NEW MAGIC SQUARE METHODS: WHEEL EXPANSION

A Discussion of new Method B Algorithm

To construct a magic square by this method 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 eventually partitioned into n+1 groups of ½(n-1) pairs including the unpaired middle number of the series. However as opposed to Method A, this partioning is done by choosing triplets of pairs (the numbers and their complements) for the "spoke" numbers and doublets of pairs for the "non-spoke" numbers and filling the entire wheel from the inside out, as shown below.

In other words, a "spoke" is generated from a triplet of pairs for a 3x3 square or a multiple of triplets {a,complement a),(b,complement b),(c,complement c)}... where the equation n - ½(n + 1) gives the number of triplets per nxn square along with the hub.
For example, for a 5x5 square II (below), the two triplets {1,2,3} and {4,5,6} along with their complements {25,24,23} and {22,21,20} are taken from the complementary table below and used to construct the "spokes".

By analysis the number of conformations is equal to the equation {(Σ ¼(n²-4n + 7) - a} × (½(n-1))! where a is varied from 0 to n-1. This method differs from method A in that each group of three pairs may take any available concentric positions on the square. On the other hand, the use of the equation { ¼(n²-4n + 7) × ½ (¼(n²-4n + 7) + 1) × (½ (n-1))!}, avoids having to use summation groups. For a 5x5 square, however, either of these equations generates 12 conformations.

Method B

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) using any of the requisite combinations. For a 5x5 square the numbers in the left diagonal are added consecutively for example using variant 1 {11,12,13,14,15} as in square I. The central square is first laid down according to the wheel algorithm, followed by the external second square (as in Square II). The "non-spoke" are finally added according to the parity table on the right.

The result is Square III and IV, where both square IV and its concentric square are also magic.

Alternatively, the "spoke" may be formed by using different combination of numbers. For example the square produced by using {4,5,6} followed by {1,2,3} gives square V then VI which may then be filled in the usual fashion.

One example of a 7x7 spinning magic square

One example (out of 168) of a 7x7 square is the combination 24 → 22 → 27 → 25 → 23 → 28 → 26. The first concentric wheel is produced using for example the first triplet (1-3) along with the last three triplets (22-24) and the central number (25) as shown in square VII. The rest of the "spoke" numbers are added so that the sum of three numbers per row is closest to 75 (the magic sum of a central 7x7 square). Thus between the numbers 1 and 3 the best choice is 1 since the sum produced is 73 as opposed to 71 with the number 3. Repeating for rows one and two using this same method gives square VIII. The "non-spoke" numbers are then filled in according to the following parity table, remembering that the allowable pair sums are n² = 49, n²+1 = 50 and n²+2 = 51 where these are compared to the possible non-allowed alternatives (note that only half of the parity table is shown since by symmetry the rest of the table is similar):

For example, although line 2 of the parity table contains an allowed O+O, no allowed E+E is available for lines 4-7. Therefore, the allowed line is 1.

We first try filling row three with "non-spoke" numbers. However, placing the sum of allowable pairs 50 and 49 is not as easy as it seems. The first thing to do is to fill in row/column 2 (as shown in square IX) since both pair of numbers are 51. This makes it easier to fill in the rest of the square since row/column three must be filled in with the pair that adds up to 49 (as shown in square X). Note that the number inserted is 31 a reverse H.

This completes method B. To continue on to brand new Loubère and wheel methods. To go back to the previous Method A-2 or to homepage.