Magic Diagonal 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 diagonal sequences on fourth-order magic tori

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

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, first explored in the "Fourth-Order Magic Torus Chart," are now more extensively detailed in "Multiplicative Magic Tori."

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 of "Magic Diagonal Sequences of Fourth-Order Magic Tori" 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.

Read more ...

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.

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 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. 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.

Read more ...

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.

Read more ...