Even Order Quadrant Magic Squares

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Index

Introduction A brief introduction to the subject with some history .
Characteristics What makes an even order-quadrant magic square?
Order-4 The one possible pattern and 4 example magic squares.
Order-8 The two symmetrical patterns and 3 example magic squares.
Order-12 The 10 possible symmetrical patterns and 3 example magic squares.
Order 16 This link is to a sub-page containing all 52 symmetrical patterns and some non-symmetrical patterns plus example squares for order-16.

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 odd-order 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_4-12.xlsx and QMS_16.xlsx. Both of these files may be downloaded from here to see the tests and to test your own squares.

Order-4 Quadrant Magic Squares (QMS)

Order-4 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 order-4 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 order-4 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 Square-3 (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.

 

Order-8 QMS

Order-8 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 order-8 there were 2 patterns, for order-12, 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 order-8 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.

 

Order-12 QMS

 Order-12 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 Most-Perfect 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.

 

Square-3

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 order-8 is pandiagonal but not compact or associated The order-4 is pandiagonal and so is quadrant magic because all order-4 pandiagonals are.


(
From P. 412 of Andrews Magic Squares and Cubes
It is the Square 3 in my QMX_4-12-Patterns.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_4-12-Patterns.xlsx

 

 

 

See the order 16 quadrant magic squares here.

This page was originally posted July 23, 2011
It was last updated July 30, 2011
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz