A Geometric Characterization of Modular Nim-Sums Through Binomial Coefficients -- Alex Kontorovich, November 11, 2003

The prototypical combinatorial game is the game of Nim. Nim-summation is an algorithm for efficiently computing a winning strategy, and this induces an addition on the integers. We will prove that the set of points where Nim-summation agrees with ordinary addition, forms a Sierpinski Triangle, and generalize this result. Our discussion will lead us through binomial coefficients, Pascal's Triangle, and modular arithmetic to the world of objects with fractional dimensions, cellular automata, and theorems of Kummer and Legendre. No prior knowledge of any of these topics will be assumed. This work is joint with Prof. Aviezri Fraenkel at the Weizmann Institute.