Knights are 24/13 times faster than the king
Christian T\'afula

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 such that is odd, define the -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 in a box, between and , where is the -fold sumset…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Artificial Intelligence in Games · Markov Chains and Monte Carlo Methods
