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 $n\equiv\pm3\pmod{18}$, which gives a partial answer to Abe's conjecture on ASMS.