Abu’l-Wafa al-Buzjani

Abu’l-Wafa al-Buzjani (940 – 997/998) was an outstanding Persian mathematician and astronomer of the Middle Ages, who among other things wrote several books on mathematics. His treatise on magic squares of different orders is one of the two oldest known works on the subject.

In many construction methods from the arab world, the squares are filled frame by frame from the outside towards the center. Abu'l-Wafa al-Buzjani also begins with the outer border, which is filled with the first 2n − 2 numbers according to a certain scheme. In the case of order n=9 this means 2 · 9 − 2=16 numbers. His method uses the following steps:

Finally, the cells that are still free are filled with the complements of the numbers already entered. These complements are always placed in the opposite end of the respective row or column. The only exception is the two upper corners. As always with bordered magic squares, their complement must be entered in the diagonally opposite corner.

  • 812141610
    15
    13
    11
    9
    5
    3
    1
    2467
  • 88078767512141610
    6715
    6913
    7111
    739
    577
    379
    181
    72246770686674

This procedure can now be continued from the outside towards the center. For the border of the inner square with order n=7, the result is shown in the next figure.

  • 88078767512141610
    672226282415
    692713
    712511
    73239
    51977
    31779
    118202181
    72246770686674
  • 88078767512141610
    672264626126282415
    69552713
    71572511
    7359239
    5196377
    3176579
    15818202156546081
    72246770686674

This principle is applied to the remaining subsquares, until the complete multi-bordered magic square is finally created.

88078767512141610
672264626126282415
695532525136342713
715747384540352511
73594943413933239
51929423744536377
31748303146506579
15818202156546081
72246770686674

Remarks:

This procedure works for all odd orders. Two further examples for n=5 and n=7 are shown in the figure below.

  • 4242386
    191017127
    211513115
    11491625
    20231822
  • 648464510128
    39163635201811
    4131222924199
    4333272523177
    3132621283747
    1321415303449
    42245403844

Variants

In arabic literature you can also find slightly modified squares, in which the number sequences coming to the top right corner are not arranged in ascending order as in the previous example, but in descending order.

The two changed areas of the outer border are marked in the following figure.

88078767516141210
712264626128262411
695732525136342513
675547384540352715
73594943413933239
51929423744536377
31748303146506579
15818202154566081
72246766687074

In another variant, the numbers in the two upper corners are changed and the numbers 4k − 2 and 4k are selected here. Of course, this also changes the other even numbers in the top row. Again, only the changes in the figure on the left are shown.

  • 14807876758101216
    672664626122242815
    695534525132362713
    715747384540352511
    73594943413933239
    51929423744536377
    31746303150486579
    15418202160585681
    66246774727068
  • 14121087576788016
    672624226162642815
    695534325152362713
    715747384540352511
    73594943413933239
    51929423744536377
    31746503130486579
    15458602120185681
    66707274764268

There are also squares like the one on the right, in which the odd and even numbers are not entered starting at a common corner, but at different corners.