10 problems

Limiting-slope conjecture. The limit

Let $k,l\in\mathbb{N}$ with $k<l$, and consider $(k,l)$-Maharaja Nim, obtained from Wythoff Nim by adjoining moves of the form $(k,l)$ and $(l,k)$. An upper P-position is a losing…

Let $\mathcal{P}_4=\\{\\{a_i,b_i\\}\mid i\in\mathbb{N}_0\\}$ be the set of $P$-positions of the game ${\rm W}^4$, indexed so that $b_i\geq a_i$. The asymptote conjecture. The limit…

Let $k\in\mathbb{N}$, and let $\mathcal{P}_k=\\{\\{a_i,b_i\\}\mid i\in\mathbb{N}_0\\}$ be the set of $P$-positions of the game ${\rm W}^k$, with $b_i\geq a_i$. Let $l(k)$ be the le…

Diagonal Wythoff-star conjecture. The position $(i,i)$ belongs to $\mathcal{P}(\mathrm{W}^\star)$ if $i\in(S_1\cup S_4)\setminus(S_2\cup S_3)$, and belongs to…

Conjecture on Fibonacci-numeration P-positions. For all $n\ge 3$ and all $i$ satisfying these conditions, both displayed positions are $P$-positions of the game under consideration…

Fix $(p,q)=(1,2)$ or $(p,q)=(2,3)$, and let $(a_n,b_n)$ be the associated sequences. Two sequences $(l_i)$ and $(u_i)$ are complementary when they are increasing and together parti…

Fix $k\in\mathbb{N}_0$ and let $Q=((p_i,q_i))_{i\in\{0,1,\ldots,k\}}$. Let $\pi_Q$ be the permutation associated with $Q$ as defined earlier, and let a $C$-fold split have the mean…

Fix $p,q\in\mathbb{N}$, let $\pi=\pi_{p,q}$, and let $a=a(p,q)$ and $b=b(p,q)$ be the associated sequences. Let $U$ and $\mathbb{N}_0$ have the meanings fixed earlier in the paper,…

Fix $p,q\bin \mathbb{N}$, and let $a=a(p,q)$ and $b=b(p,q)$ denote the associated sequences. A pair $(p,q)$ is a splitting pair if it is a Wythoff pair or a dual Wythoff pair. The…