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

TL;DR

This paper introduces a new construction method for magic squares containing all smaller magic subsquares, using extended Langford sequences, and proves their existence for certain orders, advancing understanding of the longstanding conjecture.

Contribution

The paper establishes a novel construction of ASMS using extended Langford sequences, providing partial solutions for specific orders and addressing a longstanding conjecture.

Findings

Existence of ASMS for n ≡ ±3 mod 18

Construction method based on extended Langford sequences

Partial progress on Abe's conjecture

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.