Mathematics is spectacularly successful at making generalizations: the more than 2000-year old arithmetic and geometry were developed into the monumental fields of calculus, modern algebra, topology, algebraic geometry, and so on. On the other hand, mathematics could say remarkably little about nontraditional complex systems. A good example is the notorious “3n + 1 problem. ” If n is even, take n/2, if n is odd, take (3n + 1)/2; show that, starting from an arbitrary positive integer n and applying the two rules repeatedly, eventually we end up with the periodic sequence 1,2,1,2,1,2,…. The problem was raised in the 1930s, and after 70 years of diligent research it is still completely hopeless!

Next consider some games. Tic-Tac-Toe is an easy game, so let's switch to the 3-space. The 3 × 3 × 3 Tic-Tac-Toe is a trivial first player win, the 4 × 4 × 4 Tic-Tac-Toe is a very difficult first player win (computer-assisted proof by O. Patashnik in the late 1970s), and the 5 × 5 × 5 Tic-Tac-Toe is a hopeless open problem (it is conjectured to be a draw game). Note that there is a general recipe to analyze games: perform backtracking on the game-tree (or position graph). For the 5 × 5 × 5 Tic-Tac-Toe this requires about 3125 steps, which is totally intractable.

We face the same “combinatorial chaos” with the game of Hex.