Office Applications and Entertainment, Magic Squares

15.0   Special Magic Squares, Bimagic Squares, Part 1

15.9   Bimagic Squares (16 x 16)

Harry White illustrates on his website that, under certain limiting conditions, it is possible to construct order 2n Bimagic Squares based on order n Bimagic Squares.

15.9.1 Medjig Solutions, General

As described in Section 6.8, for any integer n, a Magic Square C of order 2n can be constructed from any n x n Medjig-Square A and any n x n Magic Square B, by application of the equations:

c_j = b_i + n² a_j with i = 1, 2, ... n², j = 1, 2, ... 4n² and S_a = 3n.

To make the results for Bimagic Squares (n >= 8) comparable with the results published by other authors, following alternative convention will be applied:

c_j = 4*(b_i - 1) + a_j with i = 1, 2, ... n², j = 1, 2, ... 4n² and S_a = 5n.

as illustrated by the numerical example shown below:

with the corresponding Magic Sums S_b = 260, S_a = 40, S_c = 2056 and Sums of Squares S_b2 = 11180, S_a2 = 120, S_c2 = 351576.

Unlike the resulting Magic Sum S_c, the resulting Sum of Squares S_c2 depends from both b_i and a_j as:

S_c2 = Σ(4*(b_i - 1) + a_j)²

    = 2*16*(b_i - 1)² + 8*Σ(b_i - 1)*a_j + Σa_j²

    = 32*10668 + 8*S_ab + S_a2

    = 341376 + 8*S_ab + S_a2

With S_c2 = 351576 and S_a2 = 120 the square C will be bimagic if:

S_ab = (351576 - 341376 - 120)/8 = 1260

which should be verified while generating the Medjig Squares A for a certain Bimagic Square B.

The method of constructing Bimagic Squares of order 16, based on Medjig Squares as defined above, can be summarised as follows:

Read (preselected) 8 x 8 Bimagic Square B (192 * 136244 possibilities);

Construct 8 x 8 Medjig-Squares A, while ensuring that S_a2 = 120 and S_ab = 1260;

Construct the 16 x 16 Bimagic Squares C by applying the equations mentioned above.

The example shown above is based on Bimagic Square 1 of Attachment 17.6.41 as described in Section 17.6.7.

Attachment 15.8.1 shows 48 order 16 Bimagic Squares C, based on the applied order 8 Bimagic Square B, which could be generated with procedure MgcSqrs16a.

The same procedure generated for each order 8 Bimagic Square B shown in Attachment 17.6.41 48 order 16 Bimagic Squares C, resulting in 96 * 48 = 4608 squares.

Notes:

1. Procedure MgcSqrs16a, generating Medjig Squares A based on the equations describing the 16 x 16 Franklin Squares as described in Section 12.3, has been applied for the sake of convenience (21 parameters only).
   
   
2. With the check of S_a2 = 120 and S_ab = 1260 disabled, 53120 Magic Squares C could be generated with the same procedure for the same square B.
   
   
3. A general procedure for the generation of Medjig Squares A would require 172 parameters, which is beyond the scope of this section.
   
   
4. Each Bimagic Square B corresponds with 2⁴ * 4! / 2 = 192 transformations, which can be obtained by appropriate row and column permutations (ref. Section 6.3).
   
   Consequently each Bimagic Square C = {B, A} corresponds with 192 solutions, which can be obtained by applying the same row and column permutations on the Medjig Square A.
   
   
5. Each (essential different) Bimagic Square C corresponds with 2⁸ * 8! /2 = 5160960 transformations.

15.9.2 Pan Magic, Complete Bimagic Squares

The method of constructing Pan Magic, Complete Bimagic Squares of order 16, based on Medjig Squares as defined above, can be summarised as follows:

Read (preselected) 8 x 8 Pan Magic, Complete Bimagic Square B (29376 possibilities);

Construct 8 x 8 Pan Magic, Complete Medjig-Squares A, while ensuring that S_a2 = 120 and S_ab = 1260;

Construct the 16 x 16 Pan Magic, Complete Bimagic Squares C by applying the equations mentioned above.
(ref. Section 15.8.1)

A numerical example is shown below:

The example shown above is based on the first Bimagic Square B of a subset of unique Bimagic Squares, complete with bimagic semi-diagonals, as published by Walter Trump in 2014 (10496 ea).

Attachment 15.8.2a shows 56 order 16 Pan Magic Bimagic Squares C (8 Complete) based on the applied order 8 Bimagic Square B, which could be generated with procedure MgcSqrs16a.

The same procedure generated for 3712 of the 10496 order 8 Bimagic Squares of subject collection a total of 203776 (128 * 24 + 3584 * 56) order 16 Pan Magic Bimagic Squares C of which 28672 Complete (8 * 3584).

Attachment 15.8.2b shows 8 order 16 Pan Magic Complete Bimagic Squares C based on the applied order 8 Bimagic Square B, which could be generated with a more strict procedure MgcSqrs16b.

The same procedure generated for 3584 of the 10496 order 8 Bimagic Complete Squares of subject collection a total of 28672 (8 * 3584) order 16 Pan Magic Complete Bimagic Squares C.

Notes:

1. For all 28672 order 16 Pan Magic, Complete Bimagic Squares C found above, also the semi-diagonals are bimagic.
   
   
2. It can be noticed that the Pan Magic, Complete Medjig Square A shown in the example above is Associated as well. The resulting Square C is however Pan Magic Complete as the 8 x 8 Square B is Pan Magic Complete.

15.9.3 Associated Bimagic Squares

The method of constructing Associated Bimagic Squares of order 16, based on Medjig Squares as defined above, can be summarised as follows:

Read (preselected) 8 x 8 Associated Bimagic Square B (192 * 841 possibilities);

Construct 8 x 8 Associated Medjig-Squares A, while ensuring that S_a2 = 120 and S_ab = 1260;

Construct the 16 x 16 Associated Bimagic Squares C by applying the equations mentioned above.
(ref. Section 15.8.1)

A numerical example is shown below:

The example shown above is based on an order 8 Associated Bimagic Square B', being the Euler transformation of the order 8 Pan Magic Complete Bimagic Square B (with bimagic semi-diagonals) as used in Section 15.9.2 above.

The Associated Medjig Square A' has been obtained by a comparable transformation on the corresponding Pan Magic, Complete Medjig Square A.

Attachment 15.8.3 shows 8 order 16 Associated Bimagic Squares, based on transformation of the corresponding Pan Magic Complete Bimagic Squares as shown in Attachment 15.8.2b.

Notes:

1. The resulting order 16 Associated Bimagic Square C' could have been obtained by applying the Euler transformation directly on the order 16 Pan Magic, Complete Bimagic Square C with bimagic semi-diagonals.
   
   However the example shown above illustrates that it is possible to construct order 2n Associated Bimagic Squares based on order n Associated Bimagic Squares.
   
   
2. As the semi-diagonals of all 28672 order 16 Pan Magic, Complete Bimagic Squares C found in Section 15.9.2 above are bimagic, 28672 order 16 Associated Bimagic Squares can be constructed based on the method described above.
   
   
3. It can be noticed that the Associated Medjig Square A' shown in the example above is Pan Magic, Complete as well.
   The resulting Square C' is however Associated as the 8 x 8 Square B' is Associated.

15.9.4 Summary

The obtained results regarding the miscellaneous types of order 16 Bimagic Squares as deducted and discussed in previous sections are summarized in following table:

Next section will provide some more classical construction methods for Bimagic Squares of order 16.