Common eventual period under the selective sum

Let O=(n1n2nt,p1p2pt)O=(n_1n_2\dots n_t,p_1p_2\dots p_t) and P=(m1m2ml,q1q2ql)P=(m_1m_2\dots m_l,q_1q_2\dots q_l) be two finite octal games. Write Gs\mathcal{G}_s for the scoring-play Sprague–Grundy function, and let ++_{\ell} denote the disjunctive sum and \triangledown the selective sum.

Selective-sum period conjecture. If Gs(n+m)\mathcal{G}_s(n+_{\ell}m) eventually has period pp, then Gs(nm)\mathcal{G}_s(n\triangledown m) also eventually has period pp.

This is proposed as a stronger form of the preceding periodicity conjecture: changing from the disjunctive sum to the selective sum should not change the eventual period. The paper gives no proof or resolution.

Sources & referencesView supporting material

Primary source

Fraser Stewart, “Scoring Play Combinatorial Games Under Different Operators”, arXiv:1202.4656 (2012).

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