A NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

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

Picture of a square

Production of New Tables

This page continues from the previous Part IB. The next two tables employing numbers 288 and 289, respectively, are Tables XI and XII and the Sum of each tuple, i.e., every other line is:

S = -a2 + b2 + c2

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 289 to 288 or 288 to 289 (with or without the is). The δs are incremented by 578 for Table XI and 576 for Table XII. The δs are then added to the previous a or c. The b is calculated as previously according to equation:

[(2a/n + 2)1/2 × n]

where n is in our first case either 288 or 289. I have placed an expanded version of the first two tuples of both 288 and 289 in Part IIC.

Table XI (Even Number 288)
δ1iai b cδ2
-288i0288
289i289
i408577
867i867
868i 8161444
1445i1445
2313i12242889
2023i2023
4336i 16324912
2601i2601
6937i20407513
3179i2890
10116i 244810692
3757i3757
13873i285614449
Table XII (Odd Number 289)
δ1iai b cδ2
-289i0289
288i288
-i408577
864i864
863i 8161441
1440i1440
2303i12242881
2016i2016
4319i16324897
2592i2592
6911i20407489
3168i3168
10079i244810657
3744i3744
13823i285614401
XI or XII
Sum
0
 
499392
 
1997568
 
4494528
 
7990272
 
12484800
 
17978112
 
24470208

The light orange tuples of Table XI, whose numbers are all even, and therefore, factorable, by 4 as shown in the Table XIsubset 1. The difference between a and c, however, is now 144 compared to 576 for those tuples of Table XI. This former number while not belonging to the allowed numbers, i.e., (Δc-a = the difference between c and a, of the parent table, are allowed for the sub-tables. The tuples generated by these sub-tables provides, upon squaring, a greater number of tuples that can be used as the main diagonals in magic squares of squares.

Table XIsubset 1 (Even Number 72)
δ1iai b cδ2
-72i072
289i289
217i204361
867i867
1084i4081228
1445i1445
2529i6122673
Table XIsubset 2 (Even Number 72)
δ1iai b cδ2
-72i072
1i1
-71i1273
3i3
-68i2476
5i5
-63i3681
7i7
-56i4888
9i9
-47i6097
11i11
-36i72108
13i13
-23i84121
15i15
-8i96153
Table XIsubset 2 (Even Number 72)
δ1iai b cδ2
17i17
9i108153
19i19
28i120172
21i21
49i132193
23i23
72i144216
25i25
97i156241
27i27
124i168268
29i29
153i180297
31i31
184i192328
33i33
217i204361

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

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