A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

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

General Introduction of the New Method

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

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., (Δ_c-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 Δ_c-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 set of allowed tuples (in a sequence) into the next higher set by taking the average values of a, b or c, within a table, and converting to the average values a_avg^* b_avg^* or c_avg^* of the next higher set, within another table. That is converting all the numbers in the tuple:
   (a_avgi, b_avg, c_avg) → (a_avg^*i, b_avg^*, c_avg^*)
   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:
   x_n+1 = x_n × 5.828434...
   where x is a placeholder for a, b or c:
   x_n+1/x_n ≈ 5.828434...
   Furthermore, the ratio approaches (1 + √2)² as the numbers x_n+1 and x_n get bigger and bigger. This brings to mind the ratio of the adjacent numbers of the Fibonacci sequence:
   F_n+1/F_n ≈ 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... plus its inverse, i.e. R + 1/R = 6, is a root of the equation x² -6x + 1 just as 1.6180339... and its inverse are roots of the equation x² -x - 1). See The use of Ratio(R) for converting tuples (1,b_n,c_n) into (1,b_n+1,c_n+1).
7. Alternatively we can look at the conversion of one tuple into another by the use of a geometrical progression:
   x_n × R, x_n × R², x_n × R³, x_n × 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.

Table I (Odd Numbers) δ1i ai b c δ2 -ni 0 n 2ni 2n ni 2n 3n 6ni 6n 7ni 4n 9n 10ni 10n 17ni 6n 19n 14ni 14n 31ni 8n 33n 18ni 18n 49ni 10n 51n 22ni 22n 71ni 12n 73n

Table II (Even Numbers) δ1i ai b c δ2 -ni 0 n n/2i n/2 -n/2i n 3/2n 3/2ni 3/2n ni 2n 3n 5/2ni 5/2n 7/2ni 3n 11/2n 7/2ni 7/2n 7ni 4n 9n 9/2ni 9/2n 23/2ni 5n 27/2n 11/2ni 11/2n 17ni 6n 19n

2. 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:
[(2a/n + 2)^1/2 × n]

Table III (Odd Number 5) δ1i ai b c δ2 -5i 0 5 10i 10 5i 10 15 30i 30 35i 20 45 50i 50 85i 30 95 70i 70 155i 40 165 90i 90 245i 50 255 110i 110 355i 60 365

Table IV (Even Number 6) δ1i ai b c δ2 -6i 0 6 3i 3 -3i 6 9 9i 9 6i 12 18 15i 15 21i 18 33 21i 21 42i 24 54 27i 27 69i 30 81 33i 33 102i 36 114

3. 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.
4. 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.
5. 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.

Table III1 (Odd Number 27) δ1i ai b c δ2 -27i 0 27 54i 54 27i 54 81 162i 162 189i 108 243 270i 270 459i 162 513

Table III2 (Odd Number 27) δ1i ai b c δ2 -27i 0 27 6i 6 -21i 18 33 18i 18 -3i 36 51 30i 30 27i 54 81

6. 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.

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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