Progressive Forwards Chess

Progressive Forwards Chess is a chess variant, designed as an entry in the 32-turn contest, i.e., it lasts never more than 32 turns. Actually, the maximum number of turns this variant lasts is shorter.

Rules

The game is played with a normal set of chess pieces, by two players, starting from the usual setup from orthodox chess. A few changes are made to the usual rules of chess:

The game is played in progressive fashion (Italian style). That is, first, white makes one move, then black makes two moves, then white makes three moves, etc. The Italian rules of progressive chess are used: it is illegal to give check except at the last move of a series. When in check, one must lift it at the first move of a series - when the only possibility is to do such by giving check, one hence loses.
Pieces may not make backwards moves, and sideway moves are limited:
- Rooks, queens, and kings may make sideway capturing moves. Rooks and queens also give check in horizontal directions, and kings may not be one horizontal step apart.
- Rooks, queens, and kings may make a sideway non-capturing move if such a move is directly followed by a forwards move in the turn.
- No castling.
- No piece may move backwards, and pieces also do not give check in backward directions.
The game can be won by mating the opponent, or getting oneself in a stalemated position.

Comments

The Italian style adds to the interest of the game, I think, but is also needed for getting the bound on the number of turns. The stalemate rule gives a second way of winning. It dimishes the wish to take many pieces (in particular, pawns and knights), and an endgame may result in a race towards to opposite side of the board.

The turn count

I first count the number of single moves, and then see how this bounds the number of turns. Each player can make in total at most:

15 capturing moves (one for each piece of the opponent, except the king)
48 pawn moves (each pawn can make 6 steps. When it is promoted, it can only make capturing moves).
14 knight moves (7 per knight)
14 bishop moves (7 per bishop)
14 non-capturing moves with the king (7 forward moves, and 7 sideway moves before the forward moves)
14 non-capturing moves with the queen
28 non-capturing moves with the rooks

So, in total at most 2*(15+48+14+14+14+14+28) = 280 single moves can be made. This means that at the 24rd turn of the game (i.e., the 12th turn of black), the game must have ended, as 1+2+3+4+...+21+22+23+24 = 300, i.e., more than 280.

A more detailed analysis may reduce the number of moves somewhat. E.g., the total number of non-capturing pawn moves of white and black together can easily be seen to be at most 88 as each pair of pawns on the same column can make at most 13 (and not 14) non-capturing moves together.

Variants

Variants, that make somewhat longer games possible can also be imagined. Note however that the main variant, submitted to the contest, is described above.

Backwards capture

The main variant is altered as follows: it is allowed that pieces make capturing moves backwards. I.e., only non-capturing backward moves and some non-capturing sideway moves are allowed.

Then the count of the total number of moves per player becomes somewhat more subtle. Still, we can make 15 capturing moves in total. However, a capturing move can result in a larger possible number of forwards moves. The largest increase that a capture can give is when a rook or queen on the 8th line take a piece on the 1st line. This gives an additional 14 non-capturing forwards moves that may possibly made with that capturing piece. All other types of capture can give less increase to the number of non-capturing moves. So, the total number of single moves is at most 280 + 15*14 = 490, and that makes that the game ends on or before the 31th move (1+2+...+30+31 = 496), i.e., white's 16th turn.

I expect this game to be inferior to the one without backwards captures.

Slower turn sequences

Instead of having the sequence of moves that can be made to be 1, 2, 3, 4, etc., one could use a sequence that increases less quickly, e.g., use: 1, 1, 1, 2, 2, 2, 3, 3, 3, etc., i.e., black makes two moves on his second turn, white moves three times on his fourth turn, etc.

The sequence where after every three half-turns the number of moves is increased by one would make the standard variant last till at most the 40th turn (i.e., black's 20th turn, as 3*(1+2+..+13) = 3*91 = 273); the variant with backward capture would last till at most the 48th turn, i.e., blacks's 24th turn (3*(1+2+..+15)+2*16=291 ).

Moving each piece once

In this variant, we play that every piece may be moved only once per turn. Rooks, queens, and kings may make only capturing moves sideways (thus, dimishing the power of rooks dramatically). If all pieces have been moved, then the remainder of the moves for the turn are forfeited. A player must move as many different pieces as possible in a turn, and may check only at the last move of his sequence.

While I believe that it is impossible to give a game that lasts more than 32 turns (or 64-half-turns) for this variant, I am not yet able to give a mathematically sound proof for it.

This is a submission to the contest to design a chess variant that takes at most 32 turns.

Written by Hans Bodlaender

WWW page created: April 28, 2000.