# Most-perfect Bent Diagonal Magic Squares

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 bent-diagonal (Franklin-type) order-12 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 order-12 Most-perfect 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

• Pandiagonal

• Compact (2x2 squares sum to 4/12 of S)

• And it is Most-perfect

The HSA (and the Morris) square fails on the most-perfect diagonal test!

Strangely, if my square is transposed so that the 1 is in the upper left corner, all horizontal bent diagonals become magic!

Order-12  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

Order-12  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

Order-12 from my Most-perfect 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 Most-perfect Bent diagonal magic square! [4]

Even more surprising was this order 12 square also included in the attachment. It also is a Most-perfect 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.

Order-12 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
Order-16

On April 2, 2007, Donald Morris sent me this order-16 magic square that has almost all of the features (68) found in Franklin's unpublished order-16 on my Franklin page. [5]

And this one is Most-perfect!
Donald reports that he constructed this square in late 2005

The Morris Order-16 Most-perfect 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

Order-8

Recently Daniel Schindel, Matthew Rempel And Peter Loly (Winnipeg, Canada) counted the basic Franklin type bent-diagonal squares of order-8. [6]
There are exactly 1,105,920 of them. Two-thirds of these squares are not magic because the main diagonals do not sum correctly. Exactly one-third (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 most-perfect. The bent-diagonal pandiagonal squares all have the 2z2 feature (compact), but fail on the diagonal feature (complete) so we can assume that there are no order-8 bent-diagonal most-perfect magic squares!

Review of requirements to be classed as most-perfect:

1.      Doubly-even pandiagonal normal magic squares (i.e. order 4, 8, 12, etc using integers from 1 to n2)

2.      Every 2 x 2 block of cells (including wrap-around) sum to 2T (where T= n2 + 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 Most-perfect 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