(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 re: One Equation - equation in int and mod by Brian Smith)

I agree that the 2 part formula is simpler.

I was being pedantic; Nick's query a) appeared to be asking for a single equation as a solution.

Also I am unaware of any scientific calculators having a "ceil" function therefore I raised "Int-Mod" issue.

I have yet to acquaint myself with the "Ceil to Int-mod" translation.

Whatever, I think that it is quite apparent that Nick should find joy in these respective solutions.

Posted by Brian Nowell on 2003-04-29 17:41:58