Introduction 
A brief introduction to this subject and quadrant magic arrays. 
Order5 quadrant magic squares 
Examples of magic quadrant arrays and order5 quadrant magic squares. 
Isolike magic Stars 
An example shows the relationship to quadrantmagic squares. 
Order9 quadrant magic squares 
Examples of magic quadrant arrays and order9 quadrant magic squares. 
Order13 quadrant magic squares 
A page of magic quadrant arrays and order13 quadrant magic squares. 
Order17 quadrant magic squares 
A page of magic quadrant arrays and order17 quadrant magic squares. 
Quadrant Magic Sqrs. Summary 
Comments, questions, and a credit (and link) to Aale de Winkel. 
Some magic squares of orders m equal to 4n + 1, have arrays of m
cells appearing in each quadrant that sum to the magic constant.
If a magic square contains 4 of these arrays in the 4 quadrants, and if they are all the
same type, I call it a quadrantmagic square.
Consider a magic square of order m = 4n + 1 (i.e. 5, 9, 13, 17, etc).
Divide it into 4 quadrants such that each quadrant consists of (m + 1)/2 times (m
+ 1)/2 cells
Each array consists of m 1 cells, plus the central cell of the quadrant.
The array is considered magic if the m cells sum to the magic constant for the
square.
Quadrant Magic Arrays
The cells in the array must be arranged so that they are orthogonally and diagonally
symmetrical. This condition reduces the number of possible magic arrays to a manageable
number (at least for the smaller magic squares).
Note that the central row and the central column of the Quadrant Magic Square is common to
two orthogonally adjacent quadrants. This means that if an array has cells in the outside
row and column, these cells are shared with the adjacent quadrant.
The first 6 magic arrays were all discovered by Aale de Winkel in May, 1999. They are
the cross, plus sign, diamond, small ring, large ring and thickcross. He named them
respectively, crosmagic, plusmagic diammagic, sringmagic, lringmagic and tcrosmagic.
After I found some additional arrays, he decided to investigate the subject
systematically.
It turns out there are 10 classes of arrays, determined by their degree of symmetry.
Because of the very large number of magic arrays, we define a Quadrant Magic
Square as using only the highest order one of these, the fully symmetric
one. This class we call 'Quadrant' or quadrant magic array.
Aale enumerated the Quadrant magic patterns for orders 5, 9, 13 and 17 and labeled them
with index numbers prefixed with a 'p'. For order 13 there are 38 quadrant magic patterns,
but a total of 262,596,783,764 patterns counting all 10 classes of magic arrays.
See his Special_Magic.html page for details and listings. (The link to his site appears at
the bottom of this page.)
The first five of the above named arrays are fundamental.
They appear in all orders 4n + 1 (altered, of course, to contain the correct number
of cells).
Because the 'p' numbers for these patterns vary from order to order, these names will be
retained.
Here are the fundamental arrays for order9, with the corresponding 'p' number.
plusmagic (p1) 
sringmagic (p2) 
diammagic (p3) 
lringmagic (p5) 
crosmagic (p6) 
Quadrant Magic Squares
Some quadrant magic squares may be converted to isomorphiclike magic stars. For
order5, they are isomorphic. For the other orders, they are only pseudoisomorphic
because they cannot use all the numbers contained in the quadrant magic square.
The only magic array that may be used to form an isolike
magic star for all orders 4n+1 is the plusmagic.
Additionally, the diammagic array can form isolike
magic stars for orders 8n – 3.
The reasons for this are discussed in the isolike magic star section.
On this page, I will show examples of quadrantmagic squares, present known and
suspected characteristics, and pose a number of questions for further study.
I will also present an example of the order5 isomorphic magic star to show how the
quadrant magic square arrays are used to create these stars.
As my first example, here is an order17 pandiagonal
magic square that I show as quadrant magic four different ways by illustrating a different
array in each quadrant. However, each array actually appears in all four quadrants.
As well as the 17 rows and columns, and the 34 diagonals, each of these arrays sum to the
magic constant of 2465.

The arrays are: sringmagic (p082) plusmagic (p001) p085 lringmagic (p213) not shown is the p216 (and surely many others) The center numbers in each quadrant are: 127 256 16 145 The arrays are centered around them and they are one of the m cells of the array. Because numbers 200 and 72 are on
the center column, they are common to 2 adjacent horizontal plusmagic arrays. 
With the lringmagic arrays, there are five numbers common to each of two orthogonal
arrays.
This is LP (14, 1, 0)(1, 14, 0).
I will discuss the characteristics of these arrays and quadrant magic squares in more detail as I introduce the different orders.
Suffice to say that in order to qualify as a quadrant magic square:
Notes:
Quadrant diagrams
Order5 essentially different pandiagonal ................... crosmagic, plusmagic
Order5 1 of the 99 derivatives of above .................. crosmagic, plusmagic
Order5 normal (not pandiagonal) ............................ no quadrant magic arrays
Order5 normal (not pandiagonal) ............................ crosmagic, plusmagic
Order5 normal associative ...................................... only 2 quadrant magic
arrays
Order5 normal associative ...................................... plusmagic, & 2
crosmagic
Order5 pandiagonal associative .............................. plusmagic, & 2
crosmagic
plusmagic 
For order5 each quadrant is 3 by 3
cells. Because of the small number of cells in the order5 quadrant, there are only 5 quadrant magic arrays possible: The plusmagic and diammagic (which are the same for this order), and the crosmagic, sringmagic and lringmagic (which are the same for this order). All 36 essentially different order5 pandiagonal magic squares are plusmagic. In fact, there is a magic array of 5 cells centered around each of the 25 cells of each of these magic squares (using wraparound when necessary). From the above I think we can safely assume that all 3600 order5 pandiagonal magic squares are plusmagic. However, all order5 plusmagic are NOT pandiagonal (see examples 4 & 6 below). 

crosmagic 

This is essentially different pandiagonal magic
square # 10 (of 36). This square is
plusmagic. In fact, there is a magic array centered on each of the 25 cells of the magic
square. It may also be considered diammagic (the diamonds have sides of length 2). It is also crosmagic and may also be considered sringmagic and lringmagic, which is the same configuration for order5. 

This is one of the 99 pandiagonal derivations of
the above square. It was obtained by
transformation 13524 applied to the rows and columns: then rows and columns
interchanged with the diagonals; then another transformation 13524 applied to the rows
and columns. This one shows crosmagic arrays. However, both this and the previous magic square contain four of each of these two magic arrays. 

This is an normal (not
pandiagonal) magic square. It is not quadrant magic. There are no magic arrays. 


This is a normal (not
pandiagonal) magic square. This is a plusmagic quadrant magic square. Normal magic squares that are quadrant magic seem to be relatively rare. 


This is a normal (not
pandiagonal) associated magic square It is not quadrant magic because only two quadrants have a plusmagic array. It seems that there is always zero, two or four quadrants correct in order5 magic squares. 


This is a normal (not
pandiagonal) associated magic square This is a plusmagic quadrant magic square. There are two crosmagic arrays, but only two (see them?), so this square is not a crosmagic quadrant magic square. 


This is a pandiagonal associated crosmagic
square The crosmagic array cannot be used to form an order5 isomorphic star for 2 reasons
This square also is plusmagic (and diammagic), so an order5 isolike magic star may be made using these arrays. 
Order5 ...some conclusions and questions
Order5 has only five magic arrays because of the
small size of the quadrants, and only two of these are unique.
The plusmagic and diammagic arrays are identical, as are the crosmagic, sringmagic and
lringmagic arrays.
All 36 essentially different pandiagonal magic
squares are plusmagic on all 25 cells.
Does this apply to the 99 variations of each of these?
Only some regular order5 are plusmagic.
Are any of these plusmagic for all 25 cells?
Are all of the nine pandiagonal associated magic squares plusmagic?
Are any of the regular associated magic squares plusmagic?
A. Plusmagic

B. Crosmagic

The magic star shown below is isomorphic to
magic square A. Each number in the magic square is mapped to a location in the star. Order5 is the only
size of quadrant magic square that can be transformed to a magic star using all the
numbers contained in the square. For that reason, I use the general term 'Isolike magic
stars' to cover all orders. 

Isomorphic magic star 
Orders 9, 13, 17, etc magic squares may be
used to form this type of star and only if the square is quadrantmagic. Only 25 of the
numbers in the magic square can be used, however. Finally, for a given order, only certain
magic arrays can be used to form such a star. This order8 type B star has 12 lines of 5 numbers summing to the magic constant, the same as the order5 magic square. The outside horizontal and vertical lines (I call these the ‘square’) contain the same numbers in the same order as the outside rows and columns of the magic square. This predetermines two of the numbers, such as the 2 and 21, in each outside diagonal line (I call this the diamond). The corner numbers of the diamond are those that are common to two magic arrays in the quadrant magic square. In this case the 13, 19, 5 and 6 in the plusmagic arrays of square A. This leaves just one number in each side of the diamond to be assigned, and this is the fifth number of the corresponding magic array. 
Square B shows the crosmagic arrays. On close examination we see two reasons why an isolike magic star cannot be formed from this array.
For any order quadrant magic square the plusmagic array
may be used to convert the square to an isolike magic star.
For orders 8m  3 the diammagic array may be used but orders 8m+1
fail because of the additional diagonal cells.
All other quadrant magic arrays fail, for every order, due to either or both situations mentioned above.
Please see my Isolike Magic Stars page for more details and examples of orders 5, 9 and 13 stars.
Starting with this order, we identify the patterns by their index numbers.
p1 (plusmagic) 
Quadrant magic arrays The same array must appear in all four quadrants of the magic square for it to be called a Quadrant magic square! Quadrant magic squares using this array can form isolike
magic stars. 

p2 (sringmagic) 
A quadrant magic square with this array cannot be transformed into an
isolike magic star because of the 3 cells (instead of 1) that appear on the diagonals. So far, all Quadrant Magic squares found using this array are also lring quadrant magic. 

p3 (diammagic) 
This array also cannot be used to form an isolike magic star because of
the 3 cells (instead of 1) that appear on the diagonals. There are diammagic quadrant magic squares. 

p4 
This array cannot be used to form an isolike magic star due to both of the reasons explained above. So far, I have found no Quadrant Magic squares using this array. Aale de Winkel found this pattern on Aug. 31, 1999. He also showed mathematically that there can be only 7 totally symmetric patterns for order9. 

p5 (lringmagic) 
This array cannot be used to form an isolike magic star because of the 3 cells (instead of 1) that appear on the diagonals. Also, 3 cells instead of 1 appear in the outside rows and columns, and so are common to orthogonally adjacent quadrants.  
p6 (crosmagic) 
This array cannot be used to form an isolike magic star because of
the 5 cells (instead of 1) that appear on the diagonals. Also, 2 cells instead of 1 appear
in the outside rows and columns, and so are common to orthogonally adjacent quadrants. In any case, I have not found such magic squares in order9. 

p7 
This array cannot be used to form an isolike magic star because 2 cells
instead of 1 appear in the outside rows and columns, and so are common to orthogonally
adjacent quadrants. So far, I have found no Quadrant Magic squares using this array. 

Here the lime green cells are the center of each quadrant and are 1 of the
9 cells in each array. The yellow cells are the lringmagic arrays. I show only 2 of the 4 quadrants for clarity (in the upper left and lower right quadrants). The blue cells are the sringmagic arrays. An order9 sringmagic square cannot form an
isolike magic star because 3 cells instead of 1 are on the main diagonal. 
All ringmagic quadrant magic squares found to date are both sringmagic and lringmagic (both p2 and p5).

An order9 diammagic square cannot form an isolike magic star because 3
cells instead of 1 are on the main diagonal. These two types of order9 quadrant magic squares are the only ones found to date. 
There are 7 quadrant magic arrays (totally symmetric patterns) for
order9.
So far only 2 types of quadrant magic squares have been found.
Do sringmagic and lringmagic arrays always appear together?
Do diammagic arrays never appear with sringmagic or lringmagic squares?
Are there NO crosmagic, plusmagic, p4 or p7 order9 quadrant magic squares?
Are all order9 quadrant magic squares pandiagonal?
Orders 13 and 17 are on separate pages due to amount of material. See order13 order17
My search for quadrant magic squares was performed mostly using Latin
prescriptions.
It would be interesting to see results of searches using other methods of magic square
generation.
Most of the quadrant magic squares found (and all orders 9 and 13) are pandiagonal .
Are many regular magic squares quadrant magic?
Are many associated magic squares quadrant magic?
Can there be quadrant magic squares of even order?
(the quadrants would be n/2 with no central cell in the array.)
All quadrant magic squares found to date have been a
result of searching.
Can an algorithm be developed to generate quadrant
magic squares?
Order  Number of magic arrays on these pages  Quadrant magic  Order13 seems to have the most densely packed quadrant magic
squares. Order13 has a 14way quadrant magic square. The best I can find for order17, which should have a great many more combinations, is a 6way quadrant magic square. So far, all order13 Quadrant m. s. found are pandiagonal, although regular m. s. have been found for orders 5 and 17. 
5  5 but only 2 are unique  2 way quadrant magic  
9  7  2 way quadrant magic  
13  38  14 way quadrant magic  
17  15 ( of a total of 253)  6 way quadrant magic 
I wish to thank Aale de Winkel for discovering these fascinating magic
squares and for all the help he has given me in my attempts to consolidate the features of
quadrant magic squares and isolike magic stars.
As well as suggestions, he provided me lists of his latin prescription squares where he
searched for these features, and a program to convert any latin prescription (LP) to an
actual magic square.
Please visit his site at
http://www.magichypercubes.com/Encyclopedia/index.html then link to quadrantmagic and specialmagic.
Please send me Feedback about my Web site!
Harvey Heinz harveyheinz@shaw.ca
This page last updated
March 11, 2006
Copyright © 1999 by Harvey D. Heinz