Magic Orthogonal Sequences of 4th-Order Magic Tori

The number of fourth-order magic series is 86. The complete list of these magic series has been published by Walter Trump, who gave me confirmation that all of these magic series were present on fourth-order magic squares, even if the geometrical arrangement of the series was often chaotic. He pointed out that for example the series 7+8+9+10 does not exist in any row, column, or main diagonal of a magic 4x4-square, even when considering all the 24 possible permutations of the four numbers. However the series 7+8+9+10 occurs in the 2x2 centre square or in the four corners of certain 4x4 magic squares. As defined in MathWorld, or in Wikipedia, a magic series is not a specific sequence of numbers, but a set of numbers that add up to the magic constant.
The magic constant (MCn) of an nth order magic square (or torus) can be calculated:

MCn = (n²)² + n²  x  1    =    n(n² + 1)

                 2              n                2

For a fourth-order magic square (or magic torus) the magic constant is therefore:
MC4   =  4(4² + 1)    =   34
                     2

Magic orthogonal sequences of fourth-order magic tori

Magic orthogonal sequences on the Magic Torus Type T4.01.1 (index n° T4.194)

On any orthogonal line of a magic torus (column and row of an order 4 magic square) :
a + b + c + d = 34

This equation can be expressed differently to express the balance of the numbers and facilitate the analysis of the series:

a + b + c + d = 16+1
   2          2
Therefore:
a + b = 16 + 1 - c + d         and       c + d = 16 + 1 - a + b
   2                       2                             2                       2

Considering the continuous curved surface of a magic torus, each of the 86 4th-order magic series can be expressed in 3 essentially different sequences:

a, b, c, d  (sequence type A) - which is also the magic series
a, b, d, c  (sequence type B)
a, c, b, d  (sequence type C)

As they curve round the torus the sequences have no beginning or end. The magic torus concept is explained in a previous article "From the Magic Square to the Magic Torus." The interrelationships of the 4th-order magic tori, first explored in the "Fourth-Order Magic Torus Chart," are now more extensively detailed in "Multiplicative Magic Tori."

Rules for magic orthogonal sequences on fourth-order magic tori

Rule 1: The magic series a + b + c + d (whatever the order of the sequence) must always contain two even numbers and two odd numbers.

Rule 2: The numbers of the magic series must always be balanced this way:
             1 ≤ a ≤ 7
             2 ≤ b ≤ 8
             9 ≤ c ≤ 15
           10 ≤ d ≤ 16

Rule 3: If b - a = 1 and if b is an even number, then the series will only be magic if :
              d - c = 1, and 17-c = b.
            If d - c = 1 and if d is an even number, then the series will only be magic if :
               b - a = 1, and 17-c = b.

Exception 3a : If a third set of consecutive numbers is introduced ( c - b = 1), the series will not be magic. This excludes the magic series 7 + 8 + 9 + 10 in orthogonal sequences.

Exception 3b : If all the numbers are situated between n + 1 and 3n - 1 then only the types A and C sequences (where complementary numbers are adjacent) will be magic. This excludes the orthogonal sequence (5, 6, 12, 11).

Presentation of the list of magic sequences

In the list that follows the index numbers of the magic series are the same as those already published by Walter Trump. I have just added the suffixes A, B, and C to identify the 3 essentially different sequences that are derived from each series. I wish to emphasise that although I have always chosen the lowest number to be the first, none of the sequences have either a beginning or an end, as they each form a continuous loop round their torus.

It was difficult to decide how to present the results. I have chosen to list the different sequence types A, B, and C separately, and I have grouped the sequences by sets of number rhythms in order to facilitate comparisons and reveal patterns. I have also sorted between the complementary (a+d = b+c = n²+1) sequences and the non-complementary (a+d ≠ b+c) sequences.

Please note that if you click on the button that appears at the top right hand side of the pdf viewer below, a new window will open and full size pages of the paper "Magic Orthogonal Sequences of 4th-Order Magic Tori" will then be displayed, with options for zooming.

Observations

The complementary series (a + d = c + b = n² + 1) have, where possible, been divided into pairs of sequences that share similar characteristics. There are 4 exceptions with unique characteristics that cannot be paired. Interestingly these 4 series are inversely related :

1 + 2 + 15 + 16
3 + 4 + 13 + 14
5 + 6 + 11 + 12
7 + 8 +  9  + 10

There are many other ways of ordering the sequences to accord with the specific number rhythms of the different magic torus types. This is why an ideal list, that will suit all the torus types, remains a difficult objective.