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