A most-perfect magic square of doubly even order n = 4k is a pan-diagonal magic square containing the numbers 1 to n2 with three additional properties:
Each 2×2 subsquare, including wrap-round, sums to s/k, where s = n/2 is the magic sum.
All pairs of integers distant n/2 along any diagonal are complementary.
Examples
Specific examples of most-perfect magic squares that begin with the 2015 date demonstrate how theory and computer science are able to define this group of magic squares. Only 16 of the 64 2x2 cell blocks that sum to 130 are accented by the different colored fonts in the 8x8 example. The 12x12 square below was found by making all the 42 principal reversible squares with ReversibleSquares, running Transform1 2All on all 42, making 23040 of each,, then making the most-perfect squares from these with ReversibleMost-Perfect. These squares were then scanned for squares with 20,15 in the proper cells for any of the 8 rotations. The 2015 squares all originated with principal reversible square number #31. This square has values that sum to 35 on opposite sides of the vertical midline in the first two rows.
20
15
60
49
24
51
132
123
92
89
128
87
119
136
79
102
115
100
7
28
47
62
11
64
25
10
65
44
29
46
137
118
97
84
133
82
126
129
86
95
122
93
14
21
54
55
18
57
31
4
71
38
35
40
143
112
103
78
139
76
113
142
73
108
109
106
1
34
41
68
5
70
13
22
53
56
17
58
125
130
85
96
121
94
138
117
98
83
134
81
26
9
66
43
30
45
8
27
48
61
12
63
120
135
80
101
116
99
131
124
91
90
127
88
19
16
59
50
23
52
2
33
42
67
6
69
114
141
74
107
110
105
144
111
104
77
140
75
32
3
72
37
36
39
1
2
7
8
13
14
19
20
25
26
31
32
3
4
9
10
15
16
21
22
27
28
33
34
5
6
11
12
17
18
23
24
29
30
35
36
37
38
43
44
49
50
55
56
61
62
67
68
39
40
45
46
51
52
57
58
63
64
69
70
41
42
47
48
53
54
59
60
65
66
71
72
73
74
79
80
85
86
91
92
97
98
103
104
75
76
81
82
87
88
93
94
99
100
105
106
77
78
83
84
89
90
95
96
101
102
107
108
109
110
115
116
121
122
127
128
133
134
139
140
111
112
117
118
123
124
129
130
135
136
141
142
113
114
119
120
125
126
131
132
137
138
143
144
Properties
All most-perfect magic squares are panmagic squares. Apart from the trivial case of the first order square, most-perfect magic squares are all of order 4n. In their book, Kathleen Ollerenshaw and David S. Brée give a method of construction and enumeration of all most-perfect magic squares. They also show that there is a one-to-one correspondence between reversible squares and most-perfect magic squares. The number of essentially different most-perfect magic squares of order n for 4n = 1, 2,... form the sequence: For example, there are about 2.7 × 1044 essentially different most-perfect magic squares of order 36. All order four panmagic squares are most-perfect magic squares. The second property implies that each pair of the integers with the same background color in the 4×4 square below have the same sum, and hence any 2 such pairs sum to the magic constant.
The image below shows areas completely surrounded by larger numbers with a blue background. A water retention topographical model is one example of the physical properties of magic squares. The water retention model progressed from the specific case of the magic square to a more generalized system of random levels. A quite interesting counter-intuitive finding that a random two-level system will retain more water than a random three-level system when the size of the square is greater than 51 X 51 was discovered. This was reported in the Physical Review Letters in 2012 and referenced in the Nature article in 2018.
