Counting Up - Futility Closet

A problem from Daniel J. Velleman and Stan Wagon’s excellent 2020 book Bicycle or Unicycle?: A Collection of Intriguing Mathematical Puzzles:

A square grid measures 999×999. Each square is either black or white. Each black square that’s not on the border of the grid has exactly five white squares among its eight immediate neighbors (those that adjoin it horizontally, vertically, or diagonally). Each white square that’s not on the border has exactly four black squares among its immediate neighbors. Of the 999 × 999 = 998001 squares in the grid, how many are black and how many white?

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999 is 3 × 333, so we can divide the grid into 333² 3×3 subgrids. Each of those subgrids consists of a central square surrounded by eight neighbors. According to our information, if a subgrid’s central square is black then exactly five of its neighbors are white, and thus the subgrid contains four black and five white squares. If the central square is white, then the subgrid likewise has four black and five white squares. So the entire grid has 333² × 4 = 443556 black and 333² × 5 = 554445 white squares.

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