Abstract
A magic square of order n is an n × n square (matrix) whose entries are distinct nonnegative integers such that the sum of the numbers of any row and column is the same number, the magic constant. In this paper we introduce the concept of pseudo magic squares, i.e., magic squares defined over the ring of integers, without the restriction of distinct numbers. Additionally, we generalize this new concept by introducing a group (ring) structure over it. This new approach can provide useful tools in order to find new non-isomorphic pseudo magic squares.
Highlights
The concept of magic squares is well known in literature [2, 3, 5]
In this paper we introduce the concept of pseudo magic square (PMS)
We generalize this new concept in order to obtain a generic magic square (GMS), which is derived from a arbitrary group
Summary
The concept of magic squares is well known in literature [2, 3, 5]. In this paper we introduce the concept of pseudo magic square (PMS). We show that a PMS have a natural group structure We generalize this new concept in order to obtain a generic magic square (GMS), which is derived from a arbitrary group (ring). The structure of a group (ring) induces a group (ring) structure in the set of GMS’s Based on these facts, one can see that our approach is quite different of the ones available in literature.
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