A magic square is an arrangement of numbers
This site introduces two new methods used for the construction of semi-associated odd magic squares. The method consists of pairing numbers in complementary fashion, partitioning these complementary pairs into groups and placing these groups into a square in a certain order using parity as an aid.
New previous approaches to magic squares, stars, tesseracts have been the subject of the web especially to those interested in recreational mathematics. A good place to start is at harvey heinz webpage which gives examples of these new approaches.
To construct a magic square by these new methods the series 1..n^{2}
is paired up in complementary fashion, for example 1 is paired with
n^{2}, 2 with n^{2}-1, etc. These pair of numbers
are then partitioned into n+1 groups of
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
13 | ||||||||||||
25 | 24 | 23 | 22 | 21 | 20 | 19 | 18 | 17 | 16 | 15 | 14 |
Using method A-1 a magic square is first constructed by filling in the left diagonal with the group of numbers
These special numbers are labeled the "spoke" numbers. The remainder cells are labeled the "non-spoke" numbers and these are subsequently filled using adjacent complementary pairs using parity to choose the pairs. These may be added semi-associatively as opposed to the "spoke" pairs which are always associative, i.e diametrically equidistant from the center cell. The following shows an illustration of a partial magic square (n = 5) generated using this method along with its complementary table.
11 | 5 | 23 | ||
12 | 6 | 22 | ||
25 | 24 | 13 | 2 | 1 |
4 | 20 | 14 | ||
3 | 21 | 15 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
13 | ||||||||||||
25 | 24 | 23 | 22 | 21 | 20 | 19 | 18 | 17 | 16 | 15 | 14 |
To fill in the "non-spoke" numbers, the sum of the
"spoke" numbers on every row are subtracted from
S producing pairs of numbers. These pairs are added to the magic square using the complementary table as a guide.
Five variant examples on subsequent pages will be shown step by step in Method A-1:Variant 1, three other 5x5 variants
Method A-1:Variant 2, three 7x7 variants Method A-1:Variant 2 a 7x7,
a 9x9 square Method A-1:a 9x9 Variant and two 7x7 variants
Using this method a magic square is first constructed by filling in the left diagonal with the group of numbers
These new methods are constructed using complementary tables of (n+2)x(n+2) or greater than the magic square itself (n_{s} x n_{s}). This generates a series of De La Loubère or wheel magic squares that are related to one another via the main diagonal within the same complementary set. These n_{s} x n_{s} consists of a smaller subset of complementary numbers chosen from the larger 1..n^{2} complementary table as well a five other methods based on Loubère and wheel magic squares. These are included in the indices.
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Copyright © 2013 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com