### M is for Magic

As we have seen, Carl Brannen's QFT uses circulant matrices. By resetting a mass scale, one may renormalise a 1-circulant

XYZ

ZXY

YZX

by a constant $\lambda = \frac{1}{Y + Z - 2X}$ so that the resulting circulant obeys the condition $2X = Y + Z$. This turns the circulant into a magic square. For 2-circulants the condition is instead $2Z = X + Y$. Although perhaps not a very useful observation, it is certainly entertaining! The total number of $5 \times 5$ normal (ie. matrices built from the first few ordinals) magic squares was only computed in 1973, and the number of $6 \times 6$ ones is still unknown. There is only one $3 \times 3$ normal square, up to rotation and reflection.

A paper by A. Adler uses circulants to find an algorithm for generating higher order normal magic n-cubes, by playing with p-adic L functions. For $p = 3$, Adler constructs two cute normal magic cubes: a $3 \times 3 \times 3$ cube and a $27 \times 27 \times 27$ cube. I was further intrigued by this paper of Adler's, containing the conjecture that magic n-cubes always form a free monoid. It shows first that sets of magic squares contain prime squares, out of which all others are constructed, and then that generating functions built from cardinalities for magic cubes have the remarkable property of being everywhere divergent!

XYZ

ZXY

YZX

by a constant $\lambda = \frac{1}{Y + Z - 2X}$ so that the resulting circulant obeys the condition $2X = Y + Z$. This turns the circulant into a magic square. For 2-circulants the condition is instead $2Z = X + Y$. Although perhaps not a very useful observation, it is certainly entertaining! The total number of $5 \times 5$ normal (ie. matrices built from the first few ordinals) magic squares was only computed in 1973, and the number of $6 \times 6$ ones is still unknown. There is only one $3 \times 3$ normal square, up to rotation and reflection.

A paper by A. Adler uses circulants to find an algorithm for generating higher order normal magic n-cubes, by playing with p-adic L functions. For $p = 3$, Adler constructs two cute normal magic cubes: a $3 \times 3 \times 3$ cube and a $27 \times 27 \times 27$ cube. I was further intrigued by this paper of Adler's, containing the conjecture that magic n-cubes always form a free monoid. It shows first that sets of magic squares contain prime squares, out of which all others are constructed, and then that generating functions built from cardinalities for magic cubes have the remarkable property of being everywhere divergent!

## 2 Comments:

This monoid property of magic would be nice. The divergence of generating function as real function everywhere would mean that generating function for magic squares would exist as p-adic number for some p:s (probably very many). This would give additional strong support for the idea that p-adic generating functions are more natural than those with real argument.

Hi Matti. Unfortunately, as Adler writes and Google indicates, it appears that little research is being done on magic squares.

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