Winning strategy for Nim1 + (Nim1 ∨ (Nim1 + Nim1))

Determine the winning strategy for the impartial combinatorial game Nim(1) + (Nim(1) ∨ (Nim(1) + Nim(1))) under normal play, where '+' denotes Conway addition and '∨' denotes the selective sum. Specifically, characterize the P-positions and provide a rule or algorithm that decides optimal moves for all positions in this game, which arises as a subgame of circular Nim CN(6,2).

Background

Circular Nim CN(n,k) is a variant of Nim where n piles are arranged on a circle and each move selects k consecutive piles and removes a total of at least one stone; CN(6,2) is a notable open case. In analyzing CN(6,2), the paper observes that it contains the nested sum game Nim(1) + (Nim(1) ∨ (Nim(1) + Nim(1))) as a subgame via a natural embedding.

Understanding the winning strategy of this nested sum game—built from Conway addition and the selective sum—would inform the broader CN(6,2) problem. The authors’ categorical framework introduces Bouton monoids to synthesize game values under monoidal decompositions, suggesting a structured approach to such nested constructions, but an explicit strategy for this particular game is not yet known.