6 problems

Let $a_1$ and $a_2$ be nonnegative integers, and let $\mathcal{G}(a,b)$ denote the Sprague–Grundy value of the position $(a,b)$ in the game under consideration. Diagonal ultimate-p…

Let $x^*_0=1$ and let $x^*_1$ be even with $2^{k-1}<x^*_1<2^k$ for some $k\in\mathbb{Z}_{\geq}$. Define the defect…

Let $x=(x_0,x_1,x_2)$ be a position satisfying, for some $k\in\mathbb{Z}_{\geq}$, $2^{k-1}<x_1<x_0+x_1<2^k$. Let $u(x)=x_0+x_1+x_2$ be the upper bound. The eventual upper-bound con…

Let $x=(x_0,x_1,x_2)$ be a position satisfying, for some $k\in\mathbb{Z}_{\geq}$, $2^{k-1}<x_1<x_0+x_1<2^k$, and assume $x_1\leq x_2$. Let $u(x)=x_0+x_1+x_2$ be the upper bound. Th…

Let $x=(x_0,x_1,x_2)$ be a position with $x_1\leq x_2$, and let $u(x)=x_0+x_1+x_2$ denote the upper bound for its Sprague–Grundy value. The power-of-two interval conjecture. If…

Let $x=(x_0,x_1,x_2)$ be a position with $x_0\in\mathbb{Z}_{\geq}$ and $x_2\geq x_1=2^k$ for some $k\in\mathbb{Z}_{\geq}$. Write $\ell(x)=x_0+(x_1\mathbin{\oplus}x_2)$ for the lowe…