Office Applications and Entertainment, Magic Squares

15.0   Special Magic Squares, Bimagic Squares, Part 3

15.5   Associated Bimagic Squares (9 x 9)

Walter Trump and Holger Danielsson have recently executed the full enumeration of Associated Bimagic Squares.

Based on the 949738 Bimagic Series, as computed by Christian Boyer (May 2002), they found 1.307.729.280 Associated Bimagic Squares (6.811.090 essential different).

Following sections are adopted from a short description by Holger Danielsson about the construction of Associated Bimagic Squares (9 x 9) based on Latin Diagonal Squares (ref. Magische Quadrate, Version 2.03 d.d. 04-10-2020).

15.5.1 Introduction

Associated Bimagic Squares can be constructed with a Generator based procedure, which can be summarised as follows:

1. Generate suitable Bimagic Series
2. Construct - based on 1 - Generators with 9 Bimagic Rows and 1 Bimagic Column
                            Generators for  4 Bimagic Columns
3. Construct - based on 2 - Semi Bimagic Squares with 9 Bimagic Rows and 5 Bimagic Columns
4. Construct - based on 3 - Semi Bimagic Squares with 9 Bimagic Rows and 9 Bimagic Columns
5. Construct - based on 4 - Associated Bimagic Squares

This method will produce considerable more Associated Bimagic Squares as the mothods discussed in Section 15.2.4, Section 15.4.2 and Section 15.4.7.

15.5.2 Bimagic Series

Bimagic series suitable for the construction of Associated Bimagic Squares based on Latin Squares can be written as:

      {r_i} = 9 * {a_i} + {b_i} + 1 for i = 1 ... 9

with  a_i = 0 ... 8 and a_i ≠ a_j   for i ≠ j
      b_i = 0 ... 8 and b_i ≠ b_j   for i ≠ j

and will be further referred to as 'Euler Series'.

The construction of Associated Bimagic Squares can be based on:

  16 ea      Symmetric Euler Series (ref. Attachment 15.5.1) and
4508 ea Anti Symmetric Euler Series (ref. Attachment 15.5.2)

which limits the required processor time for the construction procedure(s) considerable.

15.5.3 Generators

Based on the 16 Symmetric Euler Series, 84 'Base Generators' can be constructed (ref. Attachment 15.5.3).

The Base Generators can be completed with 8 Anti Symmetric Euler Series as shown in following example:

Base Gen 41 2 16 24 36 46 58 66 80 4 18 20 30 52 62 64 78

Gen A 41 2 16 24 36 46 58 66 80 4 11 26 32 43 48 63 69 73 18 3 22 28 42 53 61 65 77 20 7 14 35 37 51 57 72 76 30 8 15 23 38 49 55 70 81 52 74 67 59 44 33 27 12 1 62 75 68 47 45 31 25 10 6 64 79 60 54 40 29 21 17 5 78 71 56 50 39 34 19 13 9

Based on the Base Generators and the Anti Symmetric Euler Series, 49557 Generators with 9 Bimagic Rows and 1 Bimagic Column can be constructed (Generator Set A).

Based on the Base Generators and the Anti Symmetric Euler Series, 49557 Generators for 4 Bimagic Columns can be constructed (Generator Set B).

15.5.4 Semi Bimagic Squares

Suitable sets of generators (A, B) can be found, enabling the construction of Semi Bimagic Squares by means of the exchange of integers within the rows of Gen A, as illustrated in following example:

Gen A 41 2 16 24 36 46 58 66 80 4 11 26 32 43 48 63 69 73 18 3 22 28 42 53 61 65 77 20 7 14 35 37 51 57 72 76 30 8 15 23 38 49 55 70 81 52 74 67 59 44 33 27 12 1 62 75 68 47 45 31 25 10 6 64 79 60 54 40 29 21 17 5 78 71 56 50 39 34 19 13 9

Gen B 2 14 27 31 39 53 60 70 73 16 6 19 35 40 48 59 65 81 24 8 11 28 45 50 57 67 79 36 7 10 21 44 49 56 69 77

Semi Bi (9 rws, 5 clmns) 41 2 16 24 36 46 58 66 80 4 73 48 11 69 26 63 43 32 18 53 65 28 77 3 61 22 42 20 14 35 57 7 51 37 72 76 30 70 81 8 49 38 55 23 15 52 27 59 67 44 33 74 12 1 62 31 6 45 10 75 25 47 68 64 60 40 79 21 29 54 17 5 78 39 19 50 56 34 71 13 9

Based on the method illustrated above 2232 Semi Bimagic Squares can be constructed (ref. Attachment 15.5.4).

15.5.5 Associated Bimagic Squares

The Semi Bimagic Squares constructed in Section 15.5.4 above can be transformed by means of

- row and column permutations and
- exchange of integers within the rows

into (partly) Crosswise Symmetric Semi Bimagic Squares - with 9 bimagic rows and 9 bimagic columns - as illustrated below:

Semi Bi (9 rws, 5 clmns) 41 2 16 24 36 46 58 66 80 4 73 48 11 69 26 63 43 32 18 53 65 28 77 3 61 22 42 20 14 35 57 7 51 37 72 76 30 70 81 8 49 38 55 23 15 52 27 59 67 44 33 74 12 1 62 31 6 45 10 75 25 47 68 64 60 40 79 21 29 54 17 5 78 39 19 50 56 34 71 13 9	Semi Bi (9 rws, 5 clmns) 41 2 80 16 66 24 58 36 46 4 73 32 48 43 11 63 69 26 78 39 9 19 13 50 71 56 34 18 53 42 65 22 28 61 77 3 64 60 5 40 17 79 54 21 29 20 14 76 35 72 57 37 7 51 62 31 68 6 47 45 25 10 75 30 70 15 81 23 8 55 49 38 52 27 1 59 12 67 74 44 33
Semi Bi (9 rws, 5 clmns) 41 2 80 16 66 24 58 36 46 4 73 32 48 43 11 63 69 26 78 39 9 19 13 50 71 56 34 18 53 42 65 22 28 61 77 3 64 60 5 40 17 79 54 21 29 20 14 76 35 72 57 37 7 51 62 31 68 6 47 45 25 10 75 30 70 15 81 23 8 55 49 38 52 27 1 59 12 67 74 44 33	Semi Bi (9 rws, 9 clmns) 41 2 80 16 66 24 58 36 46 4 73 43 48 63 11 32 69 26 78 39 9 19 34 50 71 56 13 18 53 22 65 42 28 3 77 61 64 60 29 40 17 79 54 21 5 20 14 51 35 76 57 37 7 72 62 31 68 6 47 45 25 10 75 30 70 55 81 23 8 15 49 38 52 27 12 59 1 67 74 44 33

A portion of the resulting Semi Bimagic Squares can be transformed into one or more Associated Bimagic Square(s) by means of row and column permutations:

Semi Bi (9 rws, 9 clmns) 41 2 80 16 66 24 58 36 46 4 73 43 48 63 11 32 69 26 78 39 9 19 34 50 71 56 13 18 53 22 65 42 28 3 77 61 64 60 29 40 17 79 54 21 5 20 14 51 35 76 57 37 7 72 62 31 68 6 47 45 25 10 75 30 70 55 81 23 8 15 49 38 52 27 12 59 1 67 74 44 33

Bimagic, Associated 69 32 48 43 4 73 63 11 26 77 3 65 22 18 53 42 28 61 7 37 35 51 20 14 76 57 72 49 15 81 55 30 70 23 8 38 36 58 16 80 41 2 66 24 46 44 74 59 12 52 27 1 67 33 10 25 6 68 62 31 47 45 75 21 54 40 29 64 60 17 79 5 56 71 19 9 78 39 34 50 13

Based on the 2232 Semi Bimagic Squares 1908 essential different Associated Bimagic Squares can be constructed (ref. Attachment 15.5.5).

15.5.6 Sub Collections

The collection of 1908 essential different Associated Bimagic Squares - as shown in Attachment 15.5.5 - contains following sub collections:

• a collection of 107 squares for which the integers of the nine regular sub squares sum to s1 = 369 (ref. Attachment 15.5.6),
• which contains a sub collection of 40 squares for which the squared integers of the nine regular sub squares sum to s2 = 20049 (ref. Attachment 15.5.7).

For the total collection of 366336 (= 192 * 1908) Associated Bimagic Squares the sub collections defined above contain respectively 10224 and 3456 squares.

15.5.7 Summary

The obtained results regarding the miscellaneous types of order 9 Associated Bimagic Squares as deducted and discussed in previous sections are summarized in following table:

Next section will provide both classical and modern construction methods for Bimagic Squares of order 6, based on non consecutive integers.