A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

THE USE OF ONE IMAGINARY NUMBER AS PART OF THE RIGHT DIAGONAL (Part IA)

Picture of a square

General Introduction of the New Method

Andrew Bremner's article on squares of squares included the 3x3 square below which is not fully magic:

Bremmer's square
5824621272
942113222
972822742

It will be shown that:

  1. Magic squares can be produced using diagonals in which one of the squares is negative, i.e., (−a2,b2,c2). We define a tuple (ai, b, c) as a major diagonal sequence of a magic square, prior to squaring, where the initial number in the tuple is imaginary.
  2. It will be shown that several different types of these diagonal sequences are possible.
    1. In the first type, all tuples can be generated using two general tables one for odd numbers and the other for even ones shown below.
    2. In a second type, the tuples are generated from the set of all squares and the set of all squares doubled. See Part IIA.
    3. In third type an infinite number of tables of tuples is generated, using a subset of squares and squares doubled from ii above, where the differences between the numbers c and a, i.e., (Δc-a) allow for only certain values. See Part IB , Part IC , Part IIC , Part IIIC , Part ID and Part IE .
    4. In the fourth type, new subtables within the original tables are generated, which when factored have a different Δc-a.
  3. The method for the generation of these tables is simple and not as complex as that which was used in Table of Right Diagonals.
  4. The Diagonal tuple may be of two types:
    (a) (-n1i,0,n2) where n1 and n2 are integers, n1 = n2 and can range from 1 to ∞
    (b) (ai, b, c) where a,b and c are integers obtained from the equation:     
    c2 = a2 + 2b2
  5. Using the new method using diagonal types of (a) produces an infinite number of magic squares of squares containing seven squares. The use of diagonal type (b), on the other hand, produces to date four magic squares of squares containing seven squares. Both types (a) and (b) are included in the next group of pages.
  6. It will also be shown that we can convert a new series of allowed tuples into a higher set by taking the average values of a, b or c, within a table, and converting to the average values aavg* bavg* or cavg* of the next higher set, within another table. That is converting all the numbers in the tuple:
    (aavgi, bavg, cavg) → (aavg*i, bavg*, cavg*)
    to the next higher tuple by multiplying by the newly discovered ratio (1 + √2)2 = 5.828434..., which I call the magic ratio (R) with an x:
    xn+1 = xn × 5.828434...
    where x is a placeholder for a, b or c:
    xn+1/xn ≈ 5.828434...
    Furthermore, the ratio approaches (1 + √2)2 as the numbers xn+1 and xn get bigger and bigger. This brings to mind the ratio of the adjacent numbers of the Fibonacci sequence:
    Fn+1/Fn ≈ ½(1 + √5) = 1.6180339...
    which is generated instead by a sum of adjacent numbers which approach the golden mean as the two numbers get bigger and bigger.
    (As an aside 5.828434... and its inverse are a roots of the equation x2 -6x + 1 just as 1.6180339... and its inverse are roots of the equation x2 -x - 1).
  7. Alternatively we can look at the conversion of one tuple into another by the use of a geometrical progression:
    xn × R, xn × R2, xn × R3, xn × R4...
    where R is a common ratio and the higher values approach the true values of x as x gets bigger and bigger. Three sequences based on a, b and c are shown in Part IVF.
  8. We can generate sequences from the table lines by employing two other novel geometric progression and recursion methods, Part VF, much simpler tha the previous method. The methods only require the initial number for the geometric progression and only the first two numbers from the tables to generate the sequences without resorting to extracting the values from the tables. The tables are then used only for comparison of the numbers with the calculated sequences.
  9. The new table of right diagonal employing imaginary numbers and the new magic ratio (R) are listed in Part IB , Part IC , Part ID and Part IE .
  10. In addition, it will be shown that in a tuple, (ai, b, c), a, b and c take on certain allowed values. Furthermore, b is always positive and the differences between a and c take on only certain allowed values. See Part IB , Part IC , Part ID and Part IE .

Generation of Tables for odd and even numbers

  1. Odd and even numbers are calculated using either Table I or Table II.
  2. Table I (Odd Numbers)
    δ1iai b cδ2
    -ni0n
    2ni2n
    ni2n3n
    6ni6n
    7ni4n9n
    10ni10n
    17ni6n19n
    14ni14n
    31ni8n33n
    18ni18n
    49ni10n51n
    22ni22n
    71ni12n73n
    Table II (Even Numbers)
    δ1iai b cδ2
    -ni0n
    n/2in/2
    -n/2in3/2n
    3/2ni3/2n
    ni2n3n
    5/2ni5/2n
    7/2ni3n11/2n
    7/2ni7/2n
    7ni4n9n
    9/2ni9/2n
    23/2ni5n27/2n
    11/2ni11/2n
    17ni6n19n
  3. Odd numbers are calculated using the left hand table while even numbers are calculated using the right hand table. For example using 5 as the odd example δ1 and δ2 are incremented by 20 and while using 6 as the even example δ1 and δ2 are incremented by 6. These values are then added to a and c, respectively, while b is calculated either using the variables in the table above or via the following equation:
  4. [(2a/n + 2)1/2 × n]
    Table III (Odd Number 5)
    δ1iai b cδ2
    -5i05
    10i10
    5i1015
    30i30
    35i2045
    50i50
    85i3095
    70i70
    155i40165
    90i90
    245i50255
    110i110
    355i60365
    Table IV (Even Number 6)
    δ1iai b cδ2
    -6i06
    3i3
    -3i69
    9i9
    6i1218
    15i15
    21i1833
    21i21
    42i2454
    27i27
    69i3081
    33i33
    102i36114
  5. It will be shown in Part IB that these tables have common factors, viz., 5 and 3 and may be further factored to produce Tables V and VI.
  6. The use of a number such as n = 27 produces Table III1 via use of template Table I where the δs are incremented by 108.
  7. We can (a) expand the first two tuples from -27i to 27i and (b) expand the δs = 54 into 6 + 18 + 30. Table III2 shows the result where the sum of the δs equal 54 and the new tuples are adjusted accordingly. At this point no further expansion is possible since b would then either be odd (not allowed!) or the sum of the various δ1s and δ2s would not be equal to the original 54. The main reason for doing this expansion is to show that there may be more tuples that are hidden beneath the surface which may be of interest.
  8. Table III1 (Odd Number 27)
    δ1iai b cδ2
    -27i027
    54i54
    27i5481
    162i162
    189i108243
    270i270
    459i162513
    Table III2 (Odd Number 27)
    δ1iai b cδ2
    -27i027
    6i6
    -21i1833
    18i18
    -3i3651
    30i30
    27i5481
  9. On the other hand, some even or odd numbers are not expandable. The even number 26 in Table IV1 below cannot be expanded further since there are no δs such that their sum would be equal to 13.

    However, the first two tuples in even number 24 of Table IV2 may be expanded using the sequence 3,9 whose sum is 12 and generates the subtable IV21 with the new tuple (-21i,12,27). This tuple is factorable by 3 into a non factorable form (7i,4,9). On squaring, these numbers affords the tuple (-49,16,81) which is an allowed right diagonal for a magic square of squares.
Table IV1 (Even Number 26)
δ1iai b cδ2
-26i026
13i13
-13i2639
39i39
26i5278
65i65
91i78143
Table IV2 (Even Number 24)
δ1ai b cδ2
-24i024
12i12
-12i2436
36i36
24i4872
60i60
84i72132
Table IV21 (Even Number 24)
δ1ai b cδ2
-24i024
3i3
-21i1227
9i9
12i2436

This concludes Part IA. To continue to Part IIA where the tuples of even and odd squares are discussed and Part IB where the magic ratio (R) is used to produce tables of allowed tuples.

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Copyright © 2016 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com