A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

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

The Concept of Tuple Conversion

I introduced the concept also that a new set of allowed tuples can be converted into the next higher set of allowed tuples for example,

using a newly discovered ratio (1 + √2)² = 5.828434... which is used to generate the next highest number in a tuple, e.g.,

so that:

in which the ratio approaches (1 + √2)² as the numbers a_n+1,b_n+1,c_n+1, a_n,b_n and c_n get bigger and bigger.

In addition, it will be shown that in a tuple, (ai, b, c), a, b and c take on certain allowed values. First b is always even. Second, the differences between c and a (Δ_c-a) take on only certain values (n₁ or n₂) (2 or 4), (16 or 18), (98 or 100), (576 or 578), (3362 or 3364),(19600 or 19602) and so on, respectfully. And where the previous value of a or c is multiplied in a sense by the magic ratio (R) of 5.828434... to get to the next value.

However, the higher values obtained are an approximation to the desired ones. The true values are actually obtained by taking the average of the two tables, for example, from above the average of 16 and 18, and multiplying this new value by the magic ratio (R) to give the true value for the next higher average which in this case is 99.0, i.e., the average of 98 and 100.

Production of New Tables of Tuples

Tuples comprised of one initial imaginary number, (ai,b,c), can upon squaring, be useful as right diagonals in magic squares. One can start with (ni,0,n), where n is a special odd or even number, and create a table by the gradual incremental buildup of that tuple to generate the next higher tuple in the table. In this method the tables are present as a paired entity where n in one table is initially odd and its pair in a second table, is even. (see Tables V and VI below).

Generation of the next higher table generates a switch between the next two paired tables where the left handed table now is even and the right handed one odd. Every pair of tables investigated thus far behaves likewise and thus, appears to be a property that goes on ad infinitum (see Table of Integers below). In addition, the initial numbers are incremented as follows, odd numbers, n, are incremented evenly and even numbers, n, are incremented oddly. Thus, the inital increment for an odd number as shown in the table is n+1 and for an even one it is n-1 which switches for the next pair of tables:

Thus we can proceed as follows. Each tuple contains three numbers which when squared becomes the sum of a magic square having three squares in its major diagonal. The equation for this sum is shown below:

The first group of two tables listed are tables V and VI below. The tables are generated only from a certain sequence of numbers. The even numbers are generated from 2n² for example 2,8,18,32,50,72,98,128,162,200... while the odd numbers are generated from (2n + 1)² for example 1², 3², 5² 7², 9².... Only these two formulas will generate the requisite diagonal numbers which work. Numbers outside these sequences will not generate true magic square diagonals having the property c² = 2b² − ( − a²).

Only a portion of these tables are shown since up to an infinite number of entries exist. The differences between the coefficient of a and c for each tuple in Tables V and VII are 2 and 4, respectively. In addition, the sum of each squared tuple is shown at the extreme right and both these values are identical for each pair of tuples, one coming from each table on the same line. If we divide the light orange tuples by 2 we see that this subset belongs to Table V. The non colored tuples, i.e., the three numbers comprising this tuple, cannot be divided by a common factor.

The next two tables in the series, VII and VIII, are shown below where the even number is now situated on the left and the odd number on the right.

In fact, both tables contain factorable tuples, the light green tuples of Table VII are divisible by 4 and the light blue of Table VIII by 9. In addition, the factored tuples of VII are subsets of Table VI, while those of Table VIII are subsets of Table V. Thus, dividing those tuples which are not in their lowest factored form, by a common factor, produces tuples which may belong to a previous table.

A second switch now places the odd number on the left and the even number to the right. All the tuple numbers of Table IX are in their lowest factored form and, those of Table X may be factored in either of two ways. The first, second and third and fifth tuples (in yellow), after division by 2 do not belong to any particular table, having a difference (Δ_c-a) of 50. This subset is shown in Table X_{subset 1}. The fourth (in light green) starting with 1175, when divided by 25 affords (47i,14,51) is part of Table VI.

Table X_{subset 1} below shows that the initial δs of 98 are obtained from the Δδ differences from table X. The differences between two δs in Table X_{subset 1} is now 192 as opposed to 98.

In addition, Table X_{subset 1} can undergo further expansion say we just expand only the first two tuples, from -25i to 73i, to produce Table X_{subset 2} (Odd Number 25) and then increment by 4. We won't attempt to expand the whole of Table X_{subset 1} since it would triple the size of the table. In addition, the sum of δ values on both right and left columns sum up to 98i and 98, respectively.

This concludes Part IB. Go to Part IC to continue on tables of allowed tuples.

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