ODDWHEEL: Wheel Magic Squares

Introduction

A magic square is an arrangement of numbers 1,2,3,... n² where every row, column and diagonal add up to the same magic sum S and n is also the order of the square. A magic square having all pairs of cells diametrically equidistant from the center of the square and equal to the sum of the first and last terms of the series n² + 1 is also called associated or symmetric. In addition, the center of this type of square must always contain the middle number of the series, i.e., ½(n² + 1).

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.

A Discussion of the general Wheel methods

To construct a magic square by these new methods the series 1..n² is paired up in complementary fashion, for example 1 is paired with n², 2 with n²-1, etc. These pair of numbers are then partitioned into n+1 groups of ½(n-1) pairs including the unpaired middle number of the series. This is depicted below using n = 5 complementary table example. We start at n = 5 since the known n = 3 consists totally of "spoke" numbers (explained below).

Methods A-1 and A-2

Using method A-1 a magic square is first constructed by filling in the left diagonal with the group of numbers ½ (n²-n+2) to ½(n²+n) in any combinatorial order, as shown below, from the numbers listed in the complementary table above. This produces a template that can be used to (1) fill in the right diagonal with n - 1 numbers in reverse order from bottom left corner to the right upper corner (2) fill in the central column by a select group of n - 1 numbers and (3) fill, in reverse order, the central row by a select fourth group of n - 1 numbers. Method A-2, on the other hand, while being similar to A-1 uses a mix of templates, the normal and its invert, and is discussed under template inversion.

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.

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 Method A-2:Template Inversion a mixed square.

Method B

Using this method a magic square is first constructed by filling in the left diagonal with the group of numbers ½ (n²-n+2) to ½(n²+n) in any combinatorial order (top left corner to the right lower corner) from the numbers listed in a n x n complementary table. A group of three adjacent pairs, as opposed to three ½(n-1) adjacent pairs used in Method A, are used to generate a partial wheel. This is followed by placing the requisite numbers into the middle column and rows to complete each concentric square until the entire wheel is built up or "expanded". Then using parity, the "non-spoke" numbers are filled similarly to Method A-1 or A-2 as shown in Method B:Wheel Expansion.

Modified Loubère, Bachet de Méziriac and Wheel Methods

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..n² 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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