3x3 Chess

(Please check 3x4 chess and 4x4 chess too, if you haven't yet)

Introduction

This page is about the game of chess on 3x3 board. This game is now solved. It means that the database of all positions is constructed, and the best move is known in every position. It is not clear how to define a starting position on 3x3 board, so all possible positions were included. Rules are same with chess, except that due to the tiny board there is no castling and no double first move of pawn (and no en passant). Pawns promote when they reach the last rank - third for white and first for black.

Does it look too simple for you? Read these pages, try to solve problems, then follow the correct line with the analysis board. Then tell me if 3x3 chess is still too simple or not. I hope you will enjoy this beautiful chess variant as much as I do.

The Answer

"Alright," said Deep Thought. "The Answer to the Great Question ..."
"Yes ...!"
"Of Life, the Universe and Everything ..." said Deep Thought.
"Yes ...!"
"Is ..." said Deep Thought, and paused.
"Yes ...!"
"Is ..."
"Yes ...!!!...?"
"Forty-two," said Deep Thought, with infinite majesty and calm.
- Douglas Adams, The Hitch Hiker's Guide to the Galaxy

The question is, of course, how long is the longest checkmate. We don't know which variant Deep Thought was solving (perhaps this or this one). But we already have the Answer for 3x3 chess: 16 moves! More details, including mate in 16 and other interesting positions, are on this page.

Problems

Test your tactical thinking, solving problems on 3x3 board. The position will be selected for you randomly. When you select your move, the best defence of computer will immediately follow. If the problem is too difficult, use the 'Give up' link at the bottom.

Analysis board

For your convenience, there is an online analysis board available. First, let's look at the example position: an empty board and two kings in the corners: w..k...K... As you see, script "3x3can.cgi" is called when you open the link. Position is specified after the question mark by a 10-character description, somewhat similar to FEN (Forsythe-Edwards Notation). The first character is 'w' or 'b', it tells which is the side to move. Other characters describe 9 squares of the board, in this order: a3 b3 c3 a2 b2 c2 a1 b1 c1. Empty square is '.' (dot), pieces are written as: K, Q, R, B, N, P for King, Queen, Rook, Bishop, Knight and Pawn. White pieces use capital characters, black pieces - small characters. Another example: wK.k..NqRn. Just open an example position and try to type your own position in the address bar.

Statistics

There are 304'545'552 legal positions in 3x3 chess (counting separately all left/right symmetric and white/black to move positions). Out of them 104'863'672 are checkmates, more than 1/3 of all positions. 3'576'856 positions are stalemates, 1.2% of all positions. So, there remain 196'105'024 in-game (non game-over) positions, where some moves are possible.

Number of drawn positions, including stalemates, is 52'268'040, or 17.2% of all legal positions. Number of drawn positions, which are not stalemates, is 48'691'184, which is 24.8% of all in-game (non game-over) positions.

Out of 195'996'550 in-game positions there are 78'022'680 positions where side to move wins (39.8%) and 69'391'160 positions where side to move loses (35.4%).

Here is the detailed statistics:

ResultNumber of
positions
% of legal
Win in 135'962'54811.809%
Win in 218'809'6686.176%
Win in 39'414'9883.091%
Win in 45'240'8061.721%
Win in 53'684'5681.210%
Win in 62'475'6920.813%
Win in 71'312'7680.431%
Win in 8637'1200.209%
Win in 9295'2200.097%
Win in 10114'2700.038%
Win in 1140'5040.013%
Win in 1222'0800.007%
Win in 137'5000.002%
Win in 142'8360.0009%
Win in 151'9280.0006%
Win in 161840.00006%
Total78'022'68025.619%
ResultNumber of
positions
% of legal
Lose in 134'465'37411.317%
Lose in 214'971'8044.916%
Lose in 37'418'0922.436%
Lose in 44'981'7681.636%
Lose in 53'532'2281.160%
Lose in 62'135'6760.701%
Lose in 71'032'2500.339%
Lose in 8495'2660.163%
Lose in 9218'1680.072%
Lose in 1082'8700.027%
Lose in 1131'1480.010%
Lose in 1214'6080.005%
Lose in 139'5160.003%
Lose in 141'5720.0005%
Lose in 157960.0003%
Lose in 16240.000008%
Total69'391'16022.785%

History

I started to think about 3x3 chess after solving one small board shogi variant in March 2003. In November of 2003 I constructed my first 3x3 chess database, and then I was improving the program and verifying my data. This web-pages, as they are now, including online analysis board, were first opened on February 16, 2004. This is the day when this study was first made public, although it still took some time until this web-site was indexed by search engines and added to some link directories.

Thanks

I would like to thank following people:

  • Takuma Yamada for ideas and discussion.
  • Chessbase, Germany, for providing a colorful chess graphics.

Feedback

I'd like to hear what you think about this study. Any questions, comments or suggestions are welcome. Also, if you'll find any mistake I would very much like to know about it. See contact details here.

Other studies of 3x3 chess

Aloril has also solved 3x3 chess, see his web-site. These two studies are totally independent, in fact we were not aware of each other existance until we both solved the game and made our web-sites. He completed his solution earlier than me - his web-page sais his first tablebase is dated 2001-05-02. However I believe 2004-02-16 is the day, when 3x3 chess analysis was first made open to the public, on these pages.

There is a slight difference between the game rules we used. Aloril's solution does not include positions where there is a white pawn on the 1-st rank, or black pawn on the last rank. This restriction makes perfect sense from the full-size chess point of view. However such positions are included into my study, for several reasons:

  1. 3x3 board is already tiny. It seems too limiting to allow pawns on second rank only.
  2. On the small size board heavy pieces are too powerful. I wanted to see more pawns for some balance.
  3. If that positions are excluded, a pawn can never take another pawn, or even just move without promotion.
  4. Since there is no defined starting position in 3x3 chess, the definition of a position being legal if it is reachable from the starting position does not apply.
  5. No pawn chains if that positions are excluded. :-)

But anyway, I think this difference is rather minor, more like a matter of taste. (although it sure makes difference in statistics). In all other senses our solutions are both complete, as far as I know.