On One-Dimensional Checkers

          On One-Dimensional Checkers

I invented one-dimensional checkers as an ironic counterpart to 3D chess. I wanted a super-simple game; but I find that complexity emerges.

1D checkers is played on a line of squares, bearing dimes on the left side and pennies on the right side. The dimes only move right, the pennies only move left. Leave one square empty in the middle. On each move, you can move one of your coins into an empty square, either one space over, or capture by jumping over the other player’s coins. Multiple jumps are allowed. The player with coins remaining at the end wins.

Consider for instance the game starting with

          DD_PP

where D = dime, first player, and P = penny, 2nd player.  I call this the 2-1-2 game; and also the 1,2;4,5 game.

Play is:

DD_PP

D_DPP

DP_ _P

_ _ D_P

_ _ DP_

_ _ _ _ D

1-dimensional checkers is really an infinite class of games, depending on the initial setup. Complexity rises with line length; for instance, length of play goes by the square of the line length.

So far I’ve investigated the 1-1-1 game, and the 2-1-2, the 3-1-3, the 4-1-4, the 5-1-5 and the 6-1-6. 1-1-1 is a win for the second player; 2-1-2 is a win for the first player; 3-1-3 is a win for the second player; 4-1-4 is a win for the first player. The next two have some freedom when the line thins out a bit, so I’m not sure yet if it continues to alternate.

Zugzwang is a persistent problem; you can be forced into making losing moves. It pays to have one of your coins posted aggressively forward; and also to have holes in your formation, to waste time in while your opponent crashes into zugzwang.

Often you trade off coins evenly until it ends with a showdown; one lone coin vs one lone coin: D_P , or D_ _P ,  or D_ _ _ P , and so on. If there’s an odd number of spaces between, then second player wins; if an even number, then first player wins.

Here is a negotiated variant of 1D checkers; the players start with an empty line, then they alternate placing their coins on the line, with only one rule; coins moving towards each other must be separated by at least one space. Once the line can take no more coins, play begins. The first player unable to move loses.