Magic Sequences of the Sub-Magic 2x2 Squares 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 sequences of the sub-magic 2x2 squares on fourth-order magic tori

On any sub-magic 2x2 square of a fourth-order magic torus (or of an order 4 magic square) :

a + b + c + d = 34

Considering a clockwise arrangement of the numbers on the sub-magic 2x2 square, each of the 86 4th-order magic series can be expressed in 3 essentially different sequences:

 

Clockwise and anti-clockwise sequences are considered equivalent, depending on whether the magic torus is being contemplated from the outside or from within. 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."

Theoretically the total number of different sub-magic 2x2 squares on fourth-order magic tori would be 86 magic series x 3 sequence types = 258. However, after checking the 255 different fourth-order magic tori I discovered that there are 134 exceptions, leaving only 124 essentially different sub-magic 2x2 square sequences. These exceptions are of course the result of combination impossibilities that can be tested and eliminated by computer calculation. However, as I do not use computer programming myself, I wondered if it might be possible to find simpler ways of distinguishing the magic sequences from their non-magic counterparts. My observations have enabled me to determine some of the rules that govern fourth-order sub-magic 2x2 square sequences:

Rules for sub-magic 2x2 square 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 there is a single occurrence of consecutive numbers where the successor number is even, then none of the sequences of the series will be magic.

         

Rule 4: If the series contains complementary numbers a + d = 17 and b + c = 17, and if there is a double occurrence of consecutive numbers where the successor number is even, only the sequences that restrict these successive numbers to the diagonals of the sub-magic 2x2 square will be magic. Some of the odd - even - odd - even series of complementary pairs have this characteristic and the only magic sequences of these series are therefore type C.

 
Rule 5: If the series contains complementary numbers a + d = 17 and b + c = 17,
             and if a - 1 = 8 - b = c - 9 = 16 - d
             implying that (c + d) - (a + b) = n²
             then only the sequence type A will be magic.
             (as this only occurs with odd-even-odd-even, and even-odd-even-odd series).

Rule 6: If the series contains complementary numbers a + d = 17 and b + c = 17,
             and if the series is odd - odd - even - even,
             and if (c + d) - (a + b) is not a multiple of 3,  
             then only the sequence C (odd - even - odd - even) will be magic.

Rule 7: If the series contains complementary numbers a + d = 17 and b + c = 17,
             and if the series is even - even - odd - odd,
             and if (c + d) - (a + b) is not a multiple of 7,  
             then only the sequence C (even - odd - even - odd) will be magic.

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 circuit within their sub-magic 2x2 square.

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.

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Observations

The complementary series (a + d = c + b = n² + 1) have, where possible, been divided into pairs of sequences that share similar characteristics. Although the sub-magic 2x2 square magic series 1 + 2 + 15 + 16, and 3 + 4 + 13 + 14 seem similar, I have preferred to stick to the same sets already used in my study of the orthogonal sequences. These include 4 exceptions with unique characteristics that cannot be easily paired:

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.