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 3x3 Nim (Posted on 2009-06-25)
A 3x3 array of counters is laid out. Players take turns removing counters. The rule for removing counters is to pick a row or column and take any 1,2 or 3 from it. Whoever removes the last counter wins.

Does the first or second player have a winning strategy?
What is this strategy?

 Submitted by Jer Rating: 4.0000 (7 votes) Solution: (Hide) The beginning position (9 counters) is a losing position so the second player has a winning strategy. This strategy is to always take the board to one of 20 losing positions shown below. Every position with 1, 3, 5, 7, or 8 is a winning position so the second player must make the board become of the following positions with 2, 4, or 6: ```X-- X-- X-- -X- -X- --X --- X-- --- --- --X --- These are positions with 2 counters on different columns/rows. XX- XX- XX- X-X X-X -X- XX- --X --- -X- --- X-X --- --X XX- -X- X-X -X- These positions are transformable to either the first or second by shifting row/columns. -XX X-X --X --X --X --X -X- -X- X-X X-X -XX -XX X-X XXX XXX -XX XXX --X XX- XX- XXX XXX -XX X-X XXX -XX XXX These positions have 3 empty spots either in 3 rows/columns or in 2 rows/columns. ``` In essence, then, there are 5 positions to memorize to be able to always force a win as the second player.

 Subject Author Date Math Strategy? Adrian Mason 2013-03-26 00:03:43 i had to... Joselyn 2010-01-01 07:43:30 nim value equivalents Brian Smith 2009-06-29 01:15:18 misère form Charlie 2009-06-28 14:13:44 simplification Charlie 2009-06-27 13:48:38 Strategery ed bottemiller 2009-06-26 23:18:15 re(3): computer solution pcbouhid 2009-06-26 18:15:51 re(2): computer solution Charlie 2009-06-26 16:03:26 re: computer solution pcbouhid 2009-06-26 12:08:12 computer solution Charlie 2009-06-26 09:50:05 re(3): tes he can revisited Charlie 2009-06-26 09:45:54 re(2): tes he can revisited Ady TZIDON 2009-06-26 06:23:56 I agree Ady TZIDON 2009-06-26 06:08:05 re: tes he can brianjn 2009-06-26 00:53:44 re(3): tes he can brianjn 2009-06-26 00:50:44 re(2): Winning Strategy (spoiler) Steve Herman 2009-06-25 21:55:12 Winning Strategy (for the wrong problem) Steve Herman 2009-06-25 21:42:42 re: Winning Strategy (spoiler) Charlie 2009-06-25 17:05:26 re: Winning Strategy (spoiler) Jer 2009-06-25 17:02:11 Winning Strategy (spoiler) Steve Herman 2009-06-25 16:18:07 re: tes he can Charlie 2009-06-25 14:33:36 re(2): tes he can pcbouhid 2009-06-25 13:42:27 re: yes he can brianjn 2009-06-25 12:50:00 re: tes he can brianjn 2009-06-25 12:45:19 yes he can Ady TZIDON 2009-06-25 11:48:26 tes he can Ady TZIDON 2009-06-25 11:45:01

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