Order 4 Magic Squares

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Contents

Introduction

Basic facts about order-4 magic squares and Dudeney groups.

The 12 Groups

Demonstrating the classification into 12 Dudeney groups.

Group I ...The pandiagonals

The 48 pandiagonal magic squares of order-4.

Group II ...The bent diagonals

The 48 bent diagonal semi-pandiagonal magic squares of order-4.

Group III …The symmetricals

The 48 associated semi-pandiagonal magic squares of order-4.

Groups XI and XII …the odd balls

The 8 magic squares for each of these groups with limited symmetry.

List of Solutions - # 1 to 200

200 of the 880 basic order-4 magic squares in index order. (A new page.)  

List of Solutions - # 201 to 400

200 of the 880 basic order-4 magic squares in index order. (A new page.) 

List of Solutions - # 401 to 600

200 of the 880 basic order-4 magic squares in index order. (A new page.) 

List of Solutions - # 601 to 880

280 of the 880 basic order-4 magic squares in index order. (A new page.) 

Introduction

There are 880 basic magic squares of order-4. The complete set was compiled by Bernard Frénicle de Bessy before 1675. [1][2]
This list has been recalculated and verified by many people since that time (myself included).

These 880 magic squares were classified into 12 groups by H. E. Dudeney and first published in The Queen, Jan. 15, 1910. The classification diagrams appeared later in Amusements in Mathematics, 1917, published by Thomas Nelson & Sons, Ltd.

Both the list of magic squares and the group classification has been more recently published in [3].

[1] Frénicle de Bessy, Des Quarrez ou Tables Magiques, including: Table generale des quarrez de quatre. Mem. de l’Acad. Roy. des Sc. 5 (1666-1699) (1729) 209-354. (Frénicle died in 1675).
[2] B. Frénicle de Bessy, et al., Divers ouvrages de mathematique et de physique (1693).
Ollerenshaw & Bondi cite a 1731 edition from The Hague??) (= Divers Ouvrages de Mathématique et de Physique par Messieurs de l’Académie des Sciences; ed. P. de la Hire; and Paris, 1693, pp. 423-507, ??NYS. (Rara, 632). Recueil de divers Ouvrages de Mathematique de Mr. Frenicle.
B. Frénicle de Bessy, Traité des triangles rectangles en nombres, dans lequel plusieurs belles proprietés de ces triangles sont demontrées par de nouveaux principes (1676) did NOT contain any magic squares. (Paul Pasles email Jan. 14, 2003).
[3] William H. Benson and Oswald Jacoby, New Recreations with Magic Squares, Dover Publ., 1976, 0-486-23236-0.

The 12 Groups

The 12 groups are classified by the patterns formed by the 8 complement pairs.
A complement pair is two numbers that together sum to n2 + 1. For order-4 that number is 17.


Group I

Group II

Group III

Group IV

Group V

Group VI

Group VII

Group VIII

Group IX

Group X

Group XI

Group XII

Groups III and VI are self-similar. That is, when each number is complemented, the same magic square is generated (only in a different orientation).

The twelve groups themselves may be grouped into four sets in which the groups in each set are strongly related. They are:

Set

Group

Number of
magic squares

Features

Symmetry

1 I 48 pandiagonal Set 1 -
1 II 48 semi-pandiagonal, bent-diagonal symmetrical around the four quadrant centers
1 III 48 semi-pandiagonal, associative symmetrical around the center point
2 IV 96 semi-pandiagonal Set 2
2 V 96 semi-pandiagonal  
2 VI-P 96 semi-pandiagonal symmetrical across the center line
3 VI-S 208 simple Set 3 - symmetrical across the center line
3 VII 56 simple  
3 VIII 56 simple  
3 IX 56 simple  
3 X 56 simple  
4 XI 8 simple Set 4
4 XII 8 simple .........

The members of each set have many features in common that become evident when working with transitions.
Also sets 1 and 2 are closely related as evidenced by the fact that 30 of the 48 transformations listed on the Transformations Summary page work for all six groups of these two sets.
Sets 2 and 3 have 2 orientations of the complementary pair pattern. 0° and 90°.
Set 4 each group has 3
orientations. Group XI has 0°, 180° and 270°. Group XII has 0°, 90° and 180° .

On this page I have posted all order-4 magic squares of the five smallest groups, which also happen to be the most interesting.

The four following pages contain the entire list of 880 solutions. They appear one solution per line, in index order. Each line includes the Dudeney group with degree of rotation required and the complement pair number and partner solution,
Each page is quite large, so be patient while it loads.

Two files are available for download:
One is sorted in index order, the other is also in index order but sorted into the 12 different groups.

Some Order-4 Features

  • The 4 corners of all order-4 squares sum to the constant S. The four central cells also sum to S.

  • All order-4 squares contain 2 even and 2 odd corners.

  • The four corner 2 x 2 arrays of all Groups I to Group VI-P squares (432 squares) sum to S. i.e. all these are gnomon-magic squares.

Group I ...The pandiagonals

A note regarding Groups I, II and III.
These are the most feature rich magic squares of order-4. In fact, pandiagonal magic squares are also known as perfect.
All the magic squares of Groups I, II and III have the feature that the corner cells of many 2x2 (i.e. all the cells), and all 3x3 and 4x4 squares sum to 34 (the magic constant).
Refer to the note following the listing for each group to realize other closely related features!

These are the 48 pandiagonal magic squares of order-4. They are scattered throughout the 880 magic squares of this order. The number above each square is the position in the indexed list. The letters A, B, C indicate which of 3 sets of 16 that square belongs to.

Following are the 48 pandiagonal magic squares of order-4.
The image on the left is the Dudeney pattern for this group, showing the complement pairs.

These 48 magic squares may be divided into 3 sets of 16 (A, B, C). Any square in a set may be transformed to any other square in the same set by moving rows and/or columns from one side of the square to the other. However, in most cases the resulting square will be rotated and/or reflected from the basic magic square shown here. Set B is shown that way on my Transformations page.

#102   A      #104   A      #107   B      #109   B      #116   C      #117   C
 1  8 10 15    1  8 10 15    1  8 11 14    1  8 11 14    1  8 13 12    1  8 13 12
12 13  3  6   14 11  5  4   12 13  2  7   15 10  5  4   14 11  2  7   15 10  3  6
 7  2 16  9    7  2 16  9    6  3 16  9    6  3 16  9    4  5 16  9    4  5 16  9
14 11  5  4   12 13  3  6   15 10  5  4   12 13  2  7   15 10  3  6   14 11  2  7
#171   B      #174   A      #177   C      #178   C      #201   A      #204   B
 1 12  6 15    1 12  7 14    1 12 13  8    1 12 13  8    1 14  7 12    1 14 11  8
14  7  9  4   15  6  9  4   14  7  2 11   15  6  3 10   15  4  9  6   15  4  5 10
11  2 16  5   10  3 16  5    4  9 16  5    4  9 16  5   10  5 16  3    6  9 16  3
 8 13  3 10    8 13  2 11   15  6  3 10   14  7  2 11    8 11  2 13   12  7  2 13
#279   A      #281   A      #292   B      #294   B      #304   C      #305   C
 2  7  9 16    2  7  9 16    2  7 12 13    2  7 12 13    2  7 14 11    2  7 14 11
11 14  4  5   13 12  6  3   11 14  1  8   16  9  6  3   13 12  1  8   16  9  4  5
 8  1 15 10    8  1 15 10    5  4 15 10    5  4 15 10    3  6 15 10    3  6 15 10
13 12  6  3   11 14  4  5   16  9  6  3   11 14  1  8   16  9  4  5   13 12  1  8
#355   B      #365   A      #372   C      #375   C      #393   A      #396   B
 2 11  5 16    2 11  8 13    2 11 14  7    2 11 14  7    2 13  8 11    2 13 12  7
13  8 10  3   16  5 10  3   13  8  1 12   16  5  4  9   16  3 10  5   16  3  6  9
12  1 15  6    9  4 15  6    3 10 15  6    3 10 15  6    9  6 15  4    5 10 15  4
 7 14  4  9    7 14  1 12   16  5  4  9   13  8  1 12    7 12  1 14   11  8  1 14
#469   B      #473   A      #483   C      #485   C      #530   A      #532   B
 3  6  9 16    3  6 12 13    3  6 15 10    3  6 15 10    3 10  5 16    3 10  8 13
13 12  7  2   16  9  7  2   13 12  1  8   16  9  4  5   13  8 11  2   16  5 11  2
 8  1 14 11    5  4 14 11    2  7 14 11    2  7 14 11   12  1 14  7    9  4 14  7
10 15  4  5   10 15  1  8   16  9  4  5   13 12  1  8    6 15  4  9    6 15  1 12
#536   C      #537   C      #560   B      #565   A      #621   B      #623   A
 3 10 15  6    3 10 15  6    3 13  8 10    3 13 12  6    4  5 10 15    4  5 11 14
13  8  1 12   16  5  4  9   16  2 11  5   16  2  7  9   14 11  8  1   15 10  8  1
 2 11 14  7    2 11 14  7    9  7 14  4    5 11 14  4    7  2 13 12    6  3 13 12
16  5  4  9   13  8  1 12    6 12  1 15   10  8  1 15    9 16  3  6    9 16  2  7
#646   C      #647   C      #690   A      #691   B      #702   C      #704   C
 4  5 16  9    4  5 16  9    4  9  6 15    4  9  7 14    4  9 16  5    4  9 16  5
14 11  2  7   15 10  3  6   14  7 12  1   15  6 12  1   14  7  2 11   15  6  3 10
 1  8 13 12    1  8 13 12   11  2 13  8   10  3 13  8    1 12 13  8    1 12 13  8
15 10  3  6   14 11  2  7    5 16  3 10    5 16  2 11   15  6  3 10   14  7  2 11
#744   B      #748   A      #785   A      #788   B      #828   A      #839   B
 4 14  7  9    4 14 11  5    5  4 14 11    5  4 15 10    6  3 13 12    6  3 16  9
15  1 12  6   15  1  8 10   16  9  7  2   16  9  6  3   15 10  8  1   15 10  5  4
10  8 13  3    6 12 13  3    3  6 12 13    2  7 12 13    4  5 11 14    1  8 11 14
 5 11  2 16    9  7  2 16   10 15  1  8   11 14  1  8    9 16  2  7   12 13  2  7

Of the 48 Group I magic squares, there are 12 pairs where lines 1 and 3 are identical. In each case, lines 2 and 4 are also identical but interchanged. Refer to the notes at end of group II and group III listings to see the close relationship between the 3 groups.

rule-w.gif (2726 bytes)

Group II ...The bent diagonals

Following are the 48  magic squares of order-4, Group II.

The magic squares in this group are all semi-pandiagonal and have the additional characteristic of magic bent diagonals. For # 21 (below) these are 1, 16, 2,15; 15, 2 11, 6, 6, 11, 5, 12; 12, 5, 16, 1 and their reverses such as 13, 4, 14, 3.

The image on the left is the Dudeney pattern for this group, showing the complement pairs.

#21           #22           #27           #28           #56           #57
 1  4 14 15    1  4 14 15    1  4 15 14    1  4 15 14    1  6 12 15    1  6 12 15
13 16  2  3   13 16  2  3   13 16  3  2   13 16  3  2   11 16  2  5   11 16  2  5
 8  5 11 10   12  9  7  6    8  5 10 11   12  9  6  7    8  3 13 10   14  9  7  4
12  9  7  6    8  5 11 10   12  9  6  7    8  5 10 11   14  9  7  4    8  3 13 10
#62           #63           #82           #83           #89           #90
 1  6 15 12    1  6 15 12    1  7 12 14    1  7 12 14    1  7 14 12    1  7 14 12
11 16  5  2   11 16  5  2   10 16  3  5   10 16  3  5   10 16  5  3   10 16  5  3
 8  3 10 13   14  9  4  7    8  2 13 11   15  9  6  4    8  2 11 13   15  9  4  6
14  9  4  7    8  3 10 13   15  9  6  4    8  2 13 11   15  9  4  6    8  2 11 13
#213          #214          #233          #234          #246          #247 
 2  3 13 16    2  3 13 16    2  3 16 13    2  3 16 13    2  5 11 16    2  5 11 16
14 15  1  4   14 15  1  4   14 15  4  1   14 15  4  1   12 15  1  6   12 15  1  6
 7  6 12  9   11 10  8  5    7  6  9 12   11 10  5  8    7  4 14  9   13 10  8  3
11 10  8  5    7  6 12  9   11 10  5  8    7  6  9 12   13 10  8  3    7  4 14  9
#269          #270          #316          #317          #323          #324
 2  5 16 11    2  5 16 11    2  8 11 13    2  8 11 13    2  8 13 11    2  8 13 11
12 15  6  1   12 15  6  1    9 15  4  6    9 15  4  6    9 15  6  4    9 15  6  4
 7  4  9 14   13 10  3  8    7  1 14 12   16 10  5  3    7  1 12 14   16 10  3  5
13 10  3  8    7  4  9 14   16 10  5  3    7  1 14 12   16 10  3  5    7  1 12 14
#421          #422          #445          #446          #450          #464
 3  2 13 16    3  2 13 16    3  2 16 13    3  2 16 13    3  5 10 16    3  5 16 10
15 14  1  4   15 14  1  4   15 14  4  1   15 14  4  1   12 14  1  7   12 14  7  1
 6  7 12  9   10 11  8  5    6  7  9 12   10 11  5  8   13 11  8  2    6  4  9 15
10 11  8  5    6  7 12  9   10 11  5  8    6  7  9 12    6  4 15  9   13 11  2  8
#465          #503          #505          #506          #583          #584
 3  5 16 10    3  8 10 13    3  8 13 10    3  8 13 10    4  1 14 15    4  1 14 15
12 14  7  1    9 14  4  7    9 14  7  4    9 14  7  4   16 13  2  3   16 13  2  3
13 11  2  8   16 11  5  2    6  1 12 15   16 11  2  5    5  8 11 10    9 12  7  6
 6  4  9 15    6  1 15 12   16 11  2  5    6  1 12 15    9 12  7  6    5  8 11 10
#591          #592          #648          #661          #662          #668
 4  1 15 14    4  1 15 14    4  6  9 15    4  6 15  9    4  6 15  9    4  7  9 14
16 13  3  2   16 13  3  2   11 13  2  8   11 13  8  2   11 13  8  2   10 13  3  8
 5  8 10 11    9 12  6  7   14 12  7  1    5  3 10 16   14 12  1  7   15 12  6  1
 9 12  6  7    5  8 10 11    5  3 16 10   14 12  1  7    5  3 10 16    5  2 16 11
#678          #679          #768          #779          #818          #844
 4  7 14  9    4  7 14  9    5  2 16 11    5  3 16 10    6  1 15 12    6  4 15  9
10 13  8  3   10 13  8  3   15 12  6  1   14 12  7  1   16 11  5  2   13 11  8  2
 5  2 11 16   15 12  1  6    4  7  9 14    4  6  9 15    3  8 10 13    3  5 10 16
15 12  1  6    5  2 11 16   10 13  3  8   11 13  2  8    9 14  4  7   12 14  1  7

Of the 48 Group II magic squares, there are 20 pairs where the first two lines are identical. In each case, lines 3 and 4 are also identical but interchanged.

Group III …The symmetricals

Following are the 48 associated magic squares of order-4. These are also semi-pandiagonal.

The image on the left is the Dudeney pattern for this group, showing the complement pairs.

#112          #113          #120          #122          #124          #126
 1  8 12 13    1  8 12 13    1  8 14 11    1  8 14 11    1  8 15 10    1  8 15 10
14 11  7  2   15 10  6  3   12 13  7  2   15 10  4  5   12 13  6  3   14 11  4  5
15 10  6  3   14 11  7  2   15 10  4  5   12 13  7  2   14 11  4  5   12 13  6  3
 4  5  9 16    4  5  9 16    6  3  9 16    6  3  9 16    7  2  9 16    7  2  9 16
#175          #176          #183          #185          #203          #206
 1 12  8 13    1 12  8 13    1 12 14  7    1 12 15  6    1 14  8 11    1 14 12  7
14  7 11  2   15  6 10  3   15  6  4  9   14  7  4  9   15  4 10  5   15  4  6  9
15  6 10  3   14  7 11  2    8 13 11  2    8 13 10  3   12  7 13  2    8 11 13  2
 4  9  5 16    4  9  5 16   10  3  5 16   11  2  5 16    6  9  3 16   10  5  3 16
#289          #290          #297          #299          #306          #308
 2  7 11 14    2  7 11 14    2  7 13 12    2  7 13 12    2  7 16  9    2  7 16  9
13 12  8  1   16  9  5  4   11 14  8  1   16  9  3  6   11 14  5  4   13 12  3  6
16  9  5  4   13 12  8  1   16  9  3  6   11 14  8  1   13 12  3  6   11 14  5  4
 3  6 10 15    3  6 10 15    5  4 10 15    5  4 10 15    8  1 10 15    8  1 10 15
#360          #361          #368          #377          #392          #395
 2 11  7 14    2 11  7 14    2 11 13  8    2 11 16  5    2 13  7 12    2 13 11  8
13  8 12  1   16  5  9  4   16  5  3 10   13  8  3 10   16  3  9  6   16  3  5 10
16  5  9  4   13  8 12  1    7 14 12  1    7 14  9  4   11  8 14  1    7 12 14  1
 3 10  6 15    3 10  6 15    9  4  6 15   12  1  6 15    5 10  4 15    9  6  4 15
#476          #478          #487          #489          #535          #539
 3  6 13 12    3  6 13 12    3  6 16  9    3  6 16  9    3 10 13  8    3 10 16  5
10 15  8  1   16  9  2  7   10 15  5  4   13 12  2  7   16  5  2 11   13  8  2 11
16  9  2  7   10 15  8  1   13 12  2  7   10 15  5  4    6 15 12  1    6 15  9  4
 5  4 11 14    5  4 11 14    8  1 11 14    8  1 11 14    9  4  7 14   12  1  7 14
#558          #562          #628          #632          #635          #637
 3 13  6 12    3 13 10  8    4  5 14 11    4  5 14 11    4  5 15 10    4  5 15 10
16  2  9  7   16  2  5 11    9 16  7  2   15 10  1  8    9 16  6  3   14 11  1  8
10  8 15  1    6 12 15  1   15 10  1  8    9 16  7  2   14 11  1  8    9 16  6  3
 5 11  4 14    9  7  4 14    6  3 12 13    6  3 12 13    7  2 12 13    7  2 12 13
#695          #698          #741          #746          #789          #790
 4  9 14  7    4  9 15  6    4 14  5 11    4 14  9  7    5  4 16  9    5  4 16  9
15  6  1 12   16  5  3 10   15  1 10  8   15  1  6 12   10 15  3  6   11 14  2  7
 5 16 11  2    1 12 14  7    9  7 16  2    5 11 16  2   11 14  2  7   10 15  3  6
10  3  8 13   13  8  2 11    6 12  3 13   10  8  3 13    8  1 13 12    8  1 13 12
#803          #808          #834          #835          #850          #860
 5 10 11  8    5 11 10  8    6  3 15 10    6  3 15 10    6  9 12  7    6 12  9  7
16  3  2 13   16  2  3 13    9 16  4  5   12 13  1  8   15  4  1 14   15  1  4 14
 4 15 14  1    4 14 15  1   12 13  1  8    9 16  4  5    3 16 13  2    3 13 16  2
 9  6  7 12    9  7  6 12    7  2 14 11    7  2 14 11   10  5  8 11   10  8  5 11

Of the 48 Group III magic squares, there are 13 pairs where lines 1 and 4 are identical. In each case, lines 2 and 3 are also identical but interchanged.
Pair 850/860 and 7 other pairs have columns 1 and 4  the same, columns 2 and 3 interchanged. There are only 6 of the 48 magic squares that do not belong to one or the other of these two sets.

Groups XI and XII …the odd balls

These two groups are the only ones not symmetrical around the horizontal and vertical center lines of the square. Consequently transformations to or from other groups will produce different results depending on the orientation of the particular magic square in these two groups.

Examining the main diagonals of the 8 group XI and eight group XII magic squares reveal the similarity between these two groups. There are a total of nine sets of four numbers that comprise the 32 main diagonals of these 16 magic squares. The 4 numbers in each set may appear in different orders.
For group XII, the first 8 appear once in the first 4 magic squares and once in the second 4 magic squares.

Group XI is not nearly as ordered as group XII, as shown by the table.

# Set of 4 numbers

Group XI

Group XII – 1st 4

Group XII – 2nd 4

1 1, 5, 13, 15

X, X, X

X

X

2 1, 9, 10, 14

X

X

X

3 2, 4, 12, 16

X, X

X

X

4 2, 9, 10, 13

X, X

X

X

5 3, 7, 8, 16

X, X

X

X

6 4, 7, 8, 15

X, X

X

X

7 4, 8, 10, 12

X, X

X

X

8 5, 7, 9, 13

X

X

X

9 6, 7, 9, 12

X

--

--

So can we say group XI is the most oddball oddball?

Group XI

#181          #202          #364          #374          #484          #643
 1 12 13  8    1 14  7 12    2 11  8 13    2 11 14  7    3  6 15 10    4  5 16  9
16  9  4  5   16  5 10  3   15  4  9  6   15 10  3  6   14  7  2 11   13  8  1 12
 2  7 14 11    9  4 15  6   10  5 16  3    1  8 13 12    4  9 16  5    3 10 15  6
15  6  3 10    8 11  2 13    7 14  1 12   16  5  4  9   13 12  1  8   14 11  2  7
#689          #724
 4  9  6 15    4 11 14  5
11  8 13  2   15  2  7 10
14  1 12  7    6 13 12  3
 5 16  3 10    9  8  1 16

Group 11 has Complementary Pair (Dudeney) patterns with 3 orientations.
No rotation:   181, 202, 364, 374, 484, 643 (the first 6).

Rotated 180°: 689.
Rotated 270°: 724.

Group XII

#3            #88           #209          #319          #449          #613
 1  2 16 15    1  7 14 12    2  1 15 16    2  8 11 13    3  4 14 13    4  3 13 14
13 14  4  3    9 15  4  6   14 13  3  4   10 16  5  3   15 16  2  1   16 15  1  2
12  7  9  6   16 10  5  3   11  8 10  5   15  9  4  6    6  9  7 12    5 10  8 11
 8 11  5 10    8  2 11 13    7 12  6  9    7  1 14 12   10  5 11  8    9  6 12  7
#650          #666
 4  6  9 15    4  6 15  9
14 12  1  7   14 12  7  1
11 13  8  2   11 13  2  8
 5  3 16 10    5  3 10 16

This group has magic squares with three CP pattern orientations.

No rotation:   88, 319.
Rotated 90°:   3, 209, 449, 613.
Rotated 180°: 650, 666.

 

This page was originally posted  June 2000
It was last updated August 06, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz