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

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

Background

The paper reduces some parameter values for S={1,4} to already studied games: for i≥5, the ruleset is isomorphic to ECN(8{1},i){\rm ECN}(8_{\{1\}},i). However, the cases with 2≤i≤4 are explicitly reported as unresolved, and no characterization of their \mathcal{P}-positions is supplied.

References

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

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