Octagon Algorithm for Partial Border Wheel Squares (Part XV)

Picture of a wheel

The Eight Node Way - (a,c,e) Squares

The previous section Part XIV looked at (a,b,c) type squares specifically the 5×5 squares where the initial, central cell number was one and the Octagon M and G algorithms were used for filling in the non spoke cells. Part XV is concerned with the generation of partial border squares where only the internal 3×3 and the outer squares are magic.

For the square to be partially or fully bordered the internal 3×3 square is constructed using a triplet of three numbers to ensure that all the rows columns and diagonals sum to the magic sum 123 for the 7th order. These triplets are labeled as (c,d,r) corresponding to the last cells in the central, diagonal and row spokes adjacent to the center cell in the square and are shown in Table Ib. Those triplets in white correspond to triplets that contain at least one common number with any of the spokes of the wheel:

Table Ib
Odd c
5,6,7
7,8,9
9,10,11
11,12,13
13,14,15
15,16,17
17,18,19
19,20,21
21,22,23

The 7×7 Wheel Squares

This section will deal with numbers (after the main diagonal is filled) that run in the order a (top center), c (bottom right) and e (left center) and whose 3×3 inner squares are filled with either of the three triplets from Table Ib. Only those triplets are that have no common numbers in the spokes are used.

The wheel portion of the square is filled according to Part X, using the blue color values from Table Ib to ensure that every row, column and diagonal of the internal 3×3 square adds up to the magic sum of 123. The construction of the square is then completed by the use the M and G algorithms, using graph theory, as was done previously in Part X.

Picture of an octagon

So using the Octagon G and M algorithms seven 7th order squares covering all seven of the blue triplets of Table Ib are constructed:

Border 7a(7,8,9)mg
4739 141 3511 28
1346 182 312738
1721 427 262933
56 925 4144 45
3430 2443 82016
3723 3248 19412
2210 3649 1540 3
   
Border 7a(9,10,11)mg
4742 141 358 28
1346 182 312738
1721 409 262933
56 1125 3944 45
3430 2441 102016
3723 3248 19412
227 3649 1543 3
   
Border 7a(11,12,13)mg
4742 141 358 28
1046 182 312741
1721 3811 262933
56 1325 3744 45
3430 2439 122016
4023 3248 1949
227 3649 1543 3
Border 7a(13,14,15)mg
4742 111 388 28
1046 182 312741
1721 3613 262933
56 1525 3544 45
3430 2437 142016
4023 3248 1949
227 3949 1243 3
   
Border 7a(15,16,17)mg
4742 111 388 28
1046 182 312741
1421 3415 262936
56 1725 3344 45
3730 2435 162013
4023 3248 1949
227 3949 1243 3
   
Border 7a(17,18,19)mg
4742 111 388 28
1046 152 342741
1421 3217 262936
56 1925 3144 45
3730 2433 182013
4023 3548 1649
227 3949 1243 3

In addition, the Border 7a(19,20,21)mg can be compared with the previous A4 Partial Border square from 2015 Part I constructed using parity where the 3×3 of the A4 square is non magic.

Border 7a(19,20,21)mg
4742 111 388 28
1046 152 342741
1418 3019 263236
56 2125 2944 45
3733 2431 201713
4023 3548 1649
227 3949 1243 3
   
7a Partial Border
4742 111 388 28
1046 152 3427 41
1418 3019 2632 36
56 2125 2944 45
3733 2431 2017 13
4023 3548 164 9
227 3949 1243 3
   
A4 Partial Border
4610 13 1 38 39 28
354519 23227 15
3431 443 2621 16
789 25 41 4243
171824 476 3033
142329 4820 536
2240 37 4912 11 4

Analysis of square A4 shows that it is composed of three algorithms: an Octagon I reflected along the vertical axis, an Octagon I rotated 180 degrees and an altogether different algorithm with two different graph structures. These three methods will be shown to produce new wheel magic squares where the octagonal algorithm graphs do not follow the usual intersection of four edges at the center of the octagon. See Part XVI for details.

This completes the Octagon M and G method for the 7×7 section. To go to Part XVI. Go back to Part XIV. Go back to homepage.


Copyright © 2022 by Eddie N Gutierrez. E-Mail: enaguti1949@gmail.com