NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES

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

Picture of a square

Tables of tuples from Square Numbers

As previously stated in Part IA all numbers may be converted into diagonal tuples using Tables I and II of Part IA. However, only certain numbers can use the methods employed on this page. These are the odd and even squares and 2× odd or 2× even squares. Thus, the even and squares can be initially be incremented by 1 or 2 which are the lowest increments possible. Therefore, incremental addition to either a square or 2 × square is carried out using the two values 1 or 2 as is shown below. Be advised that these are only partial tables and can continue on to infinity.

Notice that the tuples bearing a checkmark in the upper leftmost corner are part of the initial tuples used in the generation of Tables in Part IB. These are special tables which can be converted to the next higher table by multiplying by the magic ratio (R) (1 + √2)2 = 5.828427125...

Diagonal tuples employing squares generated from odd numbers take on the initial form of Tables A and B and are first incremented by 2:

(-o2i, 0, o2)
(-o2+2i, 0, o2+2)

which are then incremented by 6,10,14... (not shown here).

Table A (Square Odd Numbers)
δ1iai b cδ2
-i01
2i2
i23
 
-9i09
2i2
-7i611
 
-25i025
2i2
-23i1027
 
-49i049
2i2
-47i1451
 
-81i081
2i2
-79i1883
Table B (Square Odd Numbers)
δ1iai b cδ2
-121i0121
2i2
-119i22123
 
-169i0169
2i2
-167i26171
 
-225i0225
2i2
-223i30227
 
-289i0289
2i2
-287i34291
 
-361i0361
2i2
-359i38363

Diagonal tuples employing squares generated from even numbers take on the initial form of Tables C and D and are first incremented by 2:

(-e2i, 0, e2)
(-e2+2i, 0, e2+2)

which are then incremented by 6,10,14... (also not shown here).

Table C (Square Even Numbers)
δ1iai b cδ2
-4i04
2i2
-2i46
 
-16i016
2i2
-14i818
 
-36i036
2i2
-34i1238
 
-64i064
2i2
-62i1666
 
-100i0100
2i2
-98i20102
Table D (Square Even Numbers)
δ1iai b cδ2
-144i0144
2i2
-142i24146
 
-196i0196
2i2
-194i28198
 
-256i0256
2i2
-254i32258
 
-324i0324
2i2
-322i36326
 
-400i0400
2i2
-398i40402

Diagonal tuples employing squares generated from odd numbers × 2 take on the initial form of Tables E and F and are first incremented by 1:

[2 × (-o)2i, 0, 2 × o2]
[(2 ×(-o)2+1)i, 0, (2 × o2+1)]

which are then incremented by 6,10,14... (not shown here).

Table E (2 × Square Odd Numbers)
δ1iai b cδ2
-2i02
1i1
-i23
 
-18i018
1i1
-17i619
 
-50i050
1i1
-49i1051
 
-98i098
1i1
-97i1499
 
-162i0162
1i1
-161i18163
Table F (2 × Square Odd Numbers)
δ1iai b cδ2
-242i0242
1i1
-241i22243
 
-338i0338
1i1
-337i26339
 
-450i0450
1i1
-449i30451
 
-578i0578
1i1
-577i34579
 
-722i0722
1i1
-721i387233

Diagonal tuples employing squares generated from even numbers × 2 take on the initial form of Tables G and H and are first incremented by 1:

[2 × (-e)2i, 0, 2 × e2]
[(2 ×(-e)2+1)i, 0, (2 × e2+1)]

which are then incremented by 6,10,14... (not shown here).

Table G (2 × Square Even Numbers)
δ1iai b cδ2
-8i08
1i1
-7i49
 
-32i032
1i1
-31i833
 
-72i072
1i1
-71i1273
 
-128i0128
1i1
-127i16129
 
-200i0200
1i1
-199i20201
Table H (2 × Square Even Numbers)
δ1iai b cδ2
-288i0288
1i1
-287i24289
 
-392i0392
1i1
-391i28393
 
-512i0512
1i1
-511i32513
 
-648i0648
1i1
-647i36649
 
-800i0800
1i1
-799i40801

This concludes Part IIA. To continue to Part IB where the magic ratio (R) is used to produce tables of allowed tuples.

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