Sequenced Concentric Rings on Magic Squares or Magic Tori

It is well known that sequenced concentric rings are present on classic bordered magic squares or tori. The subject of bordered magic squares is already well documented, and links for further reading are suggested at the end of this post.

Geometric sequenced ring totals (25, 75, 225) on a 5x5 magic square

However, this study is about sequenced concentric rings in general, and not about bordered magic squares in particular. Sequenced concentric rings exist on many different types of magic squares or magic tori, and various cases are explored up to order-10 together with one order-18 example.

It will be seen that the classic linear arithmetic sequences of bordered magic squares can also be found on non-bordered magic squares or tori. Also, some alternative arithmetic sequences, and a geometric sequence, are brought to light using simple algebra equations, and examples of each are given in the following document:

Further Reading on Bordered Magic Squares

For those amongst you who may wish to know more about the particular cases of bordered magic squares, the sites of Harry White and Inder Taneja contain further information and examples.

Also, one of the earliest books on the subject of bordered magic squares is Bernard Violle's "Traité complet des carrés magiques pairs et impairs, simples et composés, à bordures, compartiments, croix, chassis, équerres, bandes détachées etc, suivi d'un traité des cubes magiques, et d'un essai sur les cercles magiques : avec atlas de 51 grandes feuilles, comprenant 400 figures", which was first published in 1837. From page 44 to 213, its step-by-step diagrams, that explain how to construct bordered magic squares, can be easily understood by non-French readers.