A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

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

Production of New Tables

This page continues from the previous Part IIIC. The next series of tables are Tables XVIII and XVIII, employing numbers 57121 and 57122, respectively. Again we start off with either of these two numbers and fill up the tables by adding either 57122 to 57121 or 57121 to 57122. The δs are incremented by 114244 for Table XVII and 114242 for Table XVIII. The δs are then added to the previous a or c and the b is calculated according to equation:

where n is in our first case either ±57121 or ±57122. In addition, the rightmost table lists the sum each tuple, i.e., the sum of the right major diagonal of a magic square which are identical for both tables:

The light orange tuples of Table XVIII, whose numbers are all even, are factorable by 2, as shown in Table XVIII_{subset 1}. The Δ_c-a, however, is now 57122 compared to 114244 for those tuples of Table XVIII.

This former number while not part of the allowed numbers of the parent table are allowed for the sub-tables. In addition, the tuples generated by the sub-tables provide an even greater number of tuples for use the main diagonals in magic squares of squares.

Table XVIII_{subset 1} is expanded by adding δ = 2 to ±28561 to generate the first tuple. From this tuple we may calculate b_initial, followed by division of b_final by b_initial to afford 239, the number of tuples that are required to fill in the expanded table. Consequently, using the Gauss equation:

which we rewrite to conform to our values as:

and entering in the values

Thus, the table is composed of a Δ_c-a of 57122, a b_initial = 338 and a b_final = 80782. Thus there are 239 δs starting at 2(i) and ending at 954(i) of which only three are shown. [Note n(i) is my shorthand version to stand for n or ni.]

We perform another switcheroo placing the table with even number on the left and the table with the odd number on the right. The sum of each square for each tuple is listed at the extreme right where the sums for both tuples are identical. No factors are found for the odd number table. However, the even number table contains the factor 4 for the all the numbers in the light blue tuples.

Table XIX_{subset 1} is expanded by adding δ = 1 to ±83232 to generate the first tuple as shown in Table XIX_{subset 2}. From this tuple we may calculate b_initial, followed by division of b_final by b_initial to afford 577, the number of tuples that are required to fill in the expanded table. Consequently, using the Gauss equation from above we obtain:

Thus, the table is composed of a Δ_c-a of 166464, a b_initial = 408 and a b_final = 235416. And there are 577 δs starting at 1(i) and ending at 1153(i) of which only four are shown.

This concludes Part ID. Go to Part IE to continue on tables of allowed tuples.

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