Unsolved ECN(8_{\{1,2,3\}},i) cases

Solve the extended circular Nim games ${\rm ECN}(8_{\{1,2,3\}},i)$ for $2\leq i\leq5$ by characterizing their \mathcal{P}-positions.

Background

The paper gives a complete characterization for the case i=6, but explicitly states that the cases with 2≤i≤5 remain unresolved. These games therefore constitute a concrete family of open classification problems within the extended circular Nim framework.

References

When $S = {1, 2, 3}$ and $2 \leq i \leq 5$, ${\rm ECN}(8_{{1, 2, 3}, i)$ has not been solved.

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