NEW METHOD FOR GENERATING MAGIC SQUARES OF SQUARES
THE USE OF ONE IMAGINARY NUMBER AS PART OF THE RIGHT DIAGONAL (Part IIA)
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:
(o^{2}i, 0, o^{2})
(o^{2}+2i, 0, o^{2}+2)
which are then incremented by 6,10,14... (not shown here).
Table A (Square Odd Numbers)
δ_{1}i  ai 
b  c  δ_{2} 
✓  i  0  1  
2i     2 
 i  2  3  
  
 
✓  9i  0  9  
2i     2 
 7i  6  11  
  
 
 25i  0  25  
2i     2 
 23i  10  27  
  
 
✓  49i  0  49  
2i     2 
 47i  14  51  
  
 
 81i  0  81  
2i     2 
 79i  18  83  

 
Table B (Square Odd Numbers)
δ_{1}i  ai 
b  c  δ_{2}

 121i  0  121  
2i     2 
 119i  22  123  
  
 
 169i  0  169  
2i     2 
 167i  26  171  
  
 
 225i  0  225  
2i     2 
 223i  30  227  
  
 
✓  289i  0  289  
2i     2 
 287i  34  291  
  
 
 361i  0  361  
2i     2 
 359i  38  363  

Diagonal tuples employing squares generated from even numbers take on the initial form of Tables C and D and are
first incremented by 2:
(e^{2}i, 0, e^{2})
(e^{2}+2i, 0, e^{2}+2)
which are then incremented by 6,10,14... (also not shown here).
Table C (Square Even Numbers)
δ_{1}i  ai 
b  c  δ_{2} 
 4i  0  4  
2i     2 
 2i  4  6  
  
 
 16i  0  16  
2i     2 
 14i  8  18  
  
 
 36i  0  36  
2i     2 
 34i  12  38  
  
 
 64i  0  64  
2i     2 
 62i  16  66  
  
 
 100i  0  100  
2i     2 
 98i  20  102  

 
Table D (Square Even Numbers)
δ_{1}i  ai 
b  c  δ_{2} 
 144i  0  144  
2i     2 
 142i  24  146  
  
 
 196i  0  196  
2i     2 
 194i  28  198  
  
 
 256i  0  256  
2i     2 
 254i  32  258  
  
 
 324i  0  324  
2i     2 
 322i  36  326  
  
 
 400i  0  400  
2i     2 
 398i  40  402  

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)^{2}i, 0, 2 × o^{2}]
[(2 ×(o)^{2}+1)i, 0, (2 × o^{2}+1)]
which are then incremented by 6,10,14... (not shown here).
Table E (2 × Square Odd Numbers)
δ_{1}i  ai 
b  c  δ_{2} 
✓  2i  0  2  
1i     1 
 i  2  3  
  
 
 18i  0  18  
1i     1 
 17i  6  19  
  
 
✓  50i  0  50  
1i     1 
 49i  10  51  
  
 
 98i  0  98  
1i     1 
 97i  14  99  
  
 
 162i  0  162  
1i     1 
 161i  18  163  

 
Table F (2 × Square Odd Numbers)
δ_{1}i  ai 
b  c  δ_{2} 
 242i  0  242  
1i     1 
 241i  22  243  
  
 
 338i  0  338  
1i     1 
 337i  26  339  
  
 
 450i  0  450  
1i     1 
 449i  30  451  
  
 
 578i  0  578  
1i     1 
 577i  34  579  
  
 
 722i  0  722  
1i     1 
 721i  38  7233  

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)^{2}i, 0, 2 × e^{2}]
[(2 ×(e)^{2}+1)i, 0, (2 × e^{2}+1)]
which are then incremented by 6,10,14... (not shown here).
Table G (2 × Square Even Numbers)
δ_{1}i  ai 
b  c  δ_{2} 
✓  8i  0  8  
1i     1 
 7i  4  9  
  
 
 32i  0  32  
1i     1 
 31i  8  33  
  
 
 72i  0  72  
1i     1 
 71i  12  73  
  
 
 128i  0  128  
1i     1 
 127i  16  129  
  
 
 200i  0  200  
1i     1 
 199i  20  201  

 
Table H (2 × Square Even Numbers)
δ_{1}i  ai 
b  c  δ_{2} 
✓  288i  0  288  
1i     1 
 287i  24  289  
  
 
 392i  0  392  
1i     1 
 391i  28  393  
  
 
 512i  0  512  
1i     1 
 511i  32  513  
  
 
 648i  0  648  
1i     1 
 647i  36  649  
  
 
 800i  0  800  
1i     1 
 799i  40  801  

This concludes Part IIA. To continue to Part IB where the magic ratio (R)
is used to produce tables of allowed tuples.
Go back to homepage.
Copyright © 2016 by Eddie N Gutierrez. EMail: enaguti1949@gmail.com