Unsolved ECN(8_{\{1,3,4\}},i) cases

Solve the extended circular Nim games ${\rm ECN}(8_{\{1,3,4\}},i)$ for $2\leq i\leq4$ by characterizing their \mathcal{P}-positions.

Background

The paper states that the cases with S={1,3,4} and 2≤i≤4 have not been solved. The cases with 5≤i≤6 reduce by isomorphism to games with S={1,3}, so the unresolved range is precisely the lower parameter range listed here.

References

When $S = {1, 3,4}$ and $2 \leq i \leq 4$, ${\rm ECN}(8_{{1, 3,4}, i)$ has not been solved.

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