Two Chess Geometries

Two chess variants, played on the chess board.

I can’t lay claim to the first, but I’m pretty sure that the second is my own creation – I certainly came up with it myself, but I don’t know if anyone else has thought of it in this form before. There is mention of something similarly-inspired on the Chess Variants Pages, but the game is very different to mine, and you need special equipment!

These variants both use the same board and pieces as regular chess. The movement and capture rules are the same as chess. The difference lies in the geometry of the board, where we make edge identifications to alter the space the game is played on. The first variant is called Cylindrical Chess, the second is Möbius Chess.


Cylindrical Chess

We identify the left and right edges of the board to make a cylinder.

Edge idents

Edge identifications for cylindrical chess

Pieces that move off the edge of the board arrive back on the other side of the board in a continuation of the straight line they were travelling on (of course), as it would be on a cylinder.

Diagonal moves

Diagonal movement on the cylindrical board. Different colours show the continuous (straight!) diagonals.

Of course, this means that cover is sometimes harder to find in the centre of the board.

Cylinder wrapping capture

A white rook captures a black pawn around the cylinder. That white pawn was no obstacle! Also rooks are super powerful, yo.

In general, since the edges wrap, the centre dynamic is drastically changed – it is now not crucial to have mobility from the centre, because now any file is as central as any other. It is now just important to get your pieces out anywhere into the middle ranks. Additionally, the edge pawns now have the same mobility as all the others, and the two rooks are automatically protecting each other. This makes castling less important and in fact it makes sense to play cylindrical chess without the castling rule, since this also removes the problem of deciding whether you are allowed to castle ‘around the cylinder’. Of course there’s no problem with en passant captures wrapping around the cylinder – as long as you remember to notice them!

White to mate in two

White to mate in two. Cylindrical rules.

It’s harder to achieve a checkmate under some circumstances, since now there are no corners to box the king into (though if your opponent knows what they’re doing, this is less likely to be relevant).

Mate in three

White to mate in three.

This game plays very well, it’s sufficiently similar to chess to be familiar, but sufficiently different to be surprising. The periodic boundary conditions make for a very different dynamic during play – the relative values of the pieces are quite different, but not in an overly unbalanced way. I like cylindrical chess as much as regular chess!


Möbius Chess

As you can probably guess, this time we identify the sides of the board as before but in the opposite orientation.

Edge identifications

Edge identifications for Mobius chess.

Again, pieces that travel off one edge of the board arrive back on the other edge in a continuation of the straight line they were travelling on. This requires a bit of thought (and perhaps drawing a straight line on a möbius band to test), because a straight line on such a surface can look very strange when the space is unfolded back down to a square – particularly for diagonals.

Straight diagonals

Straight diagonal lines now ‘reflect’ back towards you (and change square colour) when they cross an edge! I know it looks like you can bounce off the back wall in this diagram, but you can’t – that right angle is actually the opposite ends of the diagonal meeting, as happens for every diagonal.

“But wait!” I hear you cry, “Then the game begins with the rooks automatically attacking each other!”. I just smile gently, knowingly, and present you with the alternate starting configuration for möbius chess.

Revised starting configuration

Starting configuration for Mobius chess. Can also be played with rooks and bishops interchanged, or rooks and knights.

Yes, it looks a bit unwieldy, but it has important features:

  • The rooks are hard to mobilise, requiring at least as many moves to bring into effective play as in regular chess.
  • Similarly the bishops are behind a line of pawns and are in as central a placement as normal.
  • The knights enjoy the same instant mobility as always. The start closer to the enemy but moving them is the double-edged sword of freeing your queen and exposing your king.
  • The ranks closest to your side are So. Powerful. They provide insta-access to your opponent’s pieces, like a magical portal. But the same is true of your opponent’s home ranks – they provide access to your formation. The central ranks are less powerful in that they only allow travel amongst themselves. This particular configuration ensures that (a) it is dangerous/time-consuming to attempt to get your powerful pieces back down into your own ranks where they can exploit this, and (b) if your opponent happens to manage this, the presence of your vertical ‘pawn wall’ means that their first invading move is likely to be a costly sacrifice.
  • Of course, the whole point is to remove the auto-attacking that would occur if we kept the standard configuration – it makes the damn thing playable.

I suggest that ‘castling’ (of a sort) be allowed, where castling with the king’s rook simply interchanges them, and castling with the queen’s rook places the king in the rook’s place and the rook in the queen’s. Since getting the rooks to protect each other is mostly irrelevant, the real purpose of this move is to get the king off the diagonal ‘entryway’ into your formation that the knights are blocking and into the relative protection of the pawn wall.

Starting two ranks closer to each other other means earlier fighting and less manoeuvring, particularly in that ‘central’ area. Capturing is pretty crazy in this game, and victory is more likely to be achieved through somebody making an error rather than through tactical prowess, at least until you get used to it:

Capture wrapping

That white pawn was no cover at all. Take that, black pawn!

So this game feels quite different to chess. As I mentioned, the closeness makes battle more pronounced. Diagonal moves are somewhat less powerful but still very surprising. Knights are like fireworks of death popping back and forth all over the place. Watch out for a couple of opening gambits with them, they can catch you unaware. The board feels smaller somehow, and I would even consider moving this to a ten by ten board.

Mate in three

Black to mate in three. Silly white!

It’s not improbable that you will miss checks and checkmates everywhere until you get used to it.

Mate in two.

Black to mate in two.

I can’t really find other people to play this with very much. You have to commit to it pretty severely to get through the first few games, and I don’t think I’ve played enough of them to have a clear picture of whether or not this game is actually any good. It feels pretty perverse for some reason – blasphemous, almost.


Questions? Comments? Know any more geometry-inspired variants? Any more cylindrical or Möbius puzzles? Also big thanks to ChessVideos for their chess diagram generator. Awesome stuff.

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