# Amazons 6x5 Solution Attempt

5 . B . . B .
4 B . . . . B
3 . . . . . .
2 W . . . . W
1 . W . . W .
A B C D E F

Some work to solve 6x5 has been done in 2013. It is much harder than 5x6, which has a strong first blocking move of B1-B4xD4, similar to the B1-B3xD3 move in the 5x5 proof.

After some initial exploration, we picked a first move of B1-B4xB3 by hand. We believe that this might be the only winning move (after symmetry).

5 . B . . B .
4 B W . . . B
3 . X . . . .
2 W . . . . W
1 . . . . W .
A B C D E F

The work of proving that this move is a win was split into proving that all 238 replies by Black lose. Arrow's static evaluation function was run on all these positions that result after 2 ply. The evaluation ranged from -3 to +7.

### Solving 2-Ply Positions

We started with the easy-looking positions. In a first phase, all positions with evaluation 3 or higher were solved. In phase 2, all positions or value 2 or higher were solved. Phase 3 was started to evaluate some positions of evaluation 0. After Jiaxing's graduation in 2012, work continued for a while, then was resumed in April 2013, then suspended again. We proved one of the six worst-evaluated positions with static evaluation -3, then started to evaluate the list of remaining evaluation = 0 positions on one machine and the list of evaluation = -1 positions on another. Some of these searches were very large and ran for weeks without success. Eventually, the machine had a hardware failure and the work was suspended at that point.

It would be more practical to improve the algorithm first, then try again.

### State as of Apr 29, 2013

eval 7: 2 total, all solved
eval 6: 4 total, all solved
eval 5: 4 total, all solved
eval 4: 11 total, all solved
eval 3: 23 total, all solved
eval 2: 41 total, all solved
eval 1: 54 total, all solved
eval 0: 24 total. 5 solved, 19 unsolved
eval -1: 38 total. 2 solved, 36 unsolved
eval -2: 31 total, 0 solved, 31 unsolved
eval -3: 6 total, 1 solved, 5 unsolved
total: 238 positions, 147 solved, 91 unsolved.

Created: Apr 29, 2013 Last modified: May 11, 2016

Martin Müller