(First things first - I don't know a solution to this, but the puzzle occurred to me a few hours ago, and I thought people might be interested in it)
 
Imagine a rectangular (or square) grid of any size, every square white. If the grid is "x" squares across and "y" squares high, what is the minimum number of squares ("n") that must be shaded so that no white square is adjacent to more than 2 other white squares?
(For this puzzle, diagonally adjacent squares are not considered to be adjacent)
 
So, for example, if the grid is simply a 3x3 then the only square that needs shading is the centre one, then all others squares only touch two others - i.e. for x=3 y=3, n=1
 
a) Is there an formula to calculate "n" that will work for all paired-values of "x" and "y"?
b) If not, what is "n" for a chessboard-sized x=8 y=8 (post your suggested minimum using a standard chess-like "A7" type of description for a list of all your shaded squares)?

(In reply to Hmmm... by TomM)

As I have no definite solution yet to this problem I feel okay posting thoughts about it.

Re. your third-paragraph proposition that says a core "2 x (b + 1)" array doesn't satisfy the condition if white - as a side note, a 2x2 does. So, technically speaking shading rings for a x = y = 2a would give a valid solution as you would end up with a 2x2 in the centre, which is fine. However, I've found that even on an 8x8 I can find a smaller value for "n" than is produced using this method, so it doesn't give an optimal solution...

Posted by Nick Reed on 2002-07-22 06:59:33