Ataxx board dimensions

I've been doing some thinking on the board dimensions of rectangular (as opposed to hexagonal and other geometries) games of Ataxx. It is not clear whether a 7x7 game with no blocks is a win for red or blue. However, short game results for some smaller dimensions can easily be obtained. (I used C++ code which I created and released under GPL, written with MinGW 3.0.0, to obtain these results.)

Below I use the convention of denoting each row with sequential English letters (a, b, ...), and each column with sequential natural numbers (1, 2, ...). Passes are denoted by '0'.

rows
columns
tiles
Result
1x1 or 1x2 or 2x1
1 or 2
No moves are possible. The only question is what color of piece initially occupies each of the 1 or 2 squares.
2
2
4
Draw - score 2-2.
3
6
A 5-0 win for red: 1. b1 or 1. b2. Any other red move causes red to lose, as blue can then play 1...b1 or 1...b2 from his remaining piece.
4
8
Blue wins. 1.a1c1 leads to 1...a2c2 (blue wins), while 1.a1c2 leads to 1...a2c1 (blue wins). The longer games are 1.a2 d1b2, followed by any red move, then blue winning by shutout, or 1.b2 d1b1, followed by any red move, then blue winning by shutout.
5
10
Red wins. 1.a1c1 or 1.a1c2 results in blue winning on move 1. But 1.b1, followed by any blue move leads to a red win on move 2. 1.b2 has the same effect - red wins on move 2.
6
12
Red wins - best line (found after 10 plies of search) gives an 8-4 win as follows: 1.b1 e1 2.c1 d1 3.b1d2 e2 4.c2 0 5.b1 0 6.b2, or 1.b2 e1 2.c1 d1 3.b2d2 e2 4.c2 0 5.b1 0 6.b2
7
14
1.a1c1 loses to blue 6-8 (15-ply search). Same goes for 1.a1c2. The best lines are 1.a1c1 b1 2.f1 d1 3.e1 d2 4.e2 c2 5.f2 a1 6.0 b2 (blue wins 8-6), and 1.a1c2 b1 f1 2.d1 e1 3.d2 e2 4.c1 f2 5.a1 0 6.b2 (blue wins 8-6). Unfortunately for red, the moves 1.b1 and 1.b2 also lose (16-ply search). Blue can respond with either 1...f1 or 1...f2 in each case. The four best lines are:
  • 1.b1 f1 2.b2 e1 3.a1c1 e1c2 4.a1 e1 5.b1d1 f1d2 6.b1 e2 7.0 f1 8.0 f2 (Blue wins 8-6),
  • 1.b1 f2 2.b2 e1 3.a1c1 e1c2 4.a1 e1 5.b1d1 f2d2 6.b1 e2 7.0 f1 8.0 f2 (Blue wins 8-6),
  • 1.b2 f1 2.b1 e1 3.a1c1 e1c2 4.a1 e1 5.b1d1 f1d2 6.b1 e2 7.0 f1 8.0 f2 (Blue wins 8-6),
  • 1.b2 f2 2.b1 e1 3.a1c1 e1c2 4.a1 e1 5.b1d1 f2d2 6.b1 e2 7.0 f1 8.0 f2 (Blue wins 8-6).
Searching further (17+ plies) may reveal quicker or more punishing wins for blue, but blue will still win. I don't care to search further.
3
3


9
Another 5-0 win for red: 1. b2. Any other red move loses, resulting in 1...b2.
4
12
A win for red: 1.a1c2, followed by any blue move results in there being a shutout move for red on move 2.
5
15
Unclear at present, but likely solvable with about 22 plies of search. All three initial red hop moves to row 'c' lose on turn one to blue. However, the other four moves all show similar strength after searching ahead 10 additional plies: +2 for each of 1.a2, 1.b1 and 1.a1b3, and +3 for 1.b2.
4
4


16
Another win for red, similar to the one in the 4x3 board: 1.a1c1, followed by any blue move results in there being a shutout move for red on move 2.
5
20
Yet another similar two-move win for red: 1.a1b3, followed by any blue move results in there being a shutout move for red on move 2.
5
5
25
Unclear at present. So far, 1.a1c1 and 1.a1c2 can be shown to lose for red: 1...d2 is the response to each, resulting in a winning shutout move for blue on each of his second turns. However, interesting yet solid defences exist for 1.a1c3, and if 1.a1c3 can be defended, it is unlikely that 1.a2 or 1.b2 will fail red.
Note that symmetric moves and boards (e.g. the 2x3 board, or the move 1.d3b2 on the 4x3 board) are omitted for the sake of brevity and clarity. Also, board dimensions beyond those shown either do not interest me (e.g. 8x2), or are unlikely to provide clear results (though if I were to choose another board size to investigate, it would be 6x4).

Last updated at 2004-09(sept.)-03-2131h (UTC/GMT)