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 $k\in \{3,4,\cdots ,n-2\}$. 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 $\{d,\cdots ,d+m-1\}$. 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 $n\equiv \pm 3(\mathrm{mod}18)$, which gives a partial answer to Abe's conjecture on ASMS.