### Breakthrough on small boards Breakthrough

9 replies. Last post: 2006-11-26

Breakthrough on small boards
• Carroll at 2006-11-23

To mimic a question raised on LOA, did anybody analyse this game on small boards that would enlight some strategy elements ?

What about simple 4x4 board ? Win for black ?

• Tasmanian Devil at 2006-11-24

I wrote a program for this tonight. If I did it correctly, it appears that the maximum depth under perfect play is 14 moves (i.e., 7 moves by each side), and one of the positions that attain this is the full board (both sides have two full rows). Note that this is a second player win, since 14 is an even number, so yes, black wins. If they start with one row each (along the edges), the depth is 11 moves.

The program is not particularly user-friendly at the moment, but if you have specific questions about the positions and perfect plays, I can try to answer them.

• Ray Garrison at 2006-11-24

Yes, I have proven also (without the use of a computer) that the second player can always win on a 4x4 board with eight pieces each to start the game (white’s first move MUST be a capture), and the second player also wins if both players only start with four pieces each. You can set up a board with some pieces and come to this same conclusion relatively quickly. I wonder if the second player can force a win on a six by six board, and if so, can the second player also force a win on a full size board?

• Tasmanian Devil at 2006-11-25

Then I and my program want to challenge you to a match on the 4x4 board with one row each, and you can be second player. Our first move will be 1. d1-d2.

4 x-x-x-x
3 - - - -
2 - - - -o

1 o-o-o
A B C D

Good luck! :)

• Ray Garrison at 2006-11-25

ok, I accept and resign immediately. Your first move is a win. I can not stop your initiative. (I will play your computer and go first in the second game, if you like. My first move will be 1. a1-a2) I stand corrected, my logic was flawed due to insufficient study of the problem. the first player wins by force if each side starts with four. If each side starts with eight, however, then it still looks like a win for the second player.

• Tasmanian Devil at 2006-11-26

Then we agree. ;-)

• ypercube at 2006-11-26

What happens at 5x5 board?
With 2 or 1 row for each player?

At 5x4? 5x3?

At 4x5,4x6, 4xn with 2 rows? Is it always a black win?

• Tasmanian Devil at 2006-11-26

By 4xn, do you mean 4 rows and n coloumns? I am unable to extend the board size in my own program, but I have Breakthrough for Zillions, and it’s easy to implement these variants. I experimented a bit with these and here are my results so far.

On a board with 4 rows, it seems that Black has an easy win if the players start with 2 full rows each. White can not afford to move a piece from row 1 as long as Black occupies the entire row 3. So he has to keep caturing with pieces from row 2, but Black can answer each capture with a capture by the corresponding piece from row 4. Perfect play should give 2n + 6 moves for all n.

If they start with one row each, I haven’t proved anything, but my conjecture is that White can win in 11 moves for all n (from 4 upwards), with a1-a2 being a winning move in each case. Possible continuations are 2. d4-c3 3. b1-b2 and 2. a4-b3 3. d1-c2. It seems that Black has to reply in this region and that the size of n does not matter.

• Tasmanian Devil at 2006-11-26

According to Zillions, 5x5 with one row each is a win for the first player in 17 moves, for example, like this: 1. e1-d2, 2. e5-d4, 3. a1-b2, 4. a5-a4, 5. b1-c2, 6. b5-c4, 7. b2-a3, 8. d5-e4, 9. c2-c3, 10. d4xc3, 11. d2xc3, 12. e4-d3, 13. a3-b4, 14. c5xb4, 15. c3xb4, 16. a4-b3, 17. b4-b5.

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