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23. Aprilo 2003, 01:40:29 Temo: things Aloha all,
I would like to reply to several previous posts here. Pioneer54, outstanding post on Saturday. I myself just needed a little time away from all the discussion that, for me was getting pretty heated. I glad it’s all being resolved. My thoughts on a proof, and in response to what Gary has said: Gary, I believe you said something about you would gladly accept a challenge from anyone and prove that player 1 always wins in fun-pente. While I believe that you could certainly do that, it would not actually prove that player 1 always wins. It would certainly demonstrate that fact quite well, though. ;-) The problem is that we need to show that a win by player 1 is assured in every possible game! I have two thoughts on an actual proof. If we could prove that a whole subset of games are won by player 1, then we could reduce the proof to only showing that the remaining games are won by player 1 also. For example, let’s say that we could prove that anytime player 2’s opening move is at least 7 spaces from the center, player 1 has certain victory. If we could prove that, then we could just play all the games where player 2’s first move is within 7 spaces of the center. Still a tough task, but much easier than the original problem! We could prove that P1 always wins in another way. If we could come up with the right way of measuring the current position of the board, then all we would need to show is that P1 can always increase that measurement on his next turn. Of course, coming up with the right way to measure the board is just as difficult as the original proof!! I believe this ‘measuring of the board’ is how most artificial intelligence chess playing games work. Gary, on the subject of your challenge: (This is just food for thought) Now that you have found a game that meets your criteria, what does that say about your original premise? My two cents: There are currently eight possible variants being discussed for inclusion here. They are all the combinations of: Pente or Keyro Pente play on a 13x13 or 19x19 board play with or without the opening restriction I personally would like to see Pente on a 19x19 board without the restriction implemented. I think implementing all eight would be too much. I know implementing six has been suggested. I’m not sure if even that many is too many. Two more cents: I have seen several posts discussing how some players would like to see everyone play by the ‘correct’ rules. Wouldn’t in be better to say the official rules? I mean, with an online game, you pretty much have to play by the correct rules anyway, don’t you? Thad 23. Aprilo 2003, 18:21:56 Temo: Re: things In order to prove that p1 wins in "fun" pente (no opening restriction) either on a 13x13 board or 19x19, possibly you could exclude moves N spaces away but it would be tough to prove that different far away moves have no effect on a winning line for one of them. You have to do an exhaustive search -- taking into account every possible move by p2. But you do not have to try every possible move by p1, you can plug in an opening book of good moves by p1 and see if they will lead to a win. 24. Aprilo 2003, 00:19:37 Temo: Re: things mmammel,
The proof for game with the restriction is likely harder than the proof for game without, but the same principals should apply. And yes, we need to prove it for every possible move for player 2, and we only need to show that there is at least one 'correct' move for player 1 at each turn. The solution to the problem you mention is to create a new game, let's call it pente-X. The only rule change is that player 2's first move must be N squares from center AND that he doesn't have to declare exactly where that position is until he chooses. Clearly, pente-X includes all the possible pente games in which player 2's first move is N moves from center. Thus, once we prove that pente-X can always be won by player 1, we will also prove that the same holds true for this subset of games of pente. You wouldn’t have to do an exhaustive search if you could show that player 1 can always win AND keep the game within the N-size circle so that player 2’s first stone never affects play, no matter where outside the circle it was actually placed. Again, this is still a difficult proof, but easier than proving the game always winnable by player 1 directly. Thad 27. Aprilo 2003, 17:01:04 Temo: Re: things Thad,
yes, the pente-x idea is good. I was thinking that to make it easy you could prove a win on as small a board as possible and the win would still (effectively) be valid for any larger board -- assuming that p2 moving farther away would not be beneficial. Of course that assumption is not perfectly valid... Generally it should be easier for p1 to win on larger boards because the attacking lines are not as restricted. A win on a small board with chosen moves for p1 will not necessarily show the fastest possible win. I will try some searches on small boards. |
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