## Visual Magic Squares and Group Orbits I

**©
2004 by John Pais and Richard Singer**

**Visual
Mathematics, Volume 6 (4), 2004**

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### “I am always doing that which I
cannot do, in order that I may learn how to do it.” Pablo Picasso

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**Abstract**

This paper is an essay in visual mathematics that strives to create a guided
discovery in which the learner constructs his/her own math concepts, first
intuitively, exploring, constructing, and counting 4´4 visual magic squares, and second analytically, using
numerical representations and group orbits to classify and generate these
visual magic squares. Euler and Conway visual magic squares are introduced and
used to easily create and identify 528 of the 880 essentially different 4´4 magic squares. Euler items and their
sets of complements are used to count Euler squares. Similarity classes of Conway items are
introduced to create diagonal types and row types needed to classify, count,
and generate Conway
squares. In addition, by extending an intuitive math idea naturally constructed
in this process, it is shown how Euler, Conway,
and NonConway (4´4) magic squares can
be generated using the orbits of appropriately chosen finite groups. In
particular, it is shown that each of the six types of Conway squares
introduced, can be generated by the transitive action of the *Conway* *(square)* *group*, B(4), i.e. the
group of symmetries of the hypercube.

The intuitive-analytic duality of the learning experience is commonplace in
doing mathematics (see, e.g. [4]).
Thus, in order to develop this perspective, the reader is guided to the
manifest realization that though a visual representation may be easier to use
to see what is going on intuitively, an appropriate numerical representation
often is necessary to provide an immediate, well-developed theoretical
framework within which one can more easily state and verify important
properties and results. Moreover, this paper is designed to be actively used
for self study and math enrichment with an intended audience including both
secondary school and college level students of mathematics. However, anyone
interested in magic squares should enjoy and benefit from this new visual
approach, presented in the first two sections, which provides the most
intuitive and immediate way to create 4´4
magic squares that we know. Further, if the reader is familiar with finite
groups, then our new approach using group orbits to generate all Conway squares should
also be easily accessible. In addition, links to various MapleÔ
worksheets are provided to aid in exploring and generating Conway squares. Indeed, the fact that we had
a powerful computer algebra system available as a computational aid was crucial
to engaging the daunting task of dealing with and exploring patterns and
conjectures involving these 4´4 arrays
of visual objects and numbers. So, we consider this work a paradigm example of
the essential use of technology to productively extend ones limited
computational ability in order to approach and solve a mathematical problem.

- Euler Squares and Their Representations
- Conway Squares and Their
Representations
- Generating Conway Squares via Group
Orbits
- Acknowledgements and References