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
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.
solution of first part
