A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

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

Production of New Tables

This page continues from the previous Part IC. The next two tables employing numbers 1681 and 1682, respectively, are Tables XIII, on the left, and XIV, on the right, and the Sum of each tuple, i.e., every other line is:

shown at the extreme right and are both identical. Again we start off with either of these two numbers and fill up the tables by adding either 1682 to 1681 or 1681 to 1682 (with or without the is). The δs are incremented by 3364 for Table XIII and 3362 for Table XIV. The δs are then added to the previous a or c. The b is calculated as previously according to equation:

where n is in our first case either 1681 or 1682.

Analysis of these tables shows that only table XIV contains tuples which are divisible by 2 (in light blue) to generate Table XIV_{subset 1}.

The subtable Table XIV_{subset 1} below with odd number 841 is expanded by adding δ = 2 to ±841 to generate the first tuple. From this tuple we may calculate b_initial, followed by division of b_final by b_initial to afford 162, 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 1682, a b_initial = 58 and a b_final = 2378. Thus there are 41 δs starting at 2(i) and ending at 162(i) of which only seven are shown. [Note that n(i) is my shorthand version to stand for n or n(i)]

A second switcheroo places Table XV with an even number on the left and Table XVI with an odd number on the right. The Δ_c-a of the leftmost even table is 19600 while that of the rightmost and a second part of the odd table below it is 19602.The center table corresponds to the magic sum and is identical in both cases.

The table is split so that the portion from 9801 to 342999 is used for comparison with table XV. The extra lines (after division by 9), however, affords four more yellow lines to be used in Table XVI_{subset 1}. Furthermore, the light green tuples of Table XV were divided by 4 to afford Table XV_{subset 1}.

A portion of both tables XV and XVI can expanded further to give Table XV_{subset 1} and Table XVI_{subset 1}, with even number 2450 and odd number 1089, respectively.

Expansion of Table XV_{subset 1} from just -2450 to 2351 affords Table XV_{subset 2} with a Δ_c-a of 4900, a b_final/b_initial = 99 and final δs of 197i and 197, respectively, by the Gauss theorem.

Similarly expansion of Table XVI_{subset 1} from just -1089 to 8711 affords Table XVI_{subset 2} with a Δ_c-a of 2178, a b_final/b_initial = 70 and final δs of 278i and 278, respectively, by the Gauss theorem.

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

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