The game is called "Möbiatus", named after the unusual topology of the board. It can be simulated with two parallel boards on a flat sheet of paper, but it's more visually interesting this way.
This is a pencil-and-paper sort of Checkers from outer space. The topology of the board, which starts out confusing, gets weirder as the game progresses. The goal is to capture or immobilize all opponent pieces. The game must conclude within 70 move-pairs.
To set up the game, you'll need uniformly double-sided graph paper (cutting along the lines on one side cuts along the lines on the other- most good graph paper is like this), a pair of scissors to cut it, tape to join the board, and a differently-colored pen for each player.
Firstly, construct the board. Cut a 5x31 grid-square strip of the graph paper, then twist it into a Möbius strip (and tape it that way), carefully lining up the squares. Allow a one-square overlap- if you prefer to tape it without the overlap, cut it 5x30. (I want a smaller board, but anything less than a 6:1 ratio can't easily be twisted into a Möbius strip.)
"Place the pieces" on the board. Pick any horizontal line of squares ("horizontal" means "the thin way") and put a Red dot in each (use whatever two colors you want, I'll call them Red and Blue). Even though the Möbius strip only has one side, each square has an opposite anyway- what you get by forgetting that the strip is one-sided and considering it the back side anyway. You know what I mean. Place Red circles in each square opposite to the Red dots. Make another line of Red dots and corresponding circles, then two lines of Blue dots and corresponding circles. This is the start position.
A bit of notation. A dot indicates that there is a piece of a particular color in that location. A circle indicates that there is a piece of that color in the opposite location. (This is important, because captures occur from the opposite location!) A cross indicates a singularity, left behind and opposite to every moving piece. They'll be explained later.
On each turn, a player selects one of xir pieces and makes any legal "move" with it. Legal moves are explained later. Draw the piece in its new location, and place a circle opposite to it. Cross out its old location and the location opposite to its old location; these are now singularities. If a player has no legal move, or no pieces, that player has lost the game.
Pieces "move" vaguely like chess knights, two spaces one way and then one space at a 90-angle, jumping over the interveing spaces. A piece may not, under any circumstances, land on an occupied space; these are considered blocked, regardless of which army the blocking piece is from. Piece captures are NOT made by direct placement. A piece MAY, however, move onto a singularity- sort of. A singularity is a sort of contracted hole in space- if a piece would land on a singularity, it may land on any space adjacent to it (not diagonally). It may "slide" in this manner onto another singularity, where it "slides" again, but it must eventually come to rest. If a singularity or group of singularities (which sort of act like a single singularity) are surrounded by pieces, the move may not be made (it is blocked). A capture CAN be made from a jump onto a singularity, by sliding onto a target space.
Capturing occurs by landing on the space OPPOSITE to an opponent's piece- these are the locations marked with circles. Locations opposite to your own pieces- circles of your color- are blocked and may not be landed upon for any reason. When a piece is captured, it (and its opposite location) turns into a singularity; the piece is essentially removed form the board. The capturing piece MUST continue to make another move from the location it just moved to such that it does not move onto a location previously touched in the capture-chain. These are full moves, not singularity-style slides; again, please note that it is explicitly forbidden to touch one of the singularities you just created in your capture. If the piece would have no such move (all locations blocked, or the only way out would be to hit a location twice), it may not make the capture. This extra move may be a capture; it is quite possible for intricate chains of captures to occur.
And that's it. The last player with pieces, or with legal moves, wins the game.
Proposed rule changes:
Mandatory captures? This could, at the very least, be a variant.
It's sort of like Checkers, but written as only an alien ever could. What sorts of strategies could possibly emerge by such a bizarre set of rules- with captures that only occur at a distance (thirty spaces away), each move changing the topology of the game board in two places (two singularities- remember to make the one on the flip side!), and the potential for ridiculous multiple captures, I don't know how it will play out.
I do know the game needs a lot of work. The opening to the game is far too slow; knight moves aren't good enough. I like the way the Möbius strip works, and the way singularities work. (The origin of the "singularity on the flip side" rule, by the way, was so there wouldn't be a safe space on the board opposite to a previously occupied location. It also invalidates two spaces a move, forcing the game to an earlier conclusion). The physical difficulties of playing on a Mobius strip aren't actually too terrible- but it's possible to play it on two sides of a flat sheet of paper, as long as one remembers to do a left-right flip when moving over the edge, or even on two parallel boards. It might be easier to see that way, but it's more fun on a Möbius strip.
It needs work, and I'd love suggestions. Especially playtesting, because my only opponent so far is myself.
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