Magic squares with all subsquares of possible orders based on extended Langford sequences - Li, Wen,; Zhong, Ming, (Utilitas Mathematica, 2019)
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 6 {3,4,... ,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,..., 2m+l}\{A;} into differences {d}..., 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 = ±3 (mod 18), which gives a partial answer to Abe's conjecture on ASMS. © 2019 Utilitas Mathematica Publishing Inc.. All rights reserved.