Minesweeper

If you've ever played the classic Windows game, Minesweeper, perhaps you've wondered how geometry and other branches of math can describe the system. If you haven't, here's the Wikipedia entry: Minesweeper. If you have Windows, the game is available on the Mac App Store.

To begin to describe the game, there is a grid of discrete points on various intervals depending on difficulty. There is a randomized function on the grid, which takes each square (or point) of the grid either to an integer between 0 and 8, or a mine. The number is determined by the number of neighboring mines (or, to be mathematical, mines within a neighborhood with radius sqrt(2)). So here's my first difficulty, because it seems here that we're now dealing with two distinct types of coordinates, which I have never seen in math. Furthermore, one type of coordinate depends on proximity to the other. Considering this, it seems tempting to give up altogether the hunt for a mathematically precise description of the game, but let's see what we can come up with.