Order-5 Pandiagonal Magic Squares

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Introduction

A brief description of how 36 makes 144 makes 3600 pandiagonal magic squares.

Transforming 36 to 144

Change rows & columns in order 1-3-5-2-4, then diagonals to rows and .
finally each of these four squares to 25 by cyclical permutation of rows and columns.

List of 36 Essentially different

The 36 essentially different order-5 pandiagonal magic squares and relationship to Grogono's and Suzuki's lists of 144 'unique' or 'fundamental' squares.

It gets better!

You can construct all 3600 order-5 pandiagonal magic squares from just 1 square.

Cyclical pandiagonal m. s.

A summary and a comparison with orders 3, 7 and 11

Introduction

There are 36 ‘essentially different’ order-5 pandiagonal magic squares that can each be transformed into 3 other magic squares. The resulting 144 pandiagonal magic squares can each in turn be transformed cyclically to 24 other magic squares by successively moving a row or column from 1 side of the square to the other side. Completing these transformations on all 36 essentially different magic squares will produce the complete set of 3600 pandiagonal magic squares of order-5.

Each of the 36 essentially different magic squares is transformed to 3 others as follows.

  • Square 1 - original square
  • Square 2 - exchange rows and columns of original with diagonals
  • Square 3 - row and column transformation of original by reassembling in 1-3-5-2-4 order
  • Square 4 - exchange rows and columns of square 3 (above) with diagonals
  • 24 additional squares from each by cyclical transformations

The total number of order-5 pandiagonal basic magic squares is 36 times 4 times 25 equals 3600.

Because using a standard notation is so important when comparing magic square lists, I include here the definitions for two terms I will be using on this page:

Normalizing.............Rotating and /or reflecting a magic square or magic star to achieve the standard position so the figure may be assigned an index number.

Standard position....Any magic square may be disguised to make 7 other (apparently) different magic squares by means of rotations and reflections. These variations are NOT considered as new magic squares for purposes of enumeration. For the purpose of listing and indexing magic squares, a standard position must be defined. The magic square is then rotated and/or reflected until it is in this position. This position was defined by Frénicle in 1693 and consists of only two requirements.

  • The lowest of any corner number must be in the upper left hand corner.
  • The cell in the top row adjacent to the top left corner must be lower then the leftmost position of the second row (also adjacent to the top left corner).

Transforming 36 to 144

 Original square

1

7

13

19

25

14

20

21

2

8

22

3

9

15

16

10

11

17

23

4

18

24

5

6

12

Transform # 1 by 1-3-5-2-4
1 7 13 19 25
22 3 9 15 16
18 24 5 6 12
14 20 21 2 8
10 11 17 23 4
and then columns
1 13 25 7 19
22 9 16 3 15
18 5 12 24 6
14 21 8 20 2
10 17 4 11 23
Convert original diagonals Square # 1 (original) # 1 change order of  rows       to square # 3
From above square to # 2
1 20 9 23 12
8 22 11 5 19
15 4 18 7 21
17 6 25 14 3
24 13 2 16 10
The 4 basic transformations

An example of how four squares are obtainable from each 'essentially different' order-5 pandiagonal magic square.

From above square to # 4
1 9 12 20 23
15 18 21 4 7
24 2 10 13 16
8 11 19 22 5
17 25 3 6 14
To new rows diagonals to rows   change diagonals to rows

An essentially different order-5 magic square is defined as follows ( Benson & Jacoby).

  1. the number in the upper left corner is 1
  2. the number in cell 2 of the top row (next to the 1) is smaller then any other number in the top row, the left hand column, or the diagonal containing the 1.
  3. the number in the second row in the left hand column is less then the number in the last row in the left hand column

All references to the number of magic squares refers to the basic squares which is normally understood in magic square discussions. Alan Grogono, in contrast, uses the mathematical approach and considers the total number of squares (which includes rotations and reflections).

When making the above transformations, the resulting magic square will not necessarily be normalized. That is, the second cell in the top row may not be smaller then the first cell in the second row.  If required, this may simply be done by reflecting the square around the leading diagonal (exchanging rows and columns). No rotation will be required because the top left hand cell will always contain the number 1. It seems that if the original square is basic, about 4 of the 24 squares resulting from the cyclical transformation will be basic and the other 20 will require normalizing. If the original square is not basic, none of the other 24 will be either.

A comparison of the following derivatives from square # 31 with those above from square # 1 will show that different squares require normalization to produce a basic magic square.
Notice that the 36 essentially different squares I show on this page match the first of the 144 squares shown by Grogono and Suzuki (although all 3 sets are listed in different orders.
However, only 1 of the 6 squares derived from #1 and #31 (shown above) match a square in Grogono’s list of 144 and none of the 6 match any of Suzuki’s 144.

Original # 31

 

# 31 transformed

 

# 31 to 1-3-5-2-4

 

Square to left to

1

7

24

18

15

 

1

13

17

25

9

 

1

24

15

7

18

 

1

17

9

13

25

19

13

5

6

22

 

22

10

4

11

18

 

10

17

3

21

14

 

14

23

5

16

7

10

21

17

14

3

 

14

16

23

7

5

 

23

11

9

10

2

 

20

6

12

24

3

12

4

8

25

16

 

8

2

15

19

21

 

19

5

22

13

6

 

22

4

18

10

11

23

20

11

2

9

 

20

24

6

3

12

 

12

8

16

4

25

 

8

15

21

2

19

1 ,3, 5, 2, 4 to =>

 

Diagonals to rows

 

rows and columns

 

Diagonals to rows

After the cyclical permutations are performed, again many squares will require normalization.

Because different lists of the 144 fundamental magic squares may be compiled differently, lists by different persons will very likely not match.

Regardless of which list of the 144 fundamental squares is used, after the cyclical permutations are performed, and all resulting squares are normalized (where required), the resulting list of sorted 3600 pandiagonal magic squares will be identical. This suggests that an intermediate normalization step may be dispensed  with, and only the final normalization be performed, where required, on the finished list.

Alan Grogono refers to these 144 squares as ‘unique’. (He uses the number series from 0 to 24 instead of 1 to 25 to simplify mathematical computing.)           http://www.grogono.com/magic/text-5x5pan144.shtml#Subtop

Mutsumi Suzuki refers to them as "fundamental’.  His pages are now found in the MathForum    

The following 36 essentially different magic squares were compiled from tables in
W. H. Benson and O. Jacoby, New Recreations With Magic Squares, Dover, 1976, 0-486-23236-0

List of 36 Essentially different

The left hand number above each square is the Benson & Jacoby sequence number.
The middle number is the number in Grogono’s list of 144 unique magic squares.
The right hand number is the number of the square in Suzuki’s list of 144 fundamental magic squares.

1     2     35     2    4     31    3     1     24    4     3     20    5     6     33    6     8     27
 1  7 13 19 25     1  7 13 20 24     1  7 13 24 20     1  7 13 25 19     1  7 14 18 25     1  7 14 20 23
14 20 21  2  8    15 19 21  2  8    14 25 16  2  8    15 24 16  2  8    13 20 21  2  9    15 18 21  2  9
22  3  9 15 16    22  3 10 14 16    17  3  9 15 21    17  3 10 14 21    22  4  8 15 16    22  4 10 13 16
10 11 17 23  4     9 11 17 23  5    10 11 22 18  4     9 11 22 18  5    10 11 17 24  3     8 11 17 24  5
18 24  5  6 12    18 25  4  6 12    23 19  5  6 12    23 20  4  6 12    19 23  5  6 12    19 25  3  6 12

7     5     22    8     7     16    9     10    29    10    12    25    11    9     18    12    11    14
 1  7 14 23 20     1  7 14 25 18     1  7 15 18 24     1  7 15 19 23     1  7 15 23 19     1  7 15 24 18
13 25 16  2  9    15 23 16  2  9    13 19 21  2 10    14 18 21  2 10    13 24 16  2 10    14 23 16  2 10
17  4  8 15 21    17  4 10 13 21    22  5  8 14 16    22  5  9 13 16    17  5  8 14 21    17  5  9 13 21
10 11 22 19  3     8 11 22 19  5     9 11 17 25  3     8 11 17 25  4     9 11 22 20  3     8 11 22 20  4
24 18  5  6 12    24 20  3  6 12    20 23  4  6 12    20 24  3  6 12    25 18  4  6 12    25 19  3  6 12

13    14    34    14    13    30    15    15    12    16    16     8    17    18    36    18    17    26
 1  7 18 14 25     1  7 18 15 24     1  7 18 24 15     1  7 18 25 14     1  7 19 13 25     1  7 19 15 23
13 24  5  6 17    13 25  4  6 17    19 25 11  2  8    20 24 11  2  8    14 23  5  6 17    14 25  3  6 17
10 16 12 23  4     9 16 12 23  5    12  3  9 20 21    12  3 10 19 21    10 16 12 24  3     8 16 12 24  5
22  3  9 20 11    22  3 10 19 11    10 16 22 13  4     9 16 22 13  5    22  4  8 20 11    22  4 10 18 11
19 15 21  2  8    20 14 21  2  8    23 14  5  6 17    23 15  4  6 17    18 15 21  2  9    20 13 21  2  9

19    19    10    20     20    4    21    22    32    22    21    28    23    23     6    24    24     2
 1  7 19 23 15     1  7 19 25 13     1  7 20 13 24     1  7 20 14 23     1  7 20 23 14     1  7 20 24 13
18 25 11  2  9    20 23 11  2  9    15 23  4  6 17    15 24  3  6 17    18 24 11  2 10    19 23 11  2 10
12  4  8 20 21    12  4 10 18 21     9 16 12 25  3     8 16 12 25  4    12  5  8 19 21    12  5  9 18 21
10 16 22 14  3     8 16 22 14  5    22  5  8 19 11    22  5  9 18 11     9 16 22 15  3     8 16 22 15  4
24 13  5  6 17    24 15  3  6 17    18 14 21  2 10    19 13 21  2 10    25 13  4  6 17    25 14  3  6 17

25    26    21    26    25    17    27    28     9    28    27     5    29    30    23    30    29    13
 1  7 23 14 20     1  7 23 15 19    1   7 23 19 15     1  7 23 20 14     1  7 24 13 20     1  7 24 15 18
13 19  5  6 22    13 20  4  6 22    18 14  5  6 22    18 15  4  6 22    14 18  5  6 22    14 20  3  6 22
10 21 12 18  4     9 21 12 18  5    10 21 17 13  4     9 21 17 13  5    10 21 12 19  3     8 21 12 19  5
17  3  9 25 11    17  3 10 24 11    12  3  9 25 16    12  3 10 24 16    17  4  8 25 11    17  4 10 23 11
24 15 16  2  8    25 14 16  2  8    24 20 11  2  8    25 19 11  2  8    23 15 16  2  9    25 13 16  2  9

31    32    11    32    31     1    33    34    19    34    33    15    35    36     7    36    35     3
 1  7 24 18 15     1  7 24 20 13     1  7 25 13 19     1  7 25 14 18     1  7 25 18 14     1  7 25 19 13
19 13  5  6 22    19 15  3  6 22    15 18  4  6 22    15 19  3  6 22    20 13  4  6 22    20 14  3  6 22
10 21 17 14  3     8 21 17 14  5     9 21 12 20  3     8 21 12 20  4     9 21 17 15  3     8 21 17 15  4
12  4  8 25 16    12  4 10 23 16    17  5  8 24 11    17  5  9 23 11    12  5  8 24 16    12  5  9 23 16
23 20 11  2  9    25 18 11  2  9    23 14 16  2 10    24 13 16  2 10    23 19 11  2 10    24 18 11  2 10

It gets better!

We have defined what is an ‘essentially different’ pandiagonal magic square. Each of these 36 squares (order-5) may then be transformed by the diagonal to rows transformation, then each of these two by the 1-3-5-2-4 row and column change. Then each of these 4 by the 25 cyclical transformations to construct 100 different basic pandiagonal magic squares.

But it gets better! All 3600 order 5 pandiagonal squares may be constructed from this one ‘intermediate’ square.

A + a

B + b

C + c

D + d

E + e

C + d

D + e

E + a

A + b

B + c

E + b

A + c

B + d

C + e

D + a

B + e

C + a

D + b

E + c

A + d

D + c

E + d

A + e

B + a

C + b

From page 126 of New Recreations With Magic Squares

Notice that in this figure, each upper case letter and each lower case letter appears once in each row, column and diagonal indicating that this is a cyclical magic square. The fact that each letter appears once in each broken diagonal indicates that it is pandiagonal.

The values for the capitol letters may be assigned the values 0, 5, 10, 15 and 20 (in any order). For the lower case letters, the values are 1, 2, 3, 4, 5, again in any order. Or the Capitol letters may represent the low numbers with the high numbers assigned to the lower case letters.

Now, there are 10 ways we can select A. It may be any one of the set of 5 high numbers or any one of the set of five low numbers. For each of these 10 ways there are 4 ways to select B (any of the four remaining numbers from the set A was chosen from. For each of these 40 ways, there are 3 ways to select C. And finally two ways to select D (and 1 way only to select E). Similarly, for each of these 240 ways, there are 5 x 4 x 3 x 2 x 1 ways to select the lower case values. This gives us a total of 240 x 120 = 28,800 = 2(5!)2. Dividing this figure by 8 gives us the 3600 basic order 5 pandiagonal magic squares.

The complete list of 3600 order-5 pandiagonal magic squares are available for download.

Cyclical pandiagonal magic squares

Here I will present a condensation of what we have been discussing and a comparison with orders 7 and 11 cyclical pandiagonal magic squares.

Order-5 pandiagonal magic squares are of a type called Cyclical or Regular. This type of magic square has each cell containing a number represented by a capitol and lower case letter, and each letter appearing once in each row and column. All orders 4 and 5 pandiagonal magic squares are of this type.

Order-7 contains 38,102,400 cyclical pandiagonal magic squares. (It contains 640,120,320 non-cyclical (irregular) pandiagonal magic squares {Prof. Albert L. Candy}.) Order-7 pandiagonal magic squares, have two possible non-cyclical transformations. Namely 1-3-5-7-2-4-6 and 1-4-7-3-6-2-5.

For a given order, n, (where n is a prime number) there are (n-3)(n-4)(n!)2/8 basic cyclical pandiagonal magic squares.

[1]
Order (n)
[2]
Total Cyclical Basic Pandiagonal
[3]
Intermediate squares
[4]
Each [3] square gives
[5]
Essentially different
[6]
Each [5] square gives
3 0 0 0 0 0
5 3600 1 3600 36 100
7 38,102,400 6 6,350,400 129,600 294
11 11,153,456,455,680,000 28 398,337,730,560,000 921,773,260,800 12,100
  (n-3)(n-4)(n!)2/8 (n-3)(n-4)/2 (n!)2/4 [2]/[6] (n-1)(n2)

Column [6] figures are obtained by:
Order-5 essentially different square + 1-3-5-2-4, then the diagonal to row transformation for each of these gives 4 magic squares x the 25 cyclical transformations gives the 100 squares shown in column 6.
Order-7 is similar except there are two row/column transformations (see above). This gives us 6 magic squares times 72 = 294 magic square for each essentially different square.
The number of essentially different pandiagonal magic squares (column [5] is obtained by column [2] divided by column [6].

 

This page was originally posted March 2001
It was last updated August 10, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz