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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 eachorder $k\in\{3, 4, \cdots, n-2\}$. Since the conjecture on the existence of anASMS was proposed in 1994, much attention has been paid but very little isknown except for few sporadic examples. A $k$-extended Langford sequence ofdefect $d$ and length $m$ is equivalent to a partition of$\{1,2,\cdots,2m+1\}\backslash\{k\}$ into differences $\{d,\cdots,d+m-1\}$. Inthis paper, a construction of ASMS based on extended Langford sequence isestablished. 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 onASMS.

Magic squares with all subsquares of possible orders based on extended Langford sequences