Transfinite Nim: Extending the Winning Strategy Beyond the Finite

This article explores transfinite Nim, an extension of the classic game of Nim to transfinite ordinal numbers. The author first reviews the winning strategy for finite Nim – the balancing strategy – then introduces the binary representation of transfinite ordinal numbers, proving that every ordinal has a unique binary representation. The author further proves that the balancing strategy applies to transfinite Nim as well, demonstrating with a concrete example how to use this strategy to find the winning move.