Closed formulas for unresolved circular Nim cases

Determine closed formulas for the set of mathcal{P}-positions in the cases of circular Nim mathrm{CN}(m,k) not covered by the known characterizations.

Background

The paper surveys previously characterized cases of circular Nim, including several games with four through eight piles and selected values of k. It states that the remaining cases lack closed formulas determining the positions from which the previous player has a winning strategy. This is presented as an unresolved classification problem for circular Nim rather than as a problem solved by the paper.

References

For the other cases of {\sc circuar nim}, it is still open problems to find a closed formula for determining the set of $\mathcal{P}$-positions.

— Extended circular nim  (2501.12045 - Suetsugu, 21 Jan 2025) in Section 1, subsection “Circular nim”