The author presents a challenging game of transfinite Nim, where players strategically remove blocks from piles with ordinal heights to win. They explain the winning strategy for ordinary finite Nim based on balancing positions and extend this concept to transfinite Nim using ordinal arithmetic and binary representation. The author proves unique binary representation for all ordinals and shows how to apply the balancing strategy in a transfinite Nim game. The surprising element is the application of ordinal exponentiation and the uniqueness of the winning strategy to leave a balanced position, demonstrated through a challenge problem with ordinal heights.
https://jdh.hamkins.org/transfinite-nim/