38 problems

Let $S$ be a subtraction set, and call an integer an expansion of $S$ if adjoining it to $S$ does not change the nim-sequence. If $p$ is the period length, call $S$ non-expandable…

Let $S$ be a general three-move subtraction ruleset, and let its maximum entry be $m$. Althöfer–Bültermann's conjecture. The period length is bounded by a quadratic polynomial in t…

Let $S=\{a,b,c\}$ be a three-move subtraction set with $a<b<c$, and let the nim-value sequence have an eventual period length. The set is additive when $c=a+b$. Flammenkamp–Ward co…

Consider the truncated-support subtraction-game positions and the options described immediately before the conjecture: the source identifies a specific collection of dominated opti…

Let $G=\mathcal{T}_\tau(n;a,b)$ with $a<b$, and let $R_{\widehat\beta}=\widehat\beta(a+\tau+1)$, where $\widehat\beta=\left\lfloor\frac{b-a}{b-a-\tau}\right\rfloor$. The atomic wei…

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 $S$ 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…

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 $S$ be a two-action subtraction set, let $o_{\mathrm{zs}}(h)$ be the zero-sum scoring-play outcome at heap size $h$, and let $o^1_{\mathrm{AvA}}(h),o^2_{\mathrm{AvA}}(h)$ be th…

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 $S=\{s_2,s_1\}$ with $s_2<s_1$, and let FvF and AvA denote the symmetric friendly and antagonistic self-interest games, respectively. A friendly/antagonistic discrepancy means…

Let $S$ be a subtraction set, and let FvF denote the game in which both players use friendly tie-breaking, while AvA denotes the game in which both use antagonistic tie-breaking. F…

Consider a two-dimensional subtraction ruleset with $k$ moves and regular outcomes, and let its outcome diagram decompose into segments in the sense used for two-dimensional subtra…

Let $S$ be a subtraction set with period length $p$, let $c$ be an additional move, and let $q$ be a multiple of $p$. Consider the ruleset $S\cup\{c\}$. Zhang's conjecture. If $c$…

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.

Semilinearity conjecture. The set of P-positions of every multidimensional subtraction game with nonnegative vectors of differences is semilinear.