Al-Buni

Another method of arabic origin was published around 1200 by Ahmad al-Buni, who lived in what is now Algeria. However, it did not come from al-Buni himself, who only presented it in one of his publications. To create a bordered magic square of order n=2k + 1, proceed in the following order.

Finally, the fields that are still free are filled with the complements of the numbers already entered. It is important, however, that two places of the border remain free. Only 2n − 3 numbers were entered and the bottom left and top right corners were not filled.

  • 1513119
    7
    6
    5
    4
    3
    2
    1
    1412108
  • 741568137011729
    775
    766
    577
    784
    379
    802
    181
    671469127110738

This procedure is now continued for each border from the outside towards the center. The border of the embedded inner square of order n=7 is shown in the following figure.

  • 741568137011729
    726242275
    76206
    51977
    78184
    31779
    80162
    125232181
    671469127110738
  • 741568137011729
    761265724592275
    7620626
    5631977
    7818644
    3651779
    8016662
    156255823602181
    671469127110738

Applying these steps with the rest of the borders, you can see that all cells of the main diagonals have been left empty. They are now filled from the top right to the bottom left with the numbers that have not yet been entered until the complete multi-bordered magic square is finally created.

  • 74156813701172937
    76126572459223875
    76205233503139626
    56329473640531977
    78185434414828644
    36527424635551779
    80164349325130662
    14456255823602181
    45671469127110738
  • 74156813701172937
    76126572459223875
    76205233503139626
    56329473640531977
    78185434414828644
    36527424635551779
    80164349325130662
    14456255823602181
    45671469127110738

This procedure works for all odd orders. Two other examples are shown for n=5 and n=7.

  • 22720511
    317101223
    24813182
    11416925
    15196214
  • 441140942722
    5351833162345
    4614302124364
    3371925311347
    4812262920382
    1273217341549
    283910418436