In a 984 x 984 square grid, each square is colored either white or black. Each black square not on the border has exactly five white squares among its eight neighbors. Each white square not on the border has exactly four black squares among its eight neighbors. How many black squares are there?

I suppose that we could assume that the ratio of black to white squares is 4 to 5: that is, that 4/9 of the squares are black.

Then, 4/9 of 984*984 is 430,336 which is presumably the answer to this puzzle.

An example (or maybe the only possibility) for the pattern is:

* . . * . . * . . * . . * 
* . . * . . * . . * . . * 
. * * . * * . * * . * * .
* . . * . . * . . * . . * 
* . . * . . * . . * . . * 
. * * . * * . * * . * * .
* . . * . . * . . * . . * 
* . . * . . * . . * . . * 
. * * . * * . * * . * * .
* . . * . . * . . * . . * 
* . . * . . * . . * . . * 
. * * . * * . * * . * * .

where the stars represent black squares and periods the white. It helps that 984 is divisible by 3 and therefore 984^2 is divisible by 9, and the repeating pattern is a 3x3 pattern.

Posted by Charlie on 2013-06-18 16:02:39