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?
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