  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.
