Index
Introduction 
A brief introduction to the subject
with some history . 
Characteristics 
What makes an even orderquadrant
magic square? 
Order4 
The one possible pattern and 4
example magic squares. 
Order8 
The two symmetrical patterns and 3
example magic squares. 
Order12 
The 10 possible symmetrical
patterns and 3 example magic squares. 
Order 16 
This link is to
a subpage containing all 52 symmetrical patterns and some
nonsymmetrical patterns plus example squares for order16. 
Introduction
In 2000 and 2001
Aale de Winkel and myself investigated a new type of magic square we later named
Quadrant Magic. These squares are written up on three pages starting
with the page simply called
Quadrant Magic Squares.
These squares
were limited to odd orders. The special feature was that the square was divided
into four quadrants (quarters) with each quadrant containing a symmetrical
pattern of m cells. See the previous link for a detailed discussion on
odd order quadrant magic squares.
On May 13, 2011 I
finally sent an email to 8 friends suggesting we investigate even order quadrant
magic squares. The result exceeded my wildest expectations. From May 13 to May
24, about 90 messages were sent back and forth on this subject between members
of the group.
The original
group consisted of Christian Boyer, Arie Breedijk, Dwane Campbell, George Chen,
Francis Gaspalou, Mitsutoshu Nakamura, Walter Trump, Aale de Winkel
and myself. All
made some contribution, but Dwane Campbell, George Chen, Francis Gaspalou and
Aale de Winkel performed the lions share of the investigation!
My thanks to you all. This page summarizes the results and presents the
characteristics of even order quadrant magic squares. As usual, I will present
many examples. Hopefully I have the credits correct.
Characteristics
What makes a
magic square Quadrant magic?
Hopefully you
have previously read my original pages dealing with oddorder
quadrant magic squares. It goes into some depth on the
character of QMS.
On these pages we will
consider only even order Quadrant Magic Squares (QMS).
Conditions for
QMS
Harvey May 22/11

Only squares of
orders 4x may be quadrant magic. Orders 2, 6, 10, etc. have quadrants that are
an odd order and so cannot contain the proper patterns.

Even order
squares divided in quadrants. Each quadrant = m/2 and may, or may not
be a magic square, and/or may be compact or associated.

All 4 quadrants
must contain the same pattern of m cells summing to S.

For a normal
Quadrant Magic Square (QMS) the cells of the pattern must be arranged
symmetrically (8 fold) around the center of the quadrant.

If the square
is compact OR associated, OR the 4 quadrants are compact OR associated, then
the square is quadrant magic. It will contain all possible patterns in all 4
quadrants.

There exist the
following number of symmetrical patterns for order 4x QMS; order(patterns)
8(2), 12(10), 16(52), 20(326).
Aale de Winkel May 19/11

QMS may also
exist where the Pattern is NOT symmetrical but consists of m cells. The
identical pattern must still exist in all 4 quadrants. These are not
considered regular QMS.

Multiple
patterns may exist in the 4 quadrants of a QMS (and
usually do)!

All Compact
magic squares contain all possible symmetrical patterns for that order
(in all 4 quadrants).

Compact magic
squares are even order and pandiagonal and all 2x2 arrays sum to a constant.

QMS exist for
even order squares that are Not compact or associated. However, not all
patterns may be present (unless the quadrants are associated). All magic
squares where all quadrants are associated (not necessarily magic) contain all
possible symmetrical patterns for that order.
Dwane
Campbell May 22/11
The above points
will be demonstrated using examples.
NOTE that all
these example squares are taken from the two spreadsheets QMS_412.xlsx and
QMS_16.xlsx. Both of these files may be downloaded from here to see the tests
and to test your own squares.
Order4 Quadrant Magic Squares
(QMS)
Order4 is
the smallest magic square that can also be quadrant magic.
There is only one possible pattern for this
order because a quadrant of order 4 consist of only 4 cells. As 4 cells
are required to form a valid pattern, this means
that all 4 cells make up the pattern, and by extension all 16 cells in the
square make up the 4 required copies of the pattern.
All 48
pandiagonal cells of Dudeney’s Group I are pandiagonal and so are compact. The
48 order4 squares of Group II are all associated. So they
too
are quadrant magic.
Four
example squares.
1 




2 




3 




4 



1 
8 
10 
15 

1 
8 
12 
13 

104 
59 
77 
50 

1 
2 
16 
15 
12 
13 
3 
6 

14 
11 
7 
2 

113 
14 
140 
23 

13 
14 
4 
3 
7 
2 
16 
9 

15 
10 
6 
3 

68 
95 
41 
86 

12 
7 
9 
6 
14 
11 
5 
4 

4 
5 
9 
16 

5 
122 
32 
131 

8 
11 
5 
10 
The order4
quadrant pattern.
This is the
only pattern of 4 cells possible for the quadrants of an order 4 QMS. 
Magic Square
1
S = 34 Group 1 Index 102
This is a
pandiagonal magic square, compact, not associated.
Because it is
compact, it is a quadrant magic square with the one possible pattern
appearing in all 4 quadrants.
Magic
Square 2
S=34
Group 2, Index 112
This is a
simple magic square, associated, not compact.
It is quadrant magic because it is associated. Therefore each quadrant sums
to the constant 34. 
Magic Square
3
S = 290
This is the
center 4x4 array of cells from
Square3 (see near the end of this page).
It is a
simple magic square, Not pandiagonal, not associated and not normal.
However, it
is compact, and so is quadrant magic.
Magic
Square 4
S = 34 Group 12 No. 3
This is a
simple magic square, not associated or compact. It is not quadrant magic. In
fact not one quadrant contains the magic pattern. 
Order8 QMS
Order8 QM
squares are a little more interesting. There are two possible
symmetrical quadrant
patterns for this order.
On May 19, 2011,
Aale de Winkel predicted that for order8 there were 2 patterns, for
order12, 10 patterns, and for order 16, 52 patters. That is confirmed on
these pages (unless I have missed some, or have duplicates. For order 20,
Aale predicted 326 symmetrical patterns but I leave it to someone else to
illustrate them all!
Here I show the
two quadrant patterns as P1 and P2. I also show the four quadrants
combined in the 8 x 8 square 1 reproduced below.
In the case of the order 4 patterns it is not required to reflect them
when placing in the quadrants of the square. For the higher orders, most
patterns will require reflecting horizontally, vertically and diagonally
from the top left pattern to the top right, bottom left and bottom right
quadrants of the square.
1 
39 
10 
48 
18 
56 
25 
63 
26 
64 
17 
55 
9 
47 
2 
40 
7 
33 
16 
42 
24 
50 
31 
57 
32 
58 
23 
49 
15 
41 
8 
34 
35 
5 
44 
14 
52 
22 
59 
29 
60 
30 
51 
21 
43 
13 
36 
6 
37 
3 
46 
12 
54 
20 
61 
27 
62 
28 
53 
19 
45 
11 
38 
4 
Square 1 
1 
50 
10 
51 
21 
58 
60 
9 
45 
43 
53 
16 
38 
4 
28 
33 
49 
12 
22 
20 
32 
37 
61 
27 
14 
55 
15 
64 
56 
5 
7 
44 
30 
34 
8 
46 
24 
42 
40 
36 
54 
3 
59 
17 
47 
26 
2 
52 
48 
6 
62 
11 
13 
63 
39 
18 
19 
57 
31 
35 
29 
25 
23 
41 
Square 2 
1 
58 
3 
60 
2 
59 
16 
61 
7 
64 
5 
62 
6 
63 
4 
49 
53 
11 
55 
9 
57 
13 
47 
15 
54 
12 
56 
10 
52 
8 
50 
18 
30 
36 
17 
37 
31 
39 
32 
38 
29 
35 
28 
48 
26 
34 
27 
33 
40 
19 
51 
20 
42 
21 
43 
24 
46 
25 
45 
14 
44 
23 
41 
22 
Square 3 
This is a pandiagonal, compact magic
square. It is not associated. The quadrants also are not associated.
It contains each of the two
symmetrical patterns in all 4 quadrants so is a quadrant magic square.

This order8 square is pandiagonal magic, not
compact or associated but is a QMS.
The 4
quadrants are not magic squares but do have the associated feature.
Because of this, both patterns appear in all 4 quadrants
This square
was received from Dwane Campbell May 16/11 
This is a
pandiagonal magic square provided by Francis Gaspalou May 16/11.
Neither the
magic square or the quadrants are compact or associated.
But it is
quadrant magic with both patterns appearing in all 4 quadrants.

Order12 QMS
Order12
offers an increased variety of quadrant magic squares. As predicted
by Aale de Winkel, there are 12 symmetrical patterns.
Below are the 12 patterns and
also shown is a complete quadrant.
Here I introduce a shortcut
in the display of the patterns.. Only 1/4 of the pattern is shown.
It is the top left pattern. .Just as when applying the quadrant to
the whole square, we apply the 1/4 pattern to the whole quadrant by
Horizontal, vertical, and diagonal reflections.


The
quadrant magic square (above right) is MostPerfect and so is
Pandiagonal and compact. Being compact makes it quadrant magic!
You may check the sum of the 12 numbers in the
blue cells of a quadrant to confirm they sum to S.

Square3
In this inlaid square P1, P3, P6, and P10 appear
in all 4 quadrants, so it is a QMS!
This square is Pandiagonal, NOT associated.
It is NOT compact but has Franklin Bent diagonals!
It includes order 4 and order 8 pandiagonal magic squares
The order8 is pandiagonal but not compact or associated The order4 is
pandiagonal and so is quadrant magic because all order4 pandiagonals are.
(From P. 412 of Andrews Magic Squares
and Cubes
It is the Square 3 in my QMX_412Patterns.xlsx


This
example is NOT pandiagonal and NOT associated.
I see only one pattern (P1) and it is in only 1 quadrant (the bottom right).
So this is not a Quadrant Magic Square.
This is the Square 2 in my QMX_412Patterns.xlsx


See the order 16 quadrant magic squares
here.
