24 problems

Let $S$ be a subtraction set, and consider any two tie-breaking conventions in the associated self-interest cumulative subtraction game. For a heap size $h$, let the discrepancy be…

Eventual period-two conjecture. There exists an integer $N$ such that

Main horizontal-period conjecture. Under these assumptions,

Lengyel's conjecture. (a) If $b$ is odd and at least $3$, then $L(b;b,0;1,1)$ has period $(2b(b+1),2b)$. (b) For $b\geq 2$, $L(b;1,0;1,1)$ has period $(4b,2b)$.

Let $S=\{s_3,s_2,s_1\}$ with $s_3<s_2<s_1$. Let $o_{\mathrm{zs}}$ denote the zero-sum scoring-play outcome and let the antagonistic self-interest game use AvA tie-breaking. Three-a…

Let $S=\{s_3,s_2,s_1\}$ with $s_3<s_2<s_1$, and suppose the largest subtraction is additive: … Let a friendly/antagonistic discrepancy mean a difference between the FvF and AvA sel…

Let $S=\{s_3,s_2,s_1\}$ with $s_3<s_2<s_1$, and let a friendly/antagonistic discrepancy mean a difference between the FvF and AvA self-interest games. Dominant three-action inherit…

Let $S=\{s_2,s_1\}$ with $s_2<s_1<2s_2$, and suppose … for every $k\in\mathbb{N}$. Let FvF and AvA denote the symmetric friendly and antagonistic self-interest games. First-discrep…

Let a subtraction game be played on the non-negative integers using a finite ruleset $S\subset\mathbb{N}=\{1,2,\ldots\}$. The current player subtracts an element of $S$ when possib…

Simplicity conjecture. The roots of $\chi_S(x)$ are all simple whenever $d_t\equiv 1\pmod{2}$ and $d_i\equiv 0\pmod{2}$ for all $1\leq i\leq t-1$.

Let $A=\{a,b,c\}$ with $1<a<b<c$. Define integers by the division algorithm: … … … Here $\operatorname{Per}(A)$ and $\operatorname{PrePer}(A)$ denote the eventual period and preper…

Let $A$ be a finite subtraction set with $|A|\leq 3$, let $S\in\{0,1\}^{\alpha}$ be a seed, and write $p=\operatorname{Per}(A,S)$. An integer of the form $kp+x$ is an extension of…

Non-eventual-periodicity conjecture. Not all finite two-dimensional rulesets are eventually periodic.

Maximum segmentation conjecture. For each $k\in\mathbb N$, $k\ne 2$, there exists a ruleset of size $k$ that has a $(k+1)$-segmentation. For each $k\in\mathbb N$, there is no rules…

Three-move segmentation conjecture. If $|S|=3$, there is an outcome segmentation.

Let $d$ be a positive integer and let ${\cal D}\subset\mathbb{Z}^{d}$ be a finite difference set satisfying … for every $(a_1,\dots,a_d)\in{\cal D}$. The positions are vectors…

Consider cumulative subtraction on two piles, and fix an integer $k$. A $k$-diagonal is a diagonal half-line of positions of the form $(x,k+x)$. Diagonal periodicity conjecture. Th…

Consider cumulative subtraction on two piles, where on each turn the active player chooses a pile and plays the single-pile game on that pile. Two-pile periodicity conjecture. The…

Let a cumulative subtraction game be played with a finite action set, and suppose optimal play includes sacrifices by both players. Sacrifice-size conjecture. In such a game, Negat…

Period characterization conjecture. In Case I,

Let $(a_i)_{i=0}^n$ with $n\geq2$ be a finite representing sequence with representation word $w$, and let $m$ be an integer with $n\geq2$. Define $(b_i)$ by … and let $v$ be the re…

Period-equality conjecture. For every subtraction game, the sequences $\{\mathcal{G}(n)\}$ and $\{\mathcal{V}(n)\}$ have the same period $p$. Since subtraction-game nim-sequences a…

Let $S$ be the subtraction set of an ultimately bipartite subtraction game, meaning that its nim-sequence is eventually periodic with period two and alternates between $0$ and $1$.…

Period formula conjecture. If there exists a multiple $m$ of $2a$ satisfying