However, when looking at the limitless surface of a magic torus, which has no centre, a concentration of magic lines that intersect over a number in even-orders becomes a very interesting feature.Ī second example of extra-magic is the presence of knight move magic diagonals that occur on some magic tori (and on the magic squares that the magic tori display). Therefore, in even-orders, the cases where the intersections of magic diagonals occur over number nodes (2 magic orthogonals + 2 magic diagonals), are often ignored, because these never produce magic squares. In odd-orders the two intersections of a same pair of magic diagonals only produce a single magic square, because the second intersection (at the opposite side of the torus), occurs between number cells, and cannot be at the centre of a second magic square. In even-orders the two intersections of a same pair of magic diagonals always produce two distinct magic squares on the magic torus. When we begin to study the wrap-around characteristics of the magic tori that display the magic squares, we notice that a same pair of magic diagonals produces two intersections on a magic torus. The Definition of Extra-MagicĮxtra-Magic can be defined as the presence of any magic line intersections on Magic Tori that are not usually taken into account in the study of Magic Squares.įor example, the central intersection of the magic diagonals of a magic square takes place between the number cells of even-order magic squares, and over a number cell of odd-order magic squares.
The extra-magic tori of orders 3 and 4 are here presented in detail, together with some further observations on the extra-magic tori of orders 5 to 8, the eighth-order being particularly significant for those who are interested in chess moves.
It has also prompted me to look into the magic line graphs of magic tori, and to search for cases having a maximum number of magic line intersections. Published by Carlos Rivera on his website "Prime Puzzles" at the end of May 2019, the Puzzle 391 "9 dots and 8 lines graph" has stimulated a lot of conversation between magic square enthusiasts, and has inspired several ongoing experiments.
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