Six-pile ECN(6_{\{2\}},2) P-position characterization

Characterize the \mathcal{P}-positions of extended circular Nim ${\rm ECN}(6_{\{2\}},2)$ by determining a closed formula for its game positions.

Background

For eight piles and S={2}, the game decomposes into a disjunctive sum of two four-pile circular Nim positions. In the specific case i=2, this requires the Sprague–Grundy values of ${\rm CN}(4,2)$, for which the paper states that no closed formula is known. Consequently, the paper identifies the corresponding eight-pile extended circular Nim classification as an open problem.