CONTENTS

A brief introduction to
order13 and it's 38 quadrant magic arrays. 

This pandiagonal magic
square contains four p34 magic arrays. 

This pandiagonal square
contains four each of p05, p13 and p38 arrays. 

This pandiagonal square
has four each of p02, p09, p21 and p25 arrays.. 

This pandiagonal contains
sets of p05, p12, p14, p18, p27 and p32. 

This pandiagonal has p02,
p03, p09, p10, p15, p16, p21, p25, p26 and p30. 

This pandiagonal has p01,
p03, p05, p08, p11, p14, p19, p22, p24 and p33. 

This pandiagonal has p01,
p08, p09, p17, p18, p21, p22, p30, p32and p37. 

This pandiagonal has p01,
p04, p05, p09, p12, p16, p17, p20, p28 and p32. 

This one has p02, p04,
p05, p07, p15, p18, p21, p23, p 26, p36 and p37. 

p01, p02, p04, p05,p07,
p09, p21, p23, p24, p25,p33, p35, p36 and p37. 

Conclusions reached and
suggestions for further study. 
Introduction
My Quadrant Magic Squares page contains
general information on this subject. It also covers orders 5 and 9.
Order13 seems to be very rich in quadrant magic squares, especially squares
that are quadrant magic in multiple ways.
Following are examples that range from 1way quadrant magic to 14way quadrant
magic.
I found these by iterating through Aale de Winkel's order13 Latin
Prescriptions. This method only generates a subset of all the possible order13
magic squares. I leave the use of alternate search algorithms as an exercise for
others.
Following are the 38 magic arrays for order13. They are presented in the
order and with the index number de Winkel assigned them. I have included the
names of the original five patterns found and investigated.
Remember, to be quadrant magic, a magic square must have 4
identical arrays in the 4 quadrants.
The plusmagic (p01), and diammagic (p23), quadrant magic squares are the
only types that can transform into an order13 (or any order 8m 
3) isolike magic star.
30 of the 38 arrays have cells that are in common with the orthogonal
adjacent arrays. This is because the center row of the magic square is
common to the two top and bottom arrays.. Likewise, with the center
column, which is common to the two sidebyside arrays.
Examples: On this page I will present 10 example quadrant
magic squares. All but the first example are multipleway quadrant magic. This
means that the same magic square has multiple sets of 4 magic arrays in it's
four quadrants.
The table following shows which magic arrays are shown in each example, and
which are present as a set of 4, but not shown.
In almost all cases, I show only one of a particular magic array, but each is
one of a set of 4, inhabiting all 4 quadrants of the magic square.
I shows the latin prescription for each square displayed, but in abbreviated
form. The parentheses are left off, as is the final digit of each vector (which
is always 0 if dealing with only 2 dimensions).
The last row (extra) shows the total number of different magic arrays that are
present at least once in the square, and the number in the most populated
quadrant.
My choice of examples provides for the display of a good variety of quadrant
magic arrays The only ones not shown are p06 and p31 for which I have not yet
found quadrant magic squares for, and p29 which I could not find in suitable
combination with other sets of 4 arrays.
A p29 quadrant magic square is (1, 3, 0)(3, 1, 0). It is also p5, 13, 15, 30 and
38 quadrant magic.
The sets of 4
identical arrays that occupy the 4 quadrants of the example quadrant
magic squares 
P # 
1way
1,21,3 
3way
8,68,7 
4way
5,15,11 
6way
3,99,3 
10way
1,51,8 
10way
5,77,5 
10way
5,99,5 
10way
10,55,10 
11way
4,129,12 
14way
5,1111,5 
1 





X 
X 
X 

@ 
2 


@ 

X 



X 
X 
3 




X 
@ 




4 







X 
X 
@ 
5 

@ 

X 

X 

@ 
X 
X 
6 










7 








@ 
X 
8 





X 
@ 



9 


@ 

X 

X 
X 

X 
10 




@ 





11 





@ 




12 



@ 



X 


13 

@ 








14 



@ 

X 




15 




@ 



X 

16 




X 


@ 


17 






@ 
X 


18 



@ 


X 

X 

19 





@ 




20 







@ 


21 


@ 

X 

X 

X 
X 
22 





X 
@ 



23 








@ 
X 
24 





X 



@ 
25 


@ 

X 




X 
26 




@ 



X 

27 



@ 






28 







@ 


29 










30 




@ 

X 



31 










32 



X 


@ 
X 


33 





@ 



X 
34 
@ 









35 









@ 
36 








@ 
X 
37 






X 

@ 
X 
38 

@ 








extra 
24/20 u.l. 
26/22 u.r. 
25/19 u.r. 
26/19 l.l. 
28/19 l.r. 
35/31 u.l. 
24/17 u.l. 
36/ 25 l.l. 
32/32 u.l. 
32/32 u.l. 
@ 
= 4 patterns of this present and at least 1 shown 
X 
= 4 copies of this present but
not shown 
The last row (extra) shows the total number of different magic arrays that
are present at least once in the square, and the number in the most populated
quadrant. (u.l. = upper left quadrant, l.l. = lower left, etc.)
Notice that the 2 examples with the most magic arrays are both 10way
quadrant magic (not 11 or 14way).
Ex. 1  1 way
quadrant magic

This quadrant magic square contains 1 set of 4 of
the p34 magic arrays in it’s 4 quadrants. Therefore it is a p34
quadrant magic square.
However, a total of 24 different magic arrays each appear in the
square at least once, with 20 in the top left quadrant.
The green cells are the center cells of each quadrant. They are
included as 1 of the 13 cells of each magic array in that quadrant.
Notice that cells in the central row and central column (shown
here in yellow) are common to two adjacent arrays.
This is Latin prescription (LP) (1,2,0) (1,3,0). 
Ex. 2  3 way
quadrant magic

This quadrant magic square contains 3 sets of 4 p05
(shown twice), p13 and p38 magic arrays in it’s 4 quadrants.
Therefore it is a 3way quadrant magic square.However, a total of
26 different magic arrays each appear in the square at least once,
with 22 in the top right quadrant.
The green cells are the center cells of each quadrant. They are
included as 1 of the 13 cells of each magic array in that quadrant.
As usual, all cells in the central row and central column are
common to orthogonal adjacent arrays.
The central cell of this square (green) is the only one that is
common to 2 arrays as shown here.
LP is (8, 6, 0)(8, 7, 0) 
Ex. 3  4 way
quadrant magic

This quadrant magic square contains 4 sets of 4
magic arrays in it’s 4 quadrants. The p02, p09, p21 and p25, making
it a 4way quadrant magic square.
Of the 8 patterns (out of 38) that have no cells on the outside rows
or columns of the quadrant, 4 of them are in this square.However,
a total of 25 different magic arrays each appear in the square at
least once, with 19 in the top right quadrant.
The green cells are the center cells of each quadrant. They are
included as 1 of the 13 cells of each magic array in that quadrant.
The LP is (5, 1, 0)(5, 11, 0) 
Ex. 4  6 way
quadrant magic

This quadrant magic square contains sets of four
p12, p14, p18 and p27 magic arrays in it’s 4 quadrants. Also present
but not shown here are sets of four p05 and p 32.
Therefore it is a 6way quadrant magic square.However, a total of
26 different magic arrays each appear in the square at least once,
with 19 in the lower left quadrant.
The green cells are the center cells of each quadrant. They are
included as 1 of the 13 cells of each magic array in that quadrant.
As usual, all cells in the central row and central column are
common to orthogonal adjacent arrays. As usual, I use green to show
common cells in this diagram.
The LP is (3, 9, 0)(9, 3, 0) 
Ex. 5  10
way quadrant magic

This quadrant magic square contains sets of 4 p10,
p15, p26 and p30 magic arrays in it’s 4 quadrants. Also present but
not shown here are sets of 4 p02, p03, p09, p16, p21 and p25.
Therefore it is a 10way quadrant magic square.A total of 28
different magic arrays each appear in the square at least once, with
19 in the lower right quadrant.
The green cells are the center cells of each quadrant. They are
included as 1 of the 13 cells of each magic array in that quadrant.
As usual, all cells in the central row and central column are
common to orthogonal adjacent arrays and as usual, I use green to
show the 1 common cell in this diagram.
The LP is (1, 5, 0)(1, 8, 0). 
Ex. 6  10
way quadrant magic

This quadrant magic square contains sets of 4 p03,
p11, p19 and p33 magic arrays in it’s 4 quadrants. Also present but
not shown here are sets of 4 p01, p05, p08, p14, p22 and p24.
Therefore it is a 10way quadrant magic square.A total of 35
different magic arrays each appear in the square at least once, with
31 in the upper right quadrant.
The green cells are the center cells of each quadrant. They are
included as 1 of the 13 cells of each magic array in that quadrant.
As usual, all cells in the central row and central column are
common to orthogonal adjacent arrays and as usual, I use green to
show the 1 common cell in this diagram.
The LP is (5, 7, 0)(7, 5, 0). 
Ex. 7  10
way quadrant magic

This quadrant magic square contains sets of 4 p08,
p17,p22 and p32 magic arrays in it’s 4 quadrants. Also present but
not shown here are sets of 4 p01, p09, p18, p21, p30 and p37.
Therefore it is a 10way quadrant magic square.A total of 24
different magic arrays each appear in the square at least once, with
17 in the upper left quadrant.
The green cells are the center cells of each quadrant. They are
included as 1 of the 13 cells of each magic array in that quadrant.
The green 99 and 129 are common to the 2 right hand arrays.
The LP is (5, 9, 0)(9, 5, 0).

Ex. 8  10
way quadrant magic

This quadrant magic square contains sets of 4 p05,
p16, p20 and p28 magic arrays in it’s 4 quadrants. Also present but
not shown here are sets of 4 p01, p04, p09, p12, p17 and p32.
Therefore it is a 10way quadrant magic square.36 of the
38 magic arrays each appear in the square at least once,
with 25 in the lower left quadrant.
The green cells are the center cells of each quadrant. They are
included as 1 of the 13 cells of each magic array in that quadrant.
As usual, all cells in the central row and central column are common
to orthogonal adjacent arrays and as usual, I use green to show the
1 common cell in this diagram.
The LP is (5, 10, 0)(10, 5, 0). 
Ex. 9  11
way quadrant magic

This quadrant magic square contains sets of 4 p07,
p23, p36 and p37 magic arrays in it’s 4 quadrants. Also present but
not shown here are sets of 4 p02, p04, p05, p15, p18, p21 and p26.
Therefore it is an 11way quadrant magic square. Be aware that I
have displayed these patterns out of order to avoid overlapping
cells.
32 magic arrays each appear in the square at least once, with all
32 in the upper left and upper right quadrants.
The green cells are the center cells of each quadrant. They are
included as 1 of the 13 cells of each magic array in that quadrant.
The LP is (4, 12, 0)(9, 12, 0). 
Ex. 10  14
way quadrant magic

This quadrant magic square contains sets of 4 p01,
p04, p24 and p35 magic arrays in it’s 4 quadrants. Also present but
not shown here are sets of 4 p02, p05, p07, p09, p21, p23, p25, p33,
p36 and p37.
Therefore it is a 14way quadrant magic square.
32 magic arrays each appear in the square at least once, with all
32 in the upper left quadrant.
The green cells are the center cells of each quadrant. They are
included as 1 of the 13 cells of each magic array in that quadrant.
The green 37 is common to both the right hand arrays in this
diagram.
The LP is (5, 11, 0)(11, 5, 0). 
Here are pictures of a crossstitch project I started July
13/03 and completed on Dec. 1/03. It is a complement square to the one
shown above.
For those who are interested in crossstitch, it is on count 16 Aida
fabric and contains 18,730 crossstitches
(plus the back stitches for the text. 


Conclusions &
Questions
For order 13 there are a total of 262,596,783,764 combinations of 13
cells in the quadrant total of 49 cells.
This number is reduced to 38 patterns when we apply the restrictions that
the pattern be fully symmetric around the central cell of the quadrant. We
call these 38 arrangements of 13 cells, quadrant magic arrays (Q. M. A.).
They are identified with the numbers p01 to p38.
A quadrant magic square is any magic square that contains a set of 4
identical Q. M. A. in its four quadrants.
Many quadrant magic squares contain multiple sets of four identical Q. M.
A.
All order13 quadrant magic squares found so far are pandiagonal
(although a number of order5 and order17 quadrant magic squares found were
not pandiagonal).
Are there any regular (not pandiagonal) quadrant
magic squares of order13?
I have not yet found any order13 p06 or p31 quadrant magic squares.
Are there any such squares?
Are there any quadrant magic squares that are
greater then 14way quadrant magic?
I displayed a magic square containing 35 and another with 36 Q. M. A.
Are there any quadrant magic squares that contain at least
one of all 38 Q. M. A?
