Knights are 24/13 times faster than the king

TL;DR

This paper investigates the average speed ratio between a generalized knight and a king on an infinite chessboard, deriving a formula for their relative reachability speed based on their move sets.

Contribution

It introduces a generalized framework for comparing piece speeds on an infinite grid and provides an explicit formula for the average ratio between their minimal move counts.

Findings

Derived a formula for the average speed ratio between generalized knights and kings.

Showed the ratio depends on the move parameters a and b as 2(a+b)b^2/(a^2+3b^2).

Provides insights into the relative mobility of chess pieces in an infinite setting.

Abstract

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 \begin{equation*} \begin{aligned} \mathrm{N}_{a,b} = \{(a,b), (b,a), (-a,b), (-b,a), (-b,-a), (-a,-b), (a,-b), (b, -a)\},\newline \mathrm{K}=\{(1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1), (1,-1)\} \subseteq \mathbb{Z}^2, \end{aligned} \end{equation*} 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 $2(a+b)b^2/(a^2+3b^2)$.