Office Applications and Entertainment, Magic Squares

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15.0    Special Magic Squares, Bimagic Squares, Part 2

        Exhibit 15.2, Victor Coccoz

15.2.1  Introduction

Semi Bimagic Squares can be based on two Generators A and B for which the Magic Series have been selected from two groups (4 series/group) as illustrated in following example:

A (Bimagic Rows)
1 14 20 31 39 44 54 57
2 13 19 32 40 43 53 58
7 12 22 25 33 46 52 63
8 11 21 26 34 45 51 64
3 10 24 29 38 47 49 60
4 9 23 30 37 48 50 59
5 16 18 27 36 41 55 62
6 15 17 28 35 42 56 61
3
3
3
3
4
4
4
4
 B (Bimagic Columns)
1 4 5 8 2 3 10 11
16 13 12 9 7 6 15 14
23 22 19 18 28 25 20 17
26 27 30 31 29 32 21 24
38 39 34 35 41 44 33 36
43 42 47 46 48 45 40 37
52 49 56 53 51 50 59 58
61 64 57 60 54 55 62 63
1 1 1 1 5 5 5 5
C1 (Conjugated)
1 43 52 26 38 23 16 61
39 13 22 64 49 4 27 42
57 19 12 34 47 30 5 56
31 53 46 8 60 9 18 35
54 2 7 51 29 48 41 28
44 32 25 45 3 50 55 6
20 40 33 21 10 59 62 15
14 58 63 11 24 37 36 17
C2 (Coccoz Square)
1 26 52 43 38 16 61 23
39 64 22 13 49 27 42 4
57 34 12 19 47 5 56 30
31 8 46 53 60 18 35 9
54 51 7 2 29 41 28 48
14 11 63 58 24 36 17 37
44 45 25 32 3 55 6 50
20 21 33 40 10 62 15 59

The resulting Semi Bimagic Square C1, constructed by means of conjugation, might be converted to a ‘Coccoz Square’ C2, as described in Section 15.3.2.

However, as these ‘Coccoz Squares’ can be found by exchanging rows and columns - while checking the diagonals of the Sub Squares (2 x 2) - this is not necessarily for the construction of related Essential Different Bimagic Squares.

15.2.2  Generators

Generators for which the Magic Series have been selected from two groups, also referred to as Irregular Generators, can be constructed with routine (ref. CnstrGen01).

The resulting number of Generators are summarised in following table:

Group 2 3 4 5
1 360 360 360 360
2 - 360 360 360
3 - - 360 360
4 - - - 360

The resulting collections of Irregular Generators are further referred to as Gen i,j (e.g. Gen 3,4).

15.2.3  Semi Bimagic Squares (Alternative 3a)

Semi Bimagic Squares can be obtained by conjugation of Generators of different collections (ref. ConjSqrs1502).
The resulting number of Semi Bimagic Squares are summarised in following table:

Gen 2,3 2,4 2,5 3,4 3,5 4,5
1,2 - - - 120 120 120
1,3 - 120 120 - - 120
1,4 120 - 120 - 120 -
1,5 120 120 - 120 - -
2,3 - - - - - 120
2,4 - - - - 120 -
2,5 - - - 120 - -

The 1800 Semi Bimagic Squares found above have been put into the Standard Position and compared with the "List of SM that lead to SQ" as made available by Walter Trump.

260 Semi Bimagic Squares appeared to be Essential Different, resulting in 1050 related Essential Different Bimagic Squares, based on the productivity indicated by Walter.

15.2.4  Semi Bimagic Squares (Alternative 3b)

The Generators of Gen 1,2 thru 4,5 have been used as input for routine CnstrSqrs1502. The resulting number of Semi Magic Squares (Sm) and related Essential Different Bimagic Squares (Sq) are summarised in following table:

- Sm Sq - Sm Sq - Sm Sq - Sm Sq
Gen 1,2 170 431 Gen 2,3 212 407 Gen 3,4 291 1233 Gen 4,5 184 478
Gen 1,3 181 610 Gen 2,4 197 723 Gen 3,5 138 567 - - -
Gen 1,4 197 709 Gen 2,5 155 706 - - - - - -
Gen 1,5 150 562 - - - - - - - - -

Attachment 15.2.03 shows the 150 Semi Bimagic Squares for Gen 1,5 and the calculated related Essential Different Bimagic Squares.

The 1875 Semi Bimagic Squares of collection Gen 1,2 thru 4,5 together are however not all (essential) different from each other.

Verification:

The 1875 Semi Bimagic Squares found above have been put into the Standard Position and compared with the "List of SM that lead to SQ" as made available by Walter Trump.

1615 Semi Bimagic Squares appeared to be Essential Different, resulting in 5376 related Essential Different Bimagic Squares, based on the productivity indicated by Walter.

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