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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?

 See The Solution Submitted by Jer Rating: 4.0000 (7 votes)

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 i had to... | Comment 25 of 26 |
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

 Posted by Joselyn on 2010-01-01 07:43:30

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