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The well-known game of Ultimate Tic-Tac-Toe is played on a 9×99\times 9 board (with the rows and columns numbered by 0,1,…,80,1,\ldots,8), conceptually divided into nine 3×33\times 3 small boards. The players are denoted by "X" and "O" and take turns, starting with X.
In a now-famous 2004 article, Ben Green and Terence Tao proved that arbitrarily long arithmetic progressions exist in the primes. This holds true analogously for other sets of numbers if their density is sufficiently high, for example, numbers that are the sum of two squares.
Solving the game by hand for n=4n=4 is a good way to obtain a feel to the way a general solution works. Encode each state of the table as an element of F24\mathbb{F}_2^4, i.e. a sequence of 0 and 1 of length 4 (let 1 denote an off switch so that the goal is to reach 00000000).
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