Wednesday, 26 March 2014

Magic Diagonal Sequences of 4th-Order Magic Tori

Magic series of order 4

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 diagonal sequences on fourth-order magic tori

On any magic diagonal of a fourth-order magic torus (or of an order 4 magic square) :
a + b + c + d = 34

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 an earlier article "From the Magic Square to the Magic Torus." The interrelationships of the 4th-order magic tori are detailed in the "Fourth-Order Magic Torus Chart."

Theoretically the total number of different magic diagonal sequences 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 48 exceptions, leaving only 210 essentially different diagonal magic sequences. My observations have enabled me to determine some of the rules that govern fourth-order magic diagonal sequences:

Rules for magic diagonal sequences on fourth-order magic tori

Rule 1: If the magic series a + b + c + d has complementary pairs a + d = 17 and b + c = 17 (whatever the order of the sequence), and if consecutive numbers with an even successor number occur, then the series cannot be magic along diagonals.

Rule 2: If the magic series a + b + c + d has complementary pairs a + d = 17 and b + c = 17 (whatever the order of the sequence), and if a and b are odd numbers, then (c + d) - (a + b ) must be a multiple of 3.

Rule 3: If the magic series a + b + c + d has complementary pairs a + d = 17 and b + c = 17 (whatever the order of the sequence), and if a and b are even numbers, then (c + d) - (a + b) must be a multiple of 7.

Rule 4: If the magic series a + b + c + d has complementary pairs a + d = 17 and b + c = 17 (whatever the order of the sequence), and if a - 1 = 8 - b = c - 9 = 16 - d, then the series cannot be magic along diagonals.

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 will then be displayed, with options for zooming.



Observations
Please note that all of the torus diagonals that sum to the magic constant are taken into account - even those that never coincide with the centre of a traditional magic square - such as for example the third magic diagonal sequence (4, 6, 16, 8) of the Frénicle square n° 275 (magic torus type n°T4.04.2 - index n°T4.098).

The complementary (a + d = c + b = n² + 1) diagonal series seem quite symmetrical and regular when compared to their orthogonal and sub-magic 2x2 square cousins. I have therefore decided to use larger sets of complementary sequences in this present study.

However 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.

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