De la Hire

In an article published in 1705, Philippe de la Hire describes some methods to create magic squares of order n=4k from two auxiliary squares.

In the first auxiliary square he fills the first half of the top row with any number z from the range from 1 to n, the second half with their complementary number n + 1 − z. In the row below, he swaps the two halves. This is how he proceeds with the remaining numbers in the other pairs of rows until the entire auxiliary square is filled.

The second auxiliary square, on the other hand, is constructed using columns with slightly changed numbers. Now the upper half is filled with an arbitrary number z from the range from 0 to n − 1 and the lower half with its complementary number, which is n − 1 − z in the changed number range, since the number range starts here at 0. In the next column, the two halves are swapped. According to this principle, all other column pairs are then filled with the remaining numbers as in the figure on the right.

  • 88881111
    11118888
    55554444
    44445555
    77772222
    22227777
    66663333
    33336666
  • 70615243
    70615243
    70615243
    70615243
    07162534
    07162534
    07162534
    07162534

The numbers of the second auxiliary square are now multiplied by n=8 and added to the numbers of the first auxiliary square. The result is a magic square.

648561641173325
57149948244032
615531344203628
604521245213729
763155518422634
258105023473139
662145419432735
359115122463038

Of course, the role of the two auxiliary squares can be interchanged. This means that the rows of the first auxiliary square have been filled with the numbers from 0 to n − 1, while the rows of the second auxiliary square have been filled with the numbers from 1 to n.

  • 66661111
    11116666
    44443333
    33334444
    77770000
    00007777
    55552222
    22225555
  • 63815472
    63815472
    63815472
    63815472
    36184527
    36184527
    36184527
    36184527

Now of course all numbers of the first auxiliary square must be multiplied by n=8, before the numbers of the second auxiliary square are added. The result of this is the following magic square.

5451564913121510
141116953525550
3835403329283126
3027322537363934
596257644527
361860615863
4346414820211823
1922172444454247

You will find further variants of this method in the PDF document.