Travers

The method of J. Travers also works from the outer border towards the center. First the outer border of the square of order n is filled, then the border of the square of order n − 2 embedded in it, etc. The entire border includes exactly

2n + (2n − 2) =4n − 2

numbers. In order to compensate in pairs, he summarizes the numbers 1, … , 2n-2 and their complements in pairs, as already described by Frénicle. Since n is odd, it follows with n=2k+1

2n - 2=2 · (2k+1) - 2=4k+2 - 2=4k

that a total of 4k numbers are to be distributed. Travers divides them into four groups of equal size:

1, 2, … , k
k+1, k+2, … , 2k
2k+1, 2k+2, … , 3k
3k+1, 3k+2, … , 4k

For order n=7, the number sequences (1,2,3), (4,5,6), (7,8,9) and (10,11,12) are obtained. He now distributes them using the following steps:

  • 123
    5
    6
    7
    12
    11
    10894
  • 46123424140
    455
    446
    743
    1238
    1139
    10494847894

Once all the numbers have been entered, the complementary numbers are entered in the horizontally or vertically opposite cells. Only the two lower corners are an exception because they write the complement in the diagonally opposite corners.

After the edge of the square of order 7 is completely filled, Travers continues with the border of the embedded inner square of order n=5. Since a total of 8 numbers with their complement must be placed and the numbers 1, 2, … , 12 have already been assigned, the numbers 13, 14, … , 20 are used now.

  • 46123424140
    4513145
    44166
    71743
    122038
    1119181539
    10494847894
  • 46123424140
    4535131432315
    4434166
    7173343
    12203038
    11193736181539
    10494847894

The same procedure is followed with the embedded inner square of order n=3.

  • 46123424140
    4535131432315
    443421166
    717233343
    122024223038
    11193736181539
    10494847894
  • 46123424140
    4535131432315
    4434282126166
    71723273343
    12202429223038
    11193736181539
    10494847894

After that, only the center cell remains, in which the median value of all the numbers involved is entered, in this case 25. The following figure shows the multi-bordered magic square of the order n=7 generated by Travers using this algorithm.

  • 46123424140
    4535131432315
    4434282126166
    7172325273343
    12202429223038
    11193736181539
    10494847894
  • 46123424140
    4535131432315
    4434282126166
    7172325273343
    12202429223038
    11193736181539
    10494847894

The following figure shows further magic squares of the orders n=5 and n=9 created using the method of Travers.

  • 23122019
    22169144
    511131521
    812171018
    7252463
  • 77123472717069
    76621718195857566
    75615129304847217
    74605044374232228
    92333394143495973
    162836404538465466
    152735535234315567
    142665646324252068
    13818079781011125