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Magic Squares

Date: 10/19/2003 at 15:55:02
From: Thomas
Subject: Magic Squares

What is the size of the largest magic square that you are aware of 
that has been constructed without repeating any numbers?

I am an amateur magician and a member of American Mensa.  I have used 
the trick wherein the magician guides the audience to the number 34 
through some clever use of Hobson's choice or other ruse and then 
produces a 4 by 4 magic square using the numbers 1 thru 16 which has 
some twenty modes of symmetry all totaling 34.  Having some difficulty 
in guiding people to the number 34, I devised a method to rapidly 
calculate and construct a 4 by 4 magic square to total any even number 
using only integers and to any odd number employing mixed numbers 
including the fraction one half.  This activity may be looked upon as 
both a monumental waste of time and/or as expanding the breadth of 
human knowledge.  Magic squares can be fun, but are otherwise 
essentially useless except in magic or for their puzzling nature.

I found Ben Franklin's 8 by 8 magic square on your website and got 
entangled in the logic of its construction.  After more analysis than 
I care to admit, I borrowed some insights from the 4 by 4 square 
research that I had already done, applied it to the 8 by 8 and in less 
than five minutes, I had reconstructed his 8 by 8 square using the 
numbers 1 thru 64 into a new square that also includes a 4 by 4 magic 
square at its center and has both vertical and horizontal uniformity, 
but lacks the diagonal uniformity.  

While pondering this shortcoming, it dawned on me that I could as 
easily construct a 12 by 12 perfectly symmetrical magic square having 
at its center a perfect 4 by 4 magic square and with equal ease, I 
could construct a 16 by 16 perfect magic square containing a perfect 
8 by 8 magic square at its center and a 4 by 4 magic square at the 
center of the 8 by 8 lacking the diagonal symmetry.

Logic tells me that this can be expanded to 20 by 20, 24 by 24, 28 by 
28 and so on endlessly, limited only by the size of one's paper and 
the degree to which one is addicted to this lunacy.

At this point, the thought occured to me that no matter how big the 
largest square on record is, I could easily top it, thus attaining 
some degree of fame for accomplishing the totally useless to a 
greater degree than some other fool with an idle brain and a fair 

Not much seems to have been written on this subject.  I strongly
suspect that the same logic that I have applied to 4 by 4 squares can 
similarly be applied to 3 by 3, 5 by 5, 7 by 7 and other prime (or 
base) numbered squares.  Were I to take the time to carefully 
document my research, it would only tend to demystify magic squares 
and take the fun out of them, much as the proof that an indeterminate 
angle cannot be trisected with unmarked straight edge and compass 
took all of the fun out of that venture (some years after I began to 
attempt it courtesy of a playful geometry teacher).  

Two things about Ben Franklin's square completely bewilder me.  
First is the fact that he took the time to do it at all.  I thought 
him to be a far more practical man, in spite of his notion to fly a 
kite in a rain storm.  Second, the distribution of the numbers 
follows an almost impenetrable logic and is far different from the 
simple "cookbook" method that I developed.  I doubt that I could 
reproduce his from memory or by logic, no matter how long I study 

A 4 by 4 square containing the numbers 1 thru 16 is arranged such 
that 1 is diametrically opposed to 16.  2 is diametrically opposed to 
15 and so on.  In my 8 by 8, 1 is diametrically opposed to 64.  2 is 
diametrically opposed to 63 and so on.  In a 12 by 12, 1 and 144 are 
the starting points and for a 16 by 16, 1 and 256 are the starting 
points.  The 16 by 16 totals out to 2,056 in all directions and is 
very cumbersome to prove.  In all cases, these base numbers may be 
adjusted up or down to produce differing totals or the increment may 
be changed to alter the upper number and also change the total.  
Additionally, skipping a range of numbers in the middle allows one to 
create many different squares to the same total, each containing 
totally different numbers from the others.

Date: 10/19/2003 at 21:38:32
From: Doctor Luis
Subject: Re: Magic Squares

Hi Thomas,

Wow. You've certainly done your research. Unfortunately, nobody
can lay claim to having constructed the largest magic square simply
because there's a very easy method to construct an N x N magic square
where N is an arbitrarily large odd number.

I've constructed a 9 x 9 magic square illustrating this method:


Get it? You start on the dark orange square in the middle of the top
row, place a 1 there, and go up and to the right along the diagonal, 
putting the next number there. When you can't go up and right,
you go down. You keep going until all squares have been filled.

And that's it.

You can verify that the sums of each row, column, and long diagonal
is constant. Can you figure out what the sum is for an N x N square?

There's another method when N is a multiple of 4, but I won't go over
it right now. Can you think of a method that will work when N is even?

Well, I hope you have fun constructing larger and larger magic 
squares!  Let us know if you have any more questions.

- Doctor Luis, The Math Forum 

Date: 10/20/2003 at 14:56:15
From: Thomas
Subject: Thank you (Magic Squares)

Dear Dr. Luis:

I very much appreciate your swift and thorough reply to my question
about magic squares.  It may surprise you to know that I am also happy
to hear that there is no limit to the size of a square prior to my
insight.  Therefore, there is no need for me to check with the Guiness
folks nor to submit anything to them.  That makes one less useless
thing for me to worry about.  



Date: 10/20/2003 at 16:25:23
From: Thomas
Subject: Magic Squares

Dear Dr. Luis:

I erred when I theorized that I could manipulate and expand odd 
numbered squares in the same manner as even numbered squares.  The 
process is dependent upon the fact that any even numbered square can 
be divided into two equal number of numbers.  No odd numbered square 
can be so divided as there will always be one number left over, so to 
speak.  It MAY be possible to divide a 3 by 3 square into three groups 
and then adjust those groups up or down by adding or subtracting some 
number from each of them, such as taking 1-9 and replacing it with 1-
3, 5-7 and 9-11 or some other such scheme.  I don't know and I do 
actually "have a life" and I really need to get on with it, for the 
moment, at least.  I will likely return to this insanity later.

Your example of how to fill in a 9 by 9 square is quite interesting 
both from the simplicity of the method and from the standpoint of the 
classiness of the presentation (format of the illustration).  I wish 
that I knew just what software you used to produce the illustration.

You mentioned that you were aware of a method of dealing with 4 by 4 
squares, but chose not to elaborate upon it.  Since I also did not 
elaborate upon mine, we may both be talking about the same method and 
not be aware of it or we may have two separate methods.  

Here is my method.

Given the 4 by 4 square:

    16     2     3    13

     5    11    10     8

     9     7     6    12

     4    14    15     1

All of the rows and colums add to 34
the diagonals both add to 34
the numbers in the four corners add to 34
the numbers in the four quadrants add to 34
the four numbers in the square in the center add to 34
the two mumbers in the center and the two numbers in the center on 
the right add to 34
the two numbers in the center on the top and the two numbers in the 
center at the bottom add to 34
the diagonal pair at the lower left and the diagonal pair at the 
upper right add to 34 and the diagonal pair at the upper left and the 
diagonal pair at the lower right add to 34

This amounts to 19 symetrical combinations producing 34.  Wow!

It may also be noted that each number in the square is diametrically 
opposed by another number such that their totals will always be 17.  

So, what?

This allows for the independent substitution of another sequential 
group of numbers to replace the upper half of the numbers.

Thus replacing 9 - 16 with 10 - 17 respectively boosts the total for 
the square to 36 and still retains all of the previously stated 
symmetries.  Neato!

Now, if we construct four separate 4 by 4 magic squares using the 
following sequences:

 1 -  8, 57 - 64

 9 - 16, 49 - 56

17 - 24, 41 - 48

25 - 32, 33 - 40

We have four separate 4 by 4 squares that each total 130.

These four squares may now be joined in any fashion to produce an 8 
by 8 square.  The resultant square has all of the usual horizontal, 
vertical, and diagonal symmetry as well as containing totally 
symmetrical squares in its four quadrants and in addition, a 
vertically and horizontally symmetrical square at its center.

This seems to be the best that I can do, or that can be done.  It 
appears to me that any attempt to satisfy the diagonal symmetry in the 
4 by 4 central square automatically destroys the symmetry of the 
square as a whole.  I think that I can prove this, but I am a bit 
rusty on the logical form of a proof and I think that I will need to 
use "double reductio ad absurdem" logic to do it by virtue of the 
seeming evidence that one group of four numbers must be in two places 
at the same time to do it and since this is absurd, no can do.

My conclusions:

Odd numbered squares and even numbered squares are totally different 

I THINK that a 6 by 6 square can be created using some method similar 
to that which I have used to create the 4 by 4 squares and that it 
will also be ammenable to expansion.

I do not know how many degrees of symmetry I can squeeze out of a 6 by 
6 square.

At this point, I know how to quickly construct a 4 by 4 square 
totalling any EVEN number that one might suggest, using only integers,
and to any number using fractions - ghastly.

Sufficient study of 3 by 3 or 5 by 5 squares will likely yeild a 
simple method to quickly write squares of those types to any ODD 
number using only integers.

Armed with these two processes, I should be able to rapidly astound 
an audience by creating a magic square totalling any number they 
choose almost as fast as I can write the numbers down. 

Were they still living, my grade school and high school mathematics 
teachers would be the most astounded.

I also sometimes find myself assisting children with mathematics.  At 
times, high school age kids tend to think that they know more than I 
do and that they need not listen to me.  Requesting a number and 
producing a 4 by 4 square to that total before they have even had the 
chance to ask why I want it generally gets their attention back.

This last seems to be the most beneficial use for this madness that I 
currently have, unless this can somehow be applied to the process of 
tying "Lover's Knots" or "Turk's Head" knots.   

Have I shared anything new to you?



Date: 10/22/2003 at 15:56:06
From: Thomas
Subject: Magic Squares

Dear Dr. Luis:

This topic must be disturbing my sleep.  Having studied the 9 by 9 
square that you so kindly sent me, I applied its logic to the 3 by 3 
types and now can easily recall or recreate the magician's 3 by 3 
used in card tricks which I could not do before because of my 
apparent inability to memorize strings of numbers.  Logic sticks, 
discrete data vanishes.  

At any rate, I awoke this morning with the notion that I could 
rebuild the 9 by 9 square from nine 3 by 3 squares.  I laid out a 9 
by 9 grid and then subdivided it into nine 3 by 3 grids.  Starting 
with the number 1 in the top center small square in the top center 
large square, I filled in that large square with then numbers 1 thru 
9 using the odd square logic for a 3 by 3 square.  Then using 3 by 3 
logic I moved to the top center small square of the lower right large 
square and did the same using the numbers 10 thru 18.  Then, I moved 
on to the center left hand large square and so on in simmilar fashion 
through the rest of the grid finally ending with the number 81 in the 
center square of the bottom row.  The result is a 9 by 9 magic square 
composed of nine 3 by 3 magic squares with no numbers repeated.

I find this square easier to write as it repeatedly employs the 3 by 3 
pattern and, therefore, requires less conscious thought.

Applying this to magic, I can rapidly take any number, odd or even, 
subtract 40 from that number and starting with that result, construct 
a 9 by 9 magic square, having the given number at its exact center.  
The process is simple, but the effect may be mind boggling on an 
unsuspecting audience.  

What do you think?



PS.  In case you are curious, I am 60 years old.

Date: 10/23/2003 at 17:39:51
From: Doctor Luis
Subject: Re: Magic Squares

Hello Thomas,

Wow. Thank you for your interest. I'm glad to know that you can
now create 3x3 magic squares easily. It turns out that a lot of
people here ask about magic squares. Dr Jeremiah pointed out to
me the wealth of information in our archives, which I'll relay to you.

Here are some links of interest, 

as well as some answers involving odd and even squares,

  Finding Magic Squares 

  Largest Magic Square Ever Known 

  Even-Numbered Even Squares 

Hopefully these links will serve to answer some of your questions
and help you sleep better at night :-)

As always, let us know if you have any more questions.

- Doctor Luis, The Math Forum 
Associated Topics:
High School Puzzles

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