On March 22, 2007 I first heard
about a new magic square discovery in The Netherlands
Three Dutch secondary school pupils have created the
‘most magical magic square in 5,000 years’,…
[1]
A lively discussion
among magic square friends ensued over the next week. The result? While
these students should be complimented on their accomplishment, their
imaginative claim was slightly (?) exaggerated.
The square in question
is a bentdiagonal (Franklintype) order12 square with rows and columns
summing correctly. Also present are many other magic patterns found in
Franklin type squares. The creators excitement seemed to be due to the
fact that the main diagonals also summed correctly, making this a true
magic square. This was a feature Ben Franklin did not accomplish in his
published squares.
It is also symmetric across the central horizontal line.
I found this square with identical features
to the HSA square at Donald Morris’s Franklin Squares site
[2].
This was published in
2005 and included a complete method of construction!
The HSA square is identical to this square reflected across the leading
diagonal !!!!
Well... after first
swapping columns 5 and 7, then 6 and 8. (This pointed out by Jo Geuskens
on May 27/07 and Frans Lelieveld on May 30/07. Thanks fellows.)
Finally, I compared
the HAS square to an order12 Mostperfect magic square on my site.
[3]
In
the comparison, my square fails on

Bent diagonals (and
many other Franklin patterns)

Horizontal (and
vertical) lines of 4

Vertical symmetry
(across the center horizontal line)
My square also is
The HSA (and the
Morris) square fails on the mostperfect diagonal test!
Strangely, if my
square is transposed so that the 1 is in the upper left corner, all
horizontal bent diagonals become magic! 
Order12 HSA Square
1 
142 
11 
136 
8 
138 
5 
139 
12 
135 
2 
141 
120 
27 
110 
33 
113 
31 
116 
30 
109 
34 
119 
28 
121 
22 
131 
16 
128 
18 
125 
19 
132 
15 
122 
21 
48 
99 
38 
105 
41 
103 
44 
102 
37 
106 
47 
100 
73 
70 
83 
64 
80 
66 
77 
67 
84 
63 
74 
69 
60 
87 
50 
93 
53 
91 
56 
90 
49 
94 
59 
88 
85 
58 
95 
52 
92 
54 
89 
55 
96 
51 
86 
57 
72 
75 
62 
81 
65 
79 
68 
78 
61 
82 
71 
76 
97 
46 
107 
40 
104 
42 
101 
43 
108 
39 
98 
45 
24 
123 
14 
129 
17 
127 
20 
126 
13 
130 
23 
124 
25 
118 
35 
112 
32 
114 
29 
115 
36 
111 
26 
117 
144 
3 
134 
9 
137 
7 
140 
6 
133 
10 
143 
4 
Order12 from D. Morris page
1 
120 
121 
48 
85 
72 
73 
60 
97 
24 
25 
144 
142 
27 
22 
99 
58 
75 
70 
87 
46 
123 
118 
3 
11 
110 
131 
38 
95 
62 
83 
50 
107 
14 
35 
134 
136 
33 
16 
105 
52 
81 
64 
93 
40 
129 
112 
9 
8 
113 
128 
41 
92 
65 
80 
53 
104 
17 
32 
137 
138 
31 
18 
103 
54 
79 
66 
91 
42 
127 
114 
7 
5 
116 
125 
44 
89 
68 
77 
56 
101 
20 
29 
140 
139 
30 
19 
102 
55 
78 
67 
90 
43 
126 
115 
6 
12 
109 
132 
37 
96 
61 
84 
49 
108 
13 
36 
133 
135 
34 
15 
106 
51 
82 
63 
94 
39 
130 
111 
10 
2 
119 
122 
47 
86 
71 
74 
59 
98 
23 
26 
143 
141 
28 
21 
100 
57 
76 
69 
88 
45 
124 
117 
4 
Order12 from my Mostperfect page
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 

Announcement!On April 2, 2007 I received an email from Donald
Morris with an order 16 Mostperfect Bent diagonal magic square!
[4]
Even more surprising was this order 12 square also included in the
attachment. It also is a Mostperfect Bent diagonal magic square!
To the best of my knowledge, these are the first such squares
published!
Don tells me he constructed this square in late 2005.

Order12 From Morris email of
April 2,2007
1 
120 
85 
72 
97 
24 
133 
36 
49 
84 
37 
132 
142 
27 
58 
75 
46 
123 
10 
111 
94 
63 
106 
15 
8 
113 
92 
65 
104 
17 
140 
29 
56 
77 
44 
125 
138 
31 
54 
79 
42 
127 
6 
115 
90 
67 
102 
19 
9 
112 
93 
64 
105 
16 
141 
28 
57 
76 
45 
124 
134 
35 
50 
83 
38 
131 
2 
119 
86 
71 
98 
23 
12 
109 
96 
61 
108 
13 
144 
25 
60 
73 
48 
121 
135 
34 
51 
82 
39 
130 
3 
118 
87 
70 
99 
22 
5 
116 
89 
68 
101 
20 
137 
32 
53 
80 
41 
128 
139 
30 
55 
78 
43 
126 
7 
114 
91 
66 
103 
18 
4 
117 
88 
69 
100 
21 
136 
33 
52 
81 
40 
129 
143 
26 
59 
74 
47 
122 
11 
110 
95 
62 
107 
14 

Order16 On April 2, 2007,
Donald Morris sent me this order16 magic square that has almost all of
the features (68) found in Franklin's unpublished order16 on my Franklin
page. [5]
And this one is Mostperfect!
Donald reports that he constructed this square in late 2005

The Morris Order16
Mostperfect Bent diagonal magic square
256 
225 
48 
49 
80 
81 
160 
129 
16 
17 
224 
193 
192 
161 
112 
113 
15 
18 
223 
194 
191 
162 
111 
114 
255 
226 
47 
50 
79 
82 
159 
130 
243 
238 
35 
62 
67 
94 
147 
142 
3 
30 
211 
206 
179 
174 
99 
126 
4 
29 
212 
205 
180 
173 
100 
125 
244 
237 
36 
61 
68 
93 
148 
141 
245 
236 
37 
60 
69 
92 
149 
140 
5 
28 
213 
204 
181 
172 
101 
124 
6 
27 
214 
203 
182 
171 
102 
123 
246 
235 
38 
59 
70 
91 
150 
139 
250 
231 
42 
55 
74 
87 
154 
135 
10 
23 
218 
199 
186 
167 
106 
119 
9 
24 
217 
200 
185 
168 
105 
120 
249 
232 
41 
56 
73 
88 
153 
136 
241 
240 
33 
64 
65 
96 
145 
144 
1 
32 
209 
208 
177 
176 
97 
128 
2 
31 
210 
207 
178 
175 
98 
127 
242 
239 
34 
63 
66 
95 
146 
143 
254 
227 
46 
51 
78 
83 
158 
131 
14 
19 
222 
195 
190 
163 
110 
115 
13 
20 
221 
196 
189 
164 
109 
116 
253 
228 
45 
52 
77 
84 
157 
132 
252 
229 
44 
53 
76 
85 
156 
133 
12 
21 
220 
197 
188 
165 
108 
117 
11 
22 
219 
198 
187 
166 
107 
118 
251 
230 
43 
54 
75 
86 
155 
134 
247 
234 
39 
58 
71 
90 
151 
138 
7 
26 
215 
202 
183 
170 
103 
122 
8 
25 
216 
201 
184 
169 
104 
121 
248 
233 
40 
57 
72 
89 
152 
137 

Order8
Recently Daniel Schindel, Matthew Rempel And Peter Loly (Winnipeg, Canada)
counted the basic Franklin type bentdiagonal squares of order8.
[6]
There are exactly 1,105,920 of them. Twothirds of these squares are not magic
because the main diagonals do not sum correctly. Exactly onethird (368,640) are
pandiagonal magic.
BTW
The Peter Loly's count has been independently corroborated by other sources in
Canada and Argentina.
This figure (368,640) is in exact agreement with that reported by Dame Kathleen
Ollerenshaw as being mostperfect. The bentdiagonal pandiagonal squares all
have the 2z2 feature (compact), but fail on the diagonal feature (complete) so
we can assume that there are no order8 bentdiagonal mostperfect magic
squares!
Review of requirements to be classed as mostperfect:
1.
Doublyeven pandiagonal normal magic squares (i.e. order 4, 8, 12, etc
using integers from 1 to n^{2})
2.
Every 2 x 2 block of cells (including wraparound) sum to 2T (where T= n^{2}
+ 1) (compact)
3.
Any pair of integers distant ˝n along a diagonal sum to T (complete)
[1] Announcement of
the HSA square
http://www.eurogates.nl/?act=shownews&nid=1856
[2] Donald Morris's Franklin squares site
http://www.bestfranklinsquares.com/
[3] My Mostperfect magic squares page
[4] Donald Morris's email address (with his permission) is donald.morris4@sbcglobal.net
[5] My Franklin magic squares page
[6] Proc. R. Soc. A (2006) 462, 2271–2279, doi:10.1098/rspa.2006.1684.
Enumerating the bent diagonal squares of Dr Benjamin Franklin FRS
Published online 28 February 2006. Obtainable by download from Peter
Loly's home page
