A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

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

Picture of a square

The Concept of Tuple Conversion

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

(ani, bn, cn) → (an+1i, bn+1, cn+1)

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

an+1 = an × 5.828434...
bn+1 = bn × 5.828434...
cn+1 = cn × 5.828434...

so that:

an+1/an ≈ 5.828434...
bn+1/bn ≈ 5.828434...
cn+1/cn ≈ 5.828434...

in which the ratio approaches (1 + √2)2 as the numbers an+1,bn+1,cn+1, an,bn and cn 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 (Δc-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:

Table of Integers
TableLeftnIncrementTableRightnIncrement
Voddn+1VIeven n-1
VIIevenn+1VIIIodd n-1
IXoddn+1Xeven n-1
..................

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:

S = -a2 + b2 + c2

The first group of two tables listed are tables V and VI below. 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.

Table V (Odd Number 1)
δ1iai b cδ2
-i01
2i2
i23
6i6
7i49
10i10
17i619
14i14
31i833
18i18
49i1051
22i22
71i1273
26i26
97i1499
Table VI (Even Number 2)
δ1iai b cδ2
-2i02
i1
-i23
3i3
2i 46
5i5
7i611
7i7
14i 818
9i9
23i1027
11i11
34i 1238
13i26
47i1451
V or VI
Sum
0
 
12
 
48
 
108
 
192
 
300
 
432
 
588

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.

Table VII (Even Number 8)
δ1iai b cδ2
-8i08
9i9
i1217
27i27
28i 2444
45i45
73i3689
63i63
136i 48152
81i81
217i60233
99i99
316i 72332
117i117
433i84449
Table VIII (Odd Number 9)
δ1iai b cδ2
-9i09
8i8
-i1217
24i24
23i2441
40i40
63i 3681
56i56
119i48137
72i72
191i60209
88i88
279i 72297
104i104
383i84401
VII or VIII
Sum
0
 
432
 
1728
 
2888
 
6912
 
10800
 
15552
 
21168

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 (Δc-a) of 50. This subset is shown in Table Xsubset 1. The fourth (in light green) starting with 1175, when divided by 25 affords (47i,14,51) is part of Table VI.

Table IX (Odd Number 49)
δ1iai b cδ2
-49i049
50i50
i7099
150i150
151i140249
250i250
401i210499
350i350
751i 280849
400i400
1201i3501299
550i550
1751i4201849
650i650
2401i4902499
Table X (Even Number 50)
δ1iai b cδ2
-50i0 50
49i49
-i7099
147i147
146i 140246
245i245
391i210491
343i343
734i 280834
441i441
1175i 3501275
539i539
1714i 4201814
637i637
2351i4902451
IX or X
Sum
0
 
14700
 
58800
 
132300
 
235200
 
367500
 
529200
 
720300

Table Xsubset 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 Xsubset 1 is now 192 as opposed to 98.

In addition, Table Xsubset 1 can undergo further expansion say we just expand only the first two tuples, from -25i to 73i, to produce Table Xsubset 2 (Odd Number 25) and then increment by 4. We won't attempt to expand the whole of Table Xsubset 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.


Table Xsubset 1 (Odd Number 25)
δ1ai b cδ2
-25i025
98i98
73i70123
294i294
357i140417
440i440
857i210907
Table Xsubset 2 (Odd Number 25)
δ1iai b cδ2
-25i025
2i2
-23i1027
6i6
-17i2033
10i10
-7i3043
14i14
7i4057
18i18
25i5075
22i22
47i6097
26i26
73i70123

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

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Copyright © 2016 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com