I’m currently reading (on @drmackiver’s recommendation) Across the Board by John J Watkins, a study of the mathematical side of some chessboard-related puzzles. Among them is using knight’s tours to generate magic squares, which is the kind of useless but pleasing trick I am absolutely here for.

So I’m just as pleased that today’s entry in the DOME is also about magic squares – in particular, most-perfect magic squares.

What is the Ollerenshaw-Brée Theorem

I quote Robin Whitty of Theorem of the Day:

Let $n$, a positive integer, have prime factorisation $n = p_1^{s_1} p_2^{s_2}\dots p_r^{s_r}$, in which $p_1 = 2$ and $s_1 \ge 2$, so that $n$ is doubly even ((“A multiple of four,” as I would have it)). The number of $n \times n$ most-perfect magic squares, up to horizontal, vertical and diagonal symmetry, is given by:

$N(n) = M(n) \sum_{v=0}^{\tau(n)} W(v) \left[ W(v) + W(v+1)\right]$,

where $\tau(n)$ is the number of divisors of $n$; $M(n)$ is given by $M(n) = 2^{n−2} \left[(n/2)!\right]^2$; and $W(v) = \sum_{i=0}^{v} (-1)^{v+i} \nCr{v+1}{i+1} \Product_{j=1}^{r} \nCr{s_j+i}{i}$

What’s a most-perfect magic square?

A most-perfect magic square is one that satisfies two conditions:

  • Any cell in the grid has one natural ‘opposite’ cell, the one that’s $n/2$ units away diagonally. In a most-perfect magic square, the numbers in a cell and its opposite sum to $n^2-1$.
  • The numbers in any 2-by-2 block in the grid sum to $2\left(n^2-1\right)$

For example, Whitty gives the example (and how to construct it):

0 14 5 11
7 9 2 12
10 4 15 1
13 3 8 6

Every row, column, diagonal, split diagonal (such as 14-2-1-13 or 14-7-1-8) and 2-by-2 block in the grid sums to 30.

Why is it interesting?

Well, because magic squares are kind of cool. But beyond that, the Ollerenshaw-Brée formula was the first counting formula for any class of magic squares.

Who was Kathleen Ollerenshaw?

Kathleen Timpson was born in Withington, Manchester in 1912. She became both deaf ((She wouldn’t have an effective hearing aid until 1949)) and fascinated by mathematics at school, studying at Lady Barn House School in Cheadle and St Leonards in St Andrews. She studied maths at Somerville College, Oxford and remained there to earn her DPhil in 1945.

After the Second World War, Ollerenshaw ((She married Robert in 1939)) lectured part-time at the University of Manchester and took an active part in local and national politics. She died in 2014, aged 101.