A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

GEOMETRIC PROGRESSION OF TUPLE NUMBERS TO GENERATE NEW SEQUENCES (Part IVF)

Intro to Geometric Progressions using Table of Tuples and the Magic Ratio (R)

The tables of partial imaginary tuples listed in   Part IB,   Part IC,     Part IIIC,   Part ID   and Part IE are useful, when squared, as right diagonals for magic square of squares. I will show that the averages of each of two tuple numbers from the designated tables may be used to generate the next number of averages in the geometric progressions via successive multiplications with the common ratio R = (1 + √2)² = 5.8284268.....

Thus, for example we can generate the geometric progression:

where x_avg is a placeholder for a_avg, b_avg, or c_avg. I will show that starting with the initial x_avg and gradually replacing the starting x_avg with the next real value, there comes a point where the numbers take on the real values. From the tables below, using seven columns and ten rows, the seventh column appears to be that point. Increasing the number of rows, however, produces numbers which are initially far from the real values. To get these numbers more in line with the real values requires us to increase the number of columns as far as is required.

Generation of Tables of Geometric Progressions

Three tables are generated one for a_avg (Table GP₁), one b_avg (Table GP₂) and one for c_avg (Table GP₃). The sequences for these table were obtained from Part IIIF of the previous page which were calculated from the 8^th row of tuples for each of Tables VI to XXIV. The sequence for a_avg of tuple (a_avgi,b_avg, c_avg) is as follows:

where the last underlined number was obtained by multiplying 542868888 by R and not from a table. The numbers are arranged in columns and the initial number averages are multiplied by R over and over until the end of the column. Going across the rows we can see that the numbers approach the real number averages as the numbers get bigger and bigger.

The sequence for b_avg of tuple (a_avgi,b_avg, c_avg) is as follows:

where the last underlined number was obtained by multiplying 111960996 by R and not from a table, just as was done above.

The sequence for c_avg of tuple (a_avgi,b_avg, c_avg) is as follows:

where the last underlined number was obtained by multiplying 97022475 by R and not from a table, just as was done above.

The Table GP_c show that as we go across columns in diagonal fashion, the numbers approach and eventualy equal the real value. For example, at position (c_avg × R⁶) 2940150 increases to 2856075 at column 5 and remains constant up to the 1^strow, 7^th column, the real value.

This concludes Part IVF using the eighth rows of two tables for geometric progressions. To see a two novel methods via geometric progression and recursion Part IVF.

Finally to see what started it all go to Part IG which lists the new set of magic square of seven squares.

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