Number eight cells:

Now suppose we want to color each cell red or blue such that no three cells are in arithmetic progression — for example, we don’t want cells 1, 2, and 3 to be the same color, or 4, 6, and 8. With eight cells it’s possible to accomplish this:

But if we want to add a ninth cell we can’t avoid an arithmetic progression: If the ninth cell is blue then cells 1, 5, and 9 are evenly spaced, and if it’s red then cells 3, 6, and 9 are. Dutch mathematician B.L. van der Waerden found that there’s always such a limit: For any given positive integers *r* and *k*, there’s some number *N* such that if the integers {1, 2, …, *N*} are colored, each with one of *r* different colors, then there will be at least *k* integers in arithmetic progression whose elements are of the same color. Determining what this limit *is* (in this example it’s 9) is an open problem.

(Bonus: Alexej Kanel-Belov found this pretty theorem concerning divisibility of integer sums within an infinite grid — Martin J. Erickson, in *Beautiful Mathematics*, calls it a two-dimensional version of van der Waerden’s theorem.)