[ View Thread ] [ Post Response ] [ Return to Index ] [ Read Prev Msg ] [ Read Next Msg ]

BGonline.org Forums

Another puzzle/ next question/ SPOILER I hope)_

Posted By: Nack Ballard
Date: Sunday, 16 May 2010, at 6:55 a.m.

In Response To: Another puzzle/ next question/ SPOILER I hope)_ (Bob Koca)

I think that the player who goes second can force a win no matter the value of N. Note that it is possible to place (N-1)^2 X's on the board without making X in a row and then by pigeonhole principle the next X must make N in a row. The player who goes 2nd can force that configuration (unless the other player commits suicide earlier than that by making N in a row when it was not forced). The strategy is to scan the row and columns and find which has the fewest amount of X's. Of those choices find one which fills in a row or column with minimum number of X's. For example suppose that you have found that columns 1, 3, and 8 have only 1 X and all other columns and rows have more than one X already. Of those three columns find which open space to play in so that the row you choose has a minimum number of X's.

I agree that the maximum number of Xs that can be played onto a grid is N(N-1), and that is always an even number. But it seems like a big leap from there to conclude that with best play player B (the second player) always wins.

Designate the rows with numerals and the columns with lower-case letters, as in algebraic chess notation. If N = 3, say player A opens with a1. If B plays b2, he loses to c3, or vice versa (because A's follow-up of b1 loses to a2, or c1 loses to a3, and b3 or a3 are beaten by symmetrical variations.) If instead (in response to A's a1) B plays b3, A wins with c2, or vice versa (because b2 is met by c3 or vice versa, a2 by c1 or vice versa, and a3 by b1 or vice versa, etc.

By my reckoning, Player B can force a win by initially responding to a1 with b1, but that plays in an occupied row or column (i.e., one with one X), which is against your proposed strategy, as I interpret it. All B's initial choices (b2, b3, c2, c3) that play onto both a vacant row and column (i.e., having no X) lose as outlined in the previous paragraph.

Nack

Messages In This Thread

 

Post Response
Your Name:
Your E-Mail Address:
Subject:
Message:

If necessary, enter your password below:

Password:
 

 

[ View Thread ] [ Post Response ] [ Return to Index ] [ Read Prev Msg ] [ Read Next Msg ]

BGonline.org Forums is maintained by Stick with WebBBS 5.12.