Knights are 24/13 times faster than the king

Christian Táfula

Published 2024-05-30Version 1

On an infinite chess board, how much faster can the knight reach a square when compared to the king, in average? More generally, for coprime $b>a \in \mathbb{Z}_{\geq 1}$ such that $a+b$ is odd, define the $(a,b)$-knight and the king as \[ \mathrm{N}_{a,b}= \{(a,b), (b,a), (-a,b), (-b,a), (-b,-a), (-a,-b), (a,-b), (b, -a)\}, \] \[ \mathrm{K}=\{(1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1), (1,-1)\} \subseteq \mathbb{Z}^2, \] respectively. One way to formulate this question is by asking for the average ratio, for $\mathbf{p}\in \mathbb{Z}^2$ in a box, between $\min\{h\in \mathbb{Z}_{\geq 1} ~|~ \mathbf{p}\in h\mathrm{N}\}$ and $\min\{h\in \mathbb{Z}_{\geq 1} ~|~ \mathbf{p}\in h\mathrm{K}\}$, where $hA = \{\mathbf{a}_1+\cdots+\mathbf{a}_h ~|~ \mathbf{a}_1,\ldots, \mathbf{a}_h \in A\}$ is the $h$-fold sumset of $A$. We show that this ratio equals $2(a+b)b^2/(a^2+3b^2)$.