Magic squares with all subsquares of possible orders based on extended Langford sequences
Abstract
A magic square of order $n$ with all subsquares of possible orders (ASMS$(n)$) is a magic square which contains a general magic square of each order $k\in\{3, 4, \cdots, n2\}$. Since the conjecture on the existence of an ASMS was proposed in 1994, much attention has been paid but very little is known except for few sporadic examples. A $k$extended Langford sequence of defect $d$ and length $m$ is equivalent to a partition of $\{1,2,\cdots,2m+1\}\backslash\{k\}$ into differences $\{d,\cdots,d+m1\}$. In this paper, a construction of ASMS based on extended Langford sequence is established. As a result, it is shown that there exists an ASMS$(n)$ for $n\equiv\pm3\pmod{18}$, which gives a partial answer to Abe's conjecture on ASMS.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 DOI:
 10.48550/arXiv.1712.05560
 arXiv:
 arXiv:1712.05560
 Bibcode:
 2017arXiv171205560L
 Keywords:

 Mathematics  Combinatorics