On the existence of reflecting $n$-queens configurations
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909742168801280 |
|---|---|
| author | Dai, Tantan Kelly, Tom |
| author_facet | Dai, Tantan Kelly, Tom |
| contents | In 1967, Klarner proposed a problem concerning the existence of reflecting $n$-queens configurations. The problem considers the feasibility of placing $n$ mutually non-attacking queens on the reflecting chessboard, an $n\times n$ chessboard with a $1\times n$ "reflecting strip" of squares added along one side of the board. A queen placed on the reflecting chessboard can attack the squares in the same row, column, and diagonal, with the additional feature that its diagonal path can be reflected via the reflecting strip. Klarner noted the equivalence of this problem to a number theory problem proposed by Slater, which asks: for which $n$ is it possible to pair up the integers 1 through $n$ with the integers $n+1$ through $2n$ such that no two of the sums or differences of the $n$ pairs of integers are the same. We prove the existence of reflecting $n$-queens configurations for all sufficiently large $n$, thereby resolving both Slater's and Klarner's questions for all but a finite number of integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12742 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the existence of reflecting $n$-queens configurations Dai, Tantan Kelly, Tom Combinatorics Number Theory 05D40, 05B30, 05C15, 11P99 In 1967, Klarner proposed a problem concerning the existence of reflecting $n$-queens configurations. The problem considers the feasibility of placing $n$ mutually non-attacking queens on the reflecting chessboard, an $n\times n$ chessboard with a $1\times n$ "reflecting strip" of squares added along one side of the board. A queen placed on the reflecting chessboard can attack the squares in the same row, column, and diagonal, with the additional feature that its diagonal path can be reflected via the reflecting strip. Klarner noted the equivalence of this problem to a number theory problem proposed by Slater, which asks: for which $n$ is it possible to pair up the integers 1 through $n$ with the integers $n+1$ through $2n$ such that no two of the sums or differences of the $n$ pairs of integers are the same. We prove the existence of reflecting $n$-queens configurations for all sufficiently large $n$, thereby resolving both Slater's and Klarner's questions for all but a finite number of integers. |
| title | On the existence of reflecting $n$-queens configurations |
| topic | Combinatorics Number Theory 05D40, 05B30, 05C15, 11P99 |
| url | https://arxiv.org/abs/2407.12742 |