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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.19589 |
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| _version_ | 1866909298748030976 |
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| author | Táfula, Christian |
| author_facet | Táfula, Christian |
| contents | 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 $2(a+b)b^2/(a^2+3b^2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19589 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Knights are 24/13 times faster than the king Táfula, Christian Combinatorics 11B13, 11B75 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 $2(a+b)b^2/(a^2+3b^2)$. |
| title | Knights are 24/13 times faster than the king |
| topic | Combinatorics 11B13, 11B75 |
| url | https://arxiv.org/abs/2405.19589 |