Unsolved ECN(8_{\{1,2,4\}},2) case

Solve the extended circular Nim game ${\rm ECN}(8_{\{1,2,4\}},2)$ by characterizing its \mathcal{P}-positions.

Background

For S={1,2,4}, the paper identifies the case i=2 as unresolved. For 3≤i≤6, the rulesets are isomorphic to corresponding games with S={1,2}, so the explicitly unresolved exceptional case is ECN(8{1,2,4},2){\rm ECN}(8_{\{1,2,4\}},2).

References

When $S = {1, 2, 4}$, ${\rm ECN}(8_{{1, 2, 4}, 2)$ has not been solved.

— Extended circular nim  (2501.12045 - Suetsugu, 21 Jan 2025) in Section 5, subsection “Extended circular nim with eight piles”