Seki

The Japanese mathematician Takakazu Seki (1642 - 1708) designed a method to create multi-bordered magic squares of odd order. The method will be explained using the order n=9. With n=2k+1 this results in k=4.

The entire outer border comprises a total of 4 · (n − 1) numbers. In this example, this results in 2n − 2=16 numbers and their complements, which must first be suitably placed in the border. Seki uses the following steps:

  • 7651
    8
    4
    3
    2
  • 7651
    12
    11
    10
    9
    8
    4
    3
    161514132

The border is thus clearly defined and the cells that are still empty are filled with the complementary values ​​to the numbers already entered. As usual, the number complementary to a number x is calculated with n2 + 1 − x. This means:

80676869765166
1270
1171
1072
973
748
784
793
16151413767778812

This method can now be continued from the outer border towards the center. The inner square of order 7, which has not yet been filled, contains a total of 4 · (7 − 1)=24 cells, half of which are filled with complementary numbers. So 12 numbers in the range 17 … 28 must be placed together with their complements.

  • 80676869765166
    1221201770
    112571
    102472
    92373
    74228
    78194
    79282726183
    16151413757677812
  • 80676869765166
    126455562120175470
    11255771
    10245872
    9235973
    7460228
    7863194
    79282726616265183
    16151413757677812

Do the same with the remaining squares until you finally get the multi-bordered magic square from Seki.

  • 80676869765166
    126455562120175470
    112552473129465771
    102434443742485872
    92333394143495973
    74605040453832228
    78633635515330194
    79282726616265183
    16151413757677812
  • 80676869765166
    126455562120175470
    112552473129465771
    102434443742485872
    92333394143495973
    74605040453832228
    78633635515330194
    79282726616265183
    16151413757677812

With this method from Seki, bordered magic squares can be created for all odd orders. Two other examples for n=5 and n=7 are shown in the following figure.

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    511131521
    221217104
    8723252
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    9363115133041
    8182821263242
    7172325273343
    4434242922166
    4720193537143
    1211104546492