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 On a Grid (Posted on 2007-08-08)
Place eight 'A's and eight 'B's in a 4x4 grid such that each 'A' is orthogonally bordered by 1 or 3 'A's and each 'B' is orthogonally bordered by 0, 2, or 4 'B's.

Can a 6x6 grid be filled with 18 'A's and 18 'B's in this manner?

Note: 'orthogonally' means horizontally or vertically

 No Solution Yet Submitted by Brian Smith Rating: 4.0000 (3 votes)

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 solution of first part | Comment 1 of 6
BAAB
ABBA
ABBA
BAAB
This seems to be the single possible solution.
It can be shown (in an exthaustive way) that there are no possible solutions with a B bordered by 4 B's, with an A bordered by 3 A's, or a B bordered by 2 B's both horizontally bordering (or both vertically). The B's also can't form a 3x3 square, as the middle square would be an A which has no other A's bordering. For the A's, all of them have a single neigbouring A, so they form domino's. The only possible configuration which follows these constraints is the solution shown here.

 Posted by Robby Goetschalckx on 2007-08-08 10:55:30

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