From: "Barrie Snell"
Newsgroups: sci.math,alt.math.recreational
Subject: Re: Magic squares / Ollerenshaw
Date: Sat, 26 Sep 1998 17:52:59 +0100
On 25 September 1998 5:10 gmyerson@my-dejanews.com wrote:-
>In article ,
> L.Koene@tue.nl wrote:
> Recently, I read about Kathleen Ollerenshaw.
> She has found a formula for magic squares, or rather perfect squares.
> Could someone give the formula and explain its meaning?
>>There is a little piece about this in Nature.
>>The magic squares in question have the following properties:
>>All rows & columns & both diagonals have the same sum (the usual
>>condition);
>>All broken diagonals have the same sum ("pandiagonal");
>>All 2-by-2 subsquares have the same sum;
>>The sum of an entry and the entry n/2 diagonally away is constant.
>>Ollerenshaw & a co-worker have found out how to construct & count these
>>for all n. The article in Nature doesn't give the construction or the
>>formula, but refers readers to a forthcoming book by Ollerenshaw & her
>>co-author.
>>Gerry Myerson (gerry@mpce.mq.edu.au)
========================================================
Come on everybody!! We've all seen enough references by now firstly to the
relevant article in Nature, and secondly to the title of the forthcoming
book. Not everyone has access to Nature as most ordinary newsagents don't
stock it, and besides that, it costs 5 English Pounds a copy. For
Magic-Square enthusiasts here are some relevant excerpts from the Nature
article by Martin Gardner reviewing the forthcoming book by Kathleen
Ollerenshaw and David Bree called "Most Perfect Pandiagonal Magic Squares :
Their Construction And Enumeration" due for publication on 1st October.
Copyright? What's that?? This is the Internet for Heaven's sake!!
---------------Quotes:-
Dame Kathleen Ollerenshaw, one of England's national treasures, has solved a
long-standing, extremely difficult problem involving the construction and
enumeration (counting how many) of a certain type of magic square.
There is only 1 order-3 magic square i.e.
2 7 6 The rows, columns and diagonals add up to 15, the magic constant.
9 5 1 Rotations and mirror-reflections are all basically the same as
4 3 8 this.
At order-4, the number of magic squares jumps to 880. Among them is a
special subset of 48 squares called pandiagonal, which have three amazing
properties. This is illustrated by the example, whose constant is 30.
0 13 6 11 Transfer this to a 4x4 grid on paper, and shade it
7 10 1 12 grey and white in a chess-board fashion, with the 0
9 4 15 2 grey, 13 white, 6 grey, 11 white, 7 white, 10
14 3 8 5 grey, 1 white, and so on, finishing with 5 grey,
and this will make the explanation very clear.
First, each broken diagonal also adds up to 30. Look at your grey and white
shading pattern on paper, and look at the example sequences of 0, 3, 15, 12
and 7, 13, 8, 2. This can be expressed in another way: imagine an endless
array of this square placed side-by-side in all directions to make a
wallpaper pattern. Then every 4x4 square drawn on the pattern will also be a
pandiagonal magic square -- in other words, every straight line of four
numbers will add up to 30. Second, every 2x2 square on the wallpaper also
adds up to 30. Third, along every diagonal (including broken diagonals) any
two cells separated by one cell add up to 15.
In general, a magic square is called pandiagonal if all its broken diagonals
add up to the magic constant. Such squares can be constructed of any odd
order above three, and of any order that is a multiple of four. If a
pandiagonal square also has similar properties to the order-4 pandiagonals,
it is called 'most-perfect': for example, the most-perfect order-8 square
below has a magic constant of 252, and its 2x2 sub-squares add up to 126,
and any two numbers that are n/2=4 cells apart add up to n^2-1=63.
0 62 2 60 11 53 9 55
15 49 13 51 4 58 6 56
16 46 18 44 27 37 25 39
31 33 29 35 20 42 22 40
52 10 54 8 63 1 61 3
59 5 57 7 48 14 50 12
36 26 38 24 47 17 45 19
43 21 41 23 32 30 34 28
Put this on a chess-board pattern on paper, with the 0 grey and the 43
white.
Although all order-4 pandiagonals have been known to be most-perfect for
three centuries, little has been known about most-perfect squares of higher
orders. There was no method of constructing them all, or even determining
the number of squares of a given order.
These questions are finally settled by Kathleen Ollerenshaw and David Bree
in their new book. The authors have devised a method for constructing all
the most-perfect squares for any order, and a way of calculating their
number.
Unlike the ordinary pandiagonals, there are no most-perfect squares with odd
order, so the only possible orders are multiples of four. At each leap in
order, the number of essentially-different most-perfect squares increases
rapidly:- from 48 squares of order 4, to 368 640 of order eight, to 2.23
x10^10 of order 12. When you reach order 36, the number is 2.77 x10^44 --
around a thousand times the number of pico-pico-seconds since the Big Bang.
This solution of one of the most frustrating problems in magic-square theory
is an achievement that would have been remarkable for a mathematician of any
age. In Dame Kathleen's case it is even more remarkable, because she was 85
when she and Bree finally proved the conjectures she had earlier made. In
her own words,"The manner in which each successive application of the
properties of the binomial coefficients that characterise the Pascal
triangle led to the solution will always remain one of the most magical
revelations that I have been fortunate enough to experience. That this
should have been afforded to someone who had, with a few exceptions, been
out of active mathematics research for over 40 years will, I hope, encourage
others. The delight of discovery is not a privilege reserved solely for the
young."
-------------------- Unquote.
=====================================
Warning to Young Mathematicians.
---------------------------------------------------
Not everything that seems symmetric is symmetric.
Consider a ski-resort full of young ladies looking for
husbands, and husbands looking for young ladies.
BaZzA