(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)?

Actually, I just found 20 on a 8x8 grid.

Still searching for a uniform formula and the real smallest value.......

If the grid is set up like this:

. A B C D E F G H
1
2
3
4
5
6
7
8

then the shaded squares are (so far):
A2, A5, B1, B4, B7, C3, C6, D2, D5, D8, E1, E4, E7, F3, F6, G2, G5, G8, H4, H7.

To be continued (most probably)...

Posted by lucky on 2002-07-23 05:59:05