Most-perfect Magic Squares

Introduction	Features of Most-perfect magic squares
Features of a reversible square	Transformation of Reversible Squares
Number of principle reversible squares	Some Examples
Addendum - October, 2006	Addendum-2 - November, 2006
Most-perfect magic cubes	Most-perfect Bent diagonals (a sub-page)

Introduction

A special type of pandiagonal magic square was described in an 1897 paper [1] by Eamon McClintock of Toronto University. He used a special square array called a McClintock square as an aid in their construction. He also showed that there was a one-to-one correspondence between the most perfect magic square and the McClintock square.

Recently Dr. Ollerenshaw, looking for a way to enumerate at least a sub-set of pandiagonal magic squares, realized that there was a way to enumerate all McClintock squares of a given order. She refined the definition of this square and renamed it reversible. The resulting magic square she called Most-Perfect (with a hyphen). She was 74 years old when she published her first paper [2] on Most-perfect magic squares (in 1986). She published at least two other papers on the subject.

In 1998, Dr. Ollerenshaw co-authored a book [3] on this subject with Dr. David Brée. Dr. Brée is Professor of Artificial Intelligence at the University of Manchester (the University which Dame Ollerenshaw was associated with).

Dr. Brée's main contributions to the book was to change the method of construction, which led to a simpler method of enumeration, and to find and then prove the equation for the new method of enumerating ALL doubly even squares.

On this page I will attempt to present a simplified introduction to this type of magic square using mostly material from the above book.

[1] McClintock, E. (1897) On the most perfect forms of magic squares, with methods for their production. American Journal of Mathematics 19 p.99-120.
[2] Ollerenshaw,K. (1986) On ‘most perfect’ or ‘complete’ 8 x 8 pandiagonal magic squares. Proceedings of the Royal Society of London A407, p.259-281
[3] Kathleen Ollerenshaw and David Brée, Most-perfect Pandiagonal Magic Squares, Institute of Mathematics and its Applications, 1998, 0-905091-06-X
[4] See also Ian Stewart, Mathematical Recreations column, Scientific American, Nov. 99, p.122-123.

Features of Most-perfect magic squares

4	5	16	9
14	11	2	7
1	8	13	12
15	10	3	6

All 48 pandiagonal magic squares of order-4 are most-perfect!

For other orders, not all pandiagonal magic squares are most-perfect.

The 4 corner cells of any square array of cells in an order-4 most-perfect magic square sum to S.

Definition

Every 2 x 2 block of cells (including wrap-around) sum to 2T (where T= n² + 1) (i.e. compact)
Any pair of integers distant ½n along a diagonal sum to T (i.e. complete)
Doubly-even pandiagonal normal magic squares (i.e. order 4, 8, 12, etc using integers from 1 to n²)

Additional feature
Two integers 1/2n along any row, with the left integer in an even column, have the same sum. The same is true when the left integer is in an odd column. These two sums (for evens and odds) sum to 2T. This feature is useful in proving that any most-perfect magic square can be transformed into a reversible square.

Note:

Note2:

For mathematical convenience, the authors use the series from 0 to n²-1.
In that case S=n(n²-1)/2, T = n²-1.
I have chosen to use the series from 1 to n² to be consistent with the definition of a normal magic square with S=n(n²+1)/2.

I use the symbol S to indicate the magic sum and T to indicate the value of n² + 1) which the authors indicates with S.

Higher dimensions
Aale de Winkel reports that these same features also apply to higher dimension magic figures. Go to his magic objects site from my links page.

Features of a reversible square

1	2	3	4
5	6	7	8
9	10	11	12
13	14	15	16

Definition

Equal cross sums. (similarity)
1 + 6 = 2 + 5, 1 + 11 = 3 + 9, 1 + 16 = 4 + 13, etc.
Also 1 + 14 = 2 + 13, 2 + 12 = 4 + 10, etc.
In general, the sum of the two numbers at diagonally opposite corners of any rectangle or sub-square within the reversible square will equal the sum of the two numbers at the other pair of diagonally opposite corners.
Note also that 1 + 4 = 2 + 3, 1 + 13 = 5 + 9, 9 + 12 = 10 + 11, etc. In general the sum or the first and last numbers in each row or column equal the sum of the next and the next to last number in each row or column, etc. (reverse similarity).
Integers 1 and 2 must appear in the same row or column.

Reversible squares can be grouped into sets in which all squares can be transformed from one to another. There are M_n=2^n-2{(1/2n)!}² essentially different squares in each set.

There is a unique principle reversible square in each set in which all the rows, reading left to right , and all the columns reading top to bottom, contain integers in ascending order, and the top row begins with the integers 1and 2.
These are the 3 principle reversible squares for order-4. The other reversible squares for this order are simply the 15 rearrangements of the rows and/or columns of these three. There are 3 x 16 or 48 reversible squares, each of which may be transformed into a most-perfect magic square.

1	2	5	6
3	4	7	8
9	10	13	14
11	12	15	16

1	2	9	10
3	4	11	12
5	6	13	14
7	8	15	16

The first set of 16 essentially different reversible squares for order-4.
The principle reversible square is the one in the top left corner.

1	2	3	4	....	1	2	3	4	....	5	6	7	8	....	5	6	7	8
5	6	7	8		9	10	11	12		1	2	3	4		13	14	15	16
9	10	11	12		5	6	7	8		13	14	15	16		1	2	3	4
13	14	15	16		13	14	15	16		9	10	11	12		9	10	11	12
.
1	3	2	4		1	3	2	4		5	7	6	8		5	7	6	8
5	7	6	8		9	11	10	12		1	3	2	4		13	15	14	16
9	11	10	12		5	7	6	8		13	15	14	16		1	3	2	4
13	15	14	16		13	15	14	16		9	11	10	12		9	11	10	12

2	1	4	3	2	1	4	3	6	5	8	7	6	5	8	7
6	5	8	7	10	9	12	11	2	1	4	3	14	13	16	15
10	9	12	11	6	5	8	7	14	13	16	15	2	1	4	3
14	13	16	15	14	13	16	15	10	9	12	11	10	9	12	11
.
2	4	1	3	2	4	1	3	6	8	5	7	6	8	5	7
6	8	5	7	10	12	9	11	2	4	1	3	14	16	13	15
10	12	9	11	6	8	5	7	14	16	13	15	2	4	1	3
14	16	13	15	14	16	13	15	10	12	9	11	10	12	9	11

Transformation of Reversible Squares to Most-Perfect magic Squares

To change any reversible square to the corresponding most- perfect magic square, follow this procedure:

reverse the right half of each row
reverse the bottom half of each column
apply the transform (k = 1/2n)

In this example , the first principle reversible square for order-4 (also shown above) is shown with its transformation to a most-perfect magic square. The Transform for the last column (in this case) is Transform-2.gif (1091 bytes)

Principle reversible square				Reverse half rows				Reverse half columns				Apply Transform
1	2	3	4	1	2	4	3	1	2	4	3	1	15	4	14
5	6	7	8	5	6	8	7	5	6	8	7	8	10	5	11
9	10	11	12	9	10	12	11	13	14	16	15	13	3	16	2
13	14	15	16	13	14	16	15	9	10	12	11	12	6	9	7

Number of principle reversible squares and most-perfect magic squares

Order n	Principle reversible sqr. N_n	Variation of each M_n=2^n-2{(1/2n)!}² M_n	Total reversible squares and most-perfect magic squares N_nx M_n
4	3	16	48
8	10	36864	368640
12	42	5.30842 x 10⁸	2.22953 x 10¹⁰
16	35	2.66355 x 10¹³	9.32243 x 10¹⁴
32	126	4.70045 x 10³⁵	5.92256 x 10³⁷

Some Examples

1	16	17	32	53	60	37	44
63	50	47	34	11	6	27	22
3	14	19	30	55	58	39	42
61	52	45	36	9	8	25	24
12	5	28	21	64	49	48	33
54	59	38	43	2	15	18	31
10	7	26	23	62	51	46	35
56	57	40	41	4	13	20	29

This square is most-perfect because

Every 2 x 2 block of cells (including wrap-around) sum to 2T (where T= n² + 1)
Any pair of integers distant ½n along a diagonal sum to T
It is a doubly-even pandiagonal normal magic square using integers from 1 to 64

1	16	57	56	17	32	41	40
58	55	2	15	42	39	18	31
8	9	64	49	24	25	48	33
63	50	7	10	47	34	23	26
5	12	61	52	21	28	45	36
62	51	6	11	46	35	22	27
4	13	60	53	20	29	44	37
59	54	3	14	43	38	19	30

This pandiagonal magic square is not most-perfect. Pairs of integers distant ½n along a diagonal do not sum to T (although all 2 x 2 sets of cells sum to 2T).

However, it is interesting because it contains 32 bent diagonals that sum correctly to 260.

All most-perfect magic squares are pandiagonal. Not all pandiagonal magic squares are most-perfect.

And finally, an order-12 most-perfect magic square [1]

65	93	82	95	49	78	68	64	51	62	84	79
32	100	15	98	48	115	29	129	46	131	13	114
25	133	42	135	9	118	28	104	11	102	44	119
24	108	7	106	40	123	21	137	38	139	5	122
17	141	34	143	1	126	20	112	3	110	36	127
76	56	59	54	92	71	73	85	90	87	57	70
77	81	94	83	61	66	80	52	63	50	96	67
116	16	99	14	132	31	113	45	130	47	97	30
117	41	134	43	101	26	120	12	103	10	136	27
124	8	107	6	140	23	121	37	138	39	105	22
125	33	142	35	109	18	128	4	111	2	144	19
72	60	55	58	88	75	69	89	86	91	53	74

[1] Kathleen Ollerenshaw and David Brée, Most-perfect Pandiagonal Magic Squares, Institute of Mathematics and its Applications, 1988, 0-905091-06-X, page 20

Addendum - October, 2006 - Most-perfect Multiply magic squares

When Kathleen Ollerenshaw introduced most-perfect magic squares in 1986 [1], she was referring to additive magic squares. However, the concept may be extended to multiply magic squares with suitable adaptation of the 3 basic requirements.

This addendum, is inspired by the work on multiply magic squares in Christian Boyer’s recently posted update. [2]
In it, I will demonstrate several multiply magic squares which I consider are most-perfect. I will leave for others, a discussion and investigation of the equivalent to Ollerenshaw’s “reversible magic squares”.

Requirements for most-perfect additive M.S.	Requirements for most-perfect multiply M.S.
Every 2 x 2 block of cells (including wrap-around) sum to 2T (where T= n2 + 1) Any pair of integers distant ½n along a diagonal sum to T Doubly-even pandiagonal normal magic squares (i.e. order 4, 8, 12, etc using integers from 1 to n²)	Product of 4 cells of 2x2 array = to T2 (where T equals the product of the first and last numbers used. Product of all pairs of integers distant ½n along a diagonal equal T Because the square is not normal. (not consecutive numbers), any even order pandiagonal magic square may be most-perfect.
The sum for the 2x2 cells has the same ratio to the magic constant as 4 cells are to the order of the square. For example, an order 4 square has 4 cells so the two sums are the same. For an order 8 square, the sum of the 2z2 arrays are 4:8 or 1/2 the magic constant.	The exponent of the product for the 4 numbers in the 2x2 arrays is the ratio of the 4 cells to the order of the square. For example, for an order 4 multiply square the ratio is 1:1 so the product for each array is equal to the magic constant. For an order 6, the ratio is 4:6, so the magic constant is equal to the power 1.5 of the order 2 array. For an order 8 MMS, the product of the 4 cells of the 2x2 array squared equals the magic product of the square

Order 4 regular (additive) and multiply magic squares

Sqr 1 Additive m.s.[3]      Sqr 2 Multiplicative m.s.[4]
 1    8   10   15         01   24    10   60
12   13    3    6         30   20     3    8
 7    2   16    9         12    2   120    5
14   11    5    4         40   15     4    6
S=34                      P=14,400

Sqr 1
All 48 additive pandiagonal magic squares are most perfect. However, in higher orders, all pandiagonal magic squares are not most-perfect.

Most-perfect features (requirements)
As per condition 1, all 2x2 blocks of cells sum to 34 which equals 2(1+16).
As per condition 2, diagonal pairs (such as 13+4) sum to 17 (which is the sum of the first and last numbers used in the series).
If we add the sums of the 2 pairs we obtain the Magic sum of the square.

Sqr 2
This multiplicative magic square is not normal, so the number of such squares is infinite! I assume that all such squares of order 4 are most-perfect. Can anyone find a counter-example?

NOTE: for more theory and examples of Multiply magic hypercubes, see my cube-multiply page.

Most-perfect features
As per condition 1, product of the 4 cells in a 2x2 block = 14400 which is equal to (1x120)²
As per condition 2, the product of diagonal pairs (such as 20 x 6) is 120 (which is the product of the first and last numbers used in the series). If we multiply the two products (of the 2 pairs) we obtain the magic product of the square i.e. 120² = 14,400..

Order 6 multiply magic squares

6x6 pandiagonal additive magic squares (using consecutive integers) are impossible. But because multiply magic squares cannot use consecutive integers, 6x6 pandiagonal multiplicative magic squares are possible!

Sqr 3 Harry A. Sayles,1913 [5]
  729     192      9   46656      3    576
   32     486   2592       2   7776    162
11664      12    144    2916     48     36
    1   15552     81      64    243   5184
23328       6    288    1458     96     18
   16     972   1296       4   3888    324
P = 101,559,956,668,416, Max. # = 46,656

Most-perfect features
As per condition 1, product of the 4 cells in a 2x2 block =2,176,782,336. Because the ratio of the 4 cells in a 2x2 block to the 6 cells in a line (i.e. 4:6 or 1:1.5), 2,176,782,336 ^1.5 = 101,559,956,668,416, the magic constant.
(The previous line may not be too eligible, so, to express it differently, the product of the 4 cells is raised by the power 1.5 to equal the product of the 6 cells.)

As per condition 2, the product of diagonal pairs (such as 486 x 96) is 46,656 (which is the product of the first and last numbers used in the series). Raising this value to the 3rd power gives us the magic product. Again we are dealing with ratios, because in an order 6 additive most-perfect m. s. we would multiply the sum of the three diagonal pairs by 3.
As an added bonus, the products of all 3x3 blocks (9 cells) also equal a constant value. This, of course, has no relevance to the most-perfect designation.

Sqr 4 Christian Boyer, 2006 [2]
   5   720    160     45     80   1440
4800    12    150    192    300      6
   9   400    288     25    144    800
 320   180     10   2880     20     90
  75    48   2400      3   1200     96
 576   100     18   1600     36     50
P = 2,985,984,000,000,  Max. # = 4,800

The above multiplicative, most-perfect magic square has the smallest known product P, more than 30 times smaller than P of the Sayles's example. And it has also the same 2x2 and 3x3 sub-squares features.
Interestingly, Christian produced another square (not shown here) , also with the same features, with a smaller maximum number but a larger P (4,410 and 85,766,121,000,000).

Most perfect features of Sqr 4 are:
As per condition 1, product of the 4 cells in a 2x2 block = 207,360,000. This, when raised by the power 1.5 equals the magic product.
As per condition 2, the product of diagonal pairs (such as 12 x 1200) = 1^st times last number (3 x 4800) = 14,400. And 14,400³ = 2,985,984,000,000.

[1] Ollerenshaw,K. (1986) On ‘most perfect’ or ‘complete’ 8 x 8 pandiagonal magic squares. Proceedings of the Royal Society of
London A407, p.259-281
[2] Christian Boyer Update of October, 2006--Multiply magic squares
[3] One of the 48 order 4 pandiagonal magic squares published posthumously in 1691 as part of Frenicle de Bessy’s list of 880 order 4 magic squares.
[4] W.S. Andrews Magic Squares and Cubes, 2nd Edition, 1917. (Harry A. Sayles, p. 288. fig. 540. This was first published in : The Monist, 23, 1913, pp 631-640)
[5] ibid MS&C, 292, fig. 560

Addendum-2 - November, 2006 - Most-perfect ?

Shown here is a pandiagonal magic square published in 1917 by L. S. Pierson. [1]

Recently Gil Lamb (Thailand) pointed out to me that this square has all the features of a Most-perfect magic square, except that it does not consist of consecutive numbers (a condition impossible in a singly even pandiagonal magic square).

Note that
    it is compact (condition 1)
    it is complete (condition 2)
    it is pandiagonal
However it is not double-even and does not use consecutive numbers.

01  47  06  43  05  48
35  17  30  21  31  16
36  12  41  08  40  13
07  45  02  49  03  44
29  19  34  15  33  20
42  10  37  14  38  09

[1] W.S. Andrews, Magic Squares and Cubes, 2nd Edition, 1917, (L. S. Pierson) 238, fig. 393


	This page was originally posted May 2000 It was last updated October 19, 2010 Harvey Heinz harveyheinz@shaw.ca Copyright © 1998-2009 by Harvey D. Heinz

Most-perfect Magic Squares

Contents

Order 4 regular (additive) and multiply magic squares