Does anyone know the name of the chessboard variant where you fold a standard board along the line between the d-file and e-file, to get a double-sided 8x4 board? A piece on top can move past the edge of the board, in which case it flips and continues underneath. And vice versa. Otherwise the starting position is the same, and the rules are the same.
Many chessboards do fold this way. And if it's magnetic, then the pieces can stick to both halves, so you can play the variant as is. Folding magnetic sets are hardly new or rare -- I've had a cheap one for 30+ years.
chessvariants.com and David Pritchard's Classified Encyclopedia of Chess Variants have all sorts of other wacky warped chessboard variants. But I haven't been able to find this one there, or anywhere else online.
I don't remember this variant being called anything particular, we just "played on a folded board". Has anyone else here heard of it?
Here is an equivalent game that you could play on an 8x8 unfolded board. You might find it more convenient, even if you do have a folding magnetic chess set -- the folded board just shows you what the rules should be.
The left and right edges are considered to be joined, so that a-file and h-file are adjacent (just like in cylindrical chess). But also, the left half of the top edge joins to the right half of the top edge, in such a way that continuing up from a8 goes to h8, b8 goes to g8, c8 goes to f8, and d8 goes to e8, and vice versa. The bottom edge joins to itself as well, so that going down from a1 goes to h1 and so on. That is, the a-file and h-file form a single 16-step loop, as do b-g, c-f, and d-e (just like the 4x16 rings in circular chess).
You can check that these rules are equivalent to the folded variant.
On this board, you can check that from 56 squares the king has 8 distinct moves. From 8 corner squares (a1,d1,e1,h1,a8,d8,e8,h8), two of the eight moves end up on the same square, and one of them ends up where it started. For example, from e8 its moves are, starting from 3 o'clock and going clockwise: f8, f7, e7, d7, d8, "d9", "e9", and "f9". The first five are the usual moves, "f9" is c8, "e9" is d8 (redundant), and "d9" is e8 (null move).
On the other boards, the king has 8 distinct moves from all 64 squares. Therefore they are not equivalent variants.
(Those "spherical" chessboards are actually not simply connected. The Encyclopedia does list some spherical variants, with boards that are simply connected, compact surfaces without boundary. But the ones I saw had 66 or more cells, so obviously they are not equivalent to this.)