is V - E + F = 4/4 - 4/2 + 1 = 0. Thus an boundaryless square grid fits naturally onto a Klein bottle or torus.

Unfortunately, spheres have an Euler characteristic of 2. This means that at least one cell of a spherical chess board must be "weird" in some way. It can have fewer or more edges or vertices, becoming a triangle or other non-square polygon. But this leads to serious problems with defining consistent and intuitive movement through these cells (how does a rook travel into a triangle and exit the orthogonally opposite side? what does it mean that a king can make two orthogonal moves from a hexagon and end up still orthogonally adjacent? etc). 

Alternatively, we can dispense with the assumption that each edge is shared by two distinct cells and each corner by four. This can be done by glueing two edges of a single cell to each other (what I think of as the "waffle cone" strategy), or glueing two edges of one cell to two edges of a second cell  (the "pillow corner" strategy). In Spherical Corner Chess, both strategies are used: a1 and h8 combine to make a pillow corner, and a8 and h1 each become waffle cone points. In Throw Pillow Chess, eight squares in pairs create four pillow corners.

By messing with which edges are glued to each square while maintaining their essential "squareness," standard ideas of orthogonality and diagonality can be maintained reasonably consistently. In particular, a diagonal movement can be constructed as two perpendicular orthogonal movements. This makes knight movement much more robust, since it can be considered as an orthogonal step followed by an outward diagonal step, or a diagonal step followed by an outward orthogonal step, or two steps rookwise followed by a perpendicular orthogonal step, and each path will lead to the same destinations (which is not so for most of the Spherical Chess variants).

Last revised by Fergus Duniho.

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Chess on a Folded Magnetic Board #22903 by Fergus Duniho on 2025-05-24 17:45:25 as HTML