# 3x4 Chess

## Introduction

3x4 chess is now strongly solved. This means that game result and the best move is known for any possible position. Firstly, what is 3x4 chess? The board is 3 squares wide and 4 squares tall. There is no castling and no pawn's double first move. All other chess rules apply as usual. Starting position is not defined, instead all possible positions were analyzed, including all possible combinations of forces. Pawns are allowed on the first rank, for completeness.

In total there are 167,303,246,916 unique legal positions in 3x4 chess. 12 squares, 12 digits. All symmetry is taken into account, only one of all possible symmetrical variants of each position is counted. Also these are only positions where white is to move. This is about 3,000 times more than in 3x3 chess.

## The result

Is 3x4 chess generally won, lost, or drawn for the side to move? Since starting position is not defined, all we can do is count wins, draws and losses among various possible positions. There are 48,713,659,039 wins (29.12%), 103,995,040,255 losses (62.16%), and 14,594,547,622 draws (8.72%) in the complete database. Taking this literally we would have to conclude that 3x4 chess is won for black, with side to move (or white) scoring only 33.48%.

Of course the above score is not real since we counted *all* legal positions, which include checkmates too.
Checkmate is a position where side to move is already lost, yet it is perfectly legal.
In fact 31.11% of legal positions in 3x4 chess are checkmates — 52,048,121,059 of them.
After removing checkmates we are left with only 51,946,919,196 losses.
Now white score becomes 48.60% which looks much better.
Is this the answer then?

Not yet. For one thing, we are still counting stalemates. No moves are possible from a stalemate position, the game is over, so we probably don't want to count them. But this is not a big problem because there are very few stalemates in 3x4 chess (0.15% of legal positions) and because they are all draws anyway.

The bigger problem is with positions where side to move is in check. The identical position with opposite side to move would be illegal, so it is not counted or stored in the database. However when side to move is in check, it's totally fine, and such positions in fact constitute 43.76% of the complete database — 73,205,819,459 of them in total. The solution is to not count these positions too, restoring the symmetry.

So, only positions where side to move can in fact make a move, and is not in check, should be used to compute the side-to-move score.
There are 41,799,558,387 such positions in 3x4 chess, or 24.98% of all unique legal positions.
Among them there are 25,066,806,249 wins for the side to move (59.97%), 11,604,499,663 losses (27.76%),
and 5,128,252,475 draws (12.27%).
This means that side to move scores **66.10%**.

## Query interface

Here you can check the true value for any imaginable 3x4 chess position. Input the position in FEN format, then press "Analyze" to query the database, and "Play" - to play against a perfect opponent. Alternatively, the shorter format from 3x3 chess is also supported.

## Problems

A random problem will be picked every time you click on one of these links:

## The longest line

The longest checkmate line in 3x4 chess takes 43 moves (85 plies). White aims for the shortest mate, black for the longest. Here is one of such positions:

So, even with 11 pieces the 43-move long winning line is already possible, but not with 10 pieces.

Some of the longest lines to conversion (with DTC metric):

- 7 pieces — 26 moves — wKBk..P.BR.b.
- 9 pieces — 16 moves — wkpnpRpK..P.N
- 10 pieces — 14 moves — wRQBKB.PPPk.q
- 11 pieces — 7 moves — wN.BNRPQNqKPk

In case if you want to see the longest possible line with particular piece combination, now you can. For every piece combination I saved an example position, resulting in longest winning line. And another one, resulting in longest losing line. You can download the list of all such positions:

Just copy/paste any position into the query interface above.

## Statistics

Number of unique legal positions with every distance to mate (or conversion) in each piece configuration. Only unique legal white-to-move positions are counted.

Pieces | Piece configu- rations | Unique legal positions | Longest distance to mate | Longest distance to conversion | Download DTM statistics | Download DTC statistics |

2 | 1 | 20 | - | - | - | - |

3 | 10 | 1,746 | 8 moves | 6 moves | 1 kB | 1 kB |

4 | 55 | 69,556 | 20 moves | 20 moves | 2 kB | 2 kB |

5 | 220 | 1,635,355 | 26 moves | 16 moves | 13 kB | 7 kB |

6 | 715 | 25,117,392 | 31 moves | 20 moves | 54 kB | 24 kB |

7 | 2,002 | 264,331,909 | 40 moves | 26 moves | 188 kB | 69 kB |

8 | 5,005 | 1,936,519,455 | 40 moves | 20 moves | 554 kB | 172 kB |

9 | 11,440 | 9,770,507,866 | 41 moves | 16 moves | 1,429 kB | 380 kB |

10 | 24,310 | 32,535,116,698 | 42 moves | 14 moves | 3,260 kB | 741 kB |

11 | 48,620 | 64,617,989,055 | 43 moves | 7 moves | 6,592 kB | 1,167 kB |

12 | 92,378 | 58,151,957,864 | 43 moves | 1 move | 11,328 kB | 1,504 kB |

All | 184,756 | 167,303,246,916 | 43 moves | 26 moves | 23,420 kB | 4,066 kB |

Number of checkmates, stalemates, positions in check, and all other positions in the complete database, by piece count: