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
onetoone correspondence between the most perfect magic square and the
McClintock square.
Recently Dr. Ollerenshaw, looking for a way to enumerate at least a subset
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 MostPerfect (with a hyphen). She was 74 years old when
she published her first paper [2] on
Mostperfect magic squares (in 1986). She published at least two other papers
on the subject.
In 1998, Dr. Ollerenshaw coauthored 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.99120.
[2] Ollerenshaw,K. (1986) On ‘most perfect’ or ‘complete’ 8 x 8
pandiagonal magic squares. Proceedings of the Royal Society of London
A407, p.259281
[3] Kathleen Ollerenshaw and David Brée, Mostperfect Pandiagonal Magic
Squares, Institute of Mathematics and its Applications, 1998,
090509106X
[4] See also Ian Stewart, Mathematical Recreations column, Scientific
American, Nov. 99, p.122123.
Features of Mostperfect
magic squares
4 
5 
16 
9 
14 
11 
2 
7 
1 
8 
13 
12 
15 
10 
3 
6 

All 48 pandiagonal magic squares of order4 are
mostperfect! For other orders, not all pandiagonal magic squares
are mostperfect.
The 4 corner cells of any square array of cells in an order4
mostperfect magic square sum to S. 
Definition
 Every 2 x 2 block of cells (including wraparound) sum to 2T (where T=
n^{2} + 1) (i.e. compact)
 Any pair of integers distant ½n along a diagonal sum to
T (i.e. complete)
 Doublyeven pandiagonal normal magic squares (i.e. order 4, 8, 12, etc
using integers from 1 to n^{2})
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 mostperfect magic square can be transformed into a reversible
square.
Note:
Note2: 
For mathematical convenience,
the authors use the series from 0 to n^{2}1.
In that case S=n(n^{2}1)/2, T = n^{2}1.
I have chosen to use the series from 1 to n^{2} to be
consistent with the definition of a normal magic square with S=n(n^{2}+1)/2.
I use the symbol S to indicate the magic sum and T to indicate the value
of n^{2} + 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 subsquare 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^{n2}{(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 order4. 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
mostperfect 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 order4.
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 MostPerfect 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 order4 (also
shown above) is shown with its transformation to a mostperfect magic square.
The Transform for the last column (in this case) is
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 mostperfect magic squares
Order
n 
Principle
reversible sqr.
N_{n} 
Variation of each
M_{n}=2^{n2}{(1/2n)!}^{2}
M_{n} 
Total reversible squares and
mostperfect magic squares
N_{n }x M_{n} 
4 
3 
16 
48 
8 
10 
36864 
368640 
12 
42 
5.30842 x 10^{8} 
2.22953 x 10^{10} 
16 
35 
2.66355 x 10^{13} 
9.32243 x 10^{14} 
32 
126 
4.70045 x 10^{35} 
5.92256 x 10^{37} 
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 mostperfect because
 Every 2 x 2 block of cells (including wraparound) sum to 2T
(where T= n^{2} + 1)
 Any pair of integers distant ½n along a diagonal sum to
T
 It is a doublyeven 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
mostperfect. 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 mostperfect magic squares are pandiagonal. Not all
pandiagonal magic squares are mostperfect. 
And finally, an order12 mostperfect 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, Mostperfect Pandiagonal Magic Squares, Institute of
Mathematics and its Applications, 1988, 090509106X, page 20
Addendum  October, 2006
 Mostperfect Multiply magic squares
When Kathleen
Ollerenshaw introduced mostperfect 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
mostperfect. I will leave for others, a discussion and investigation of the
equivalent to Ollerenshaw’s “reversible magic squares”.
Requirements for mostperfect
additive M.S. 
Requirements for mostperfect multiply M.S. 

Every 2 x 2 block of
cells (including wraparound) sum to 2T (where T= n2 + 1)

Any pair of integers
distant ½n along a diagonal sum to T

Doublyeven
pandiagonal normal magic squares (i.e. order 4, 8, 12, etc using
integers from 1 to n^{2})


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 mostperfect.

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 mostperfect.
Mostperfect 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 mostperfect. Can anyone find a counterexample?
NOTE: for more theory and examples of Multiply
magic hypercubes, see my cubemultiply
page.
Mostperfect 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. 120^{2} = 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
Mostperfect 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 mostperfect 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 mostperfect
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, mostperfect 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 subsquares 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 14,400^{3} =
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.259281
[2] Christian Boyer Update of October, 2006Multiply 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 631640)
[5] ibid MS&C, 292, fig. 560
Addendum2  November,
2006  Mostperfect ?
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 Mostperfect 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 doubleeven 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
