Office Applications and Entertainment, Magic Squares

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15.0    Special Magic Squares, Bimagic Squares

15.5    Bimagic Squares (10 x 10)

A Magic Square is Bimagic if it remains magic after each of the numbers have been squared.

The construction of order 10 Bimagic Squares is challenging as the number of bimagic series is 24.643.236 as determined by Chistian Boyer (May 2002).

15.5.1  Historical Background

The first Bimagic Square of order 10 was constructed by Fredrik Jansson (2004), who applied a bit vector method (128 bits) on the collection of 24.643.236 series.

The applied program was written in C and executed on an Athlon 2800 computer. The square was found within 24 hours.

A few years later order 10 Bimagic Squares were published by Christian Boyer (2006) and Pan Fengchu (2007).

The three Historical Bimagic Squares of order 10 mentioned above are shown in Attachment 15,5.1.

It can be noticed that the square of Fredrik Jansson and the square of Pan Fengchu are both based on the same 10 horizontal bimagic series.

Following section will describe and illustrate how comparable square(s) can be constructed in a more conventional way.

15.5.2  Half Generator Method

By applying the half generator method, as described in detail in Section 15.3.1 for order 8 Bimagic Squares (Achille Rilly, 1901), the number of applicable bimagic series can be reduced considerable.

A sub collection of 16603 complement free bimagic series (s1 = 505, s2 = 33835) can be generated based on following integers:

    { 1,  3,  5,  6,  8, 10, 14, 15, 17, 18, 19, 22, 24, 26, 28, 30, 35, 39
     40, 42, 44, 45, 47, 49, 51, 53, 55, 58, 60, 63, 64, 65, 67, 68, 69, 70
     72, 74, 76, 78, 80, 81, 85, 88, 89, 90, 92, 94, 97, 99}

retrieved from the five bottom rows of an order 10 Almost Associated Magic Square (ref. Section 10.1.5).

The Half Generator Method as applicable for order 10 Bimagic Squares can be summarised as follows:

  • Based on the sub collection described above numerous complement free Generators with 5 Bimagic Rows can be obtained, which can be completed with their complements (blue);
  • Based on the Generators obtained above, Semi Bimagic Squares with 10 Bimagic Rows and Columns can be constructed by permutating the numbers within the rows of the Generators;
  • By permutation of the rows and columns within the Semi Bimagic Squares, Bimagic Squares might be obtained (ref. CnstrSqrs10).

Subject procedure is illustrated below for the first occurring suitable Generator and resulting Semi- and Simple Bimagic Square.

Generator
1 5 44 45 47 49 63 76 78 97
3 6 18 53 60 64 68 69 74 90
8 10 24 39 40 65 70 72 88 89
14 15 17 35 42 55 67 81 85 94
19 22 26 28 30 51 58 80 92 99
100 96 57 56 54 52 38 25 23 4
98 95 83 48 41 37 33 32 27 11
93 91 77 62 61 36 31 29 13 12
87 86 84 66 59 46 34 20 16 7
82 79 75 73 71 50 43 21 9 2
Semi Bimagic Square
1 5 44 45 47 49 63 76 78 97
3 6 18 68 69 90 60 53 74 64
40 39 24 72 88 65 89 70 8 10
67 81 35 85 94 55 14 15 42 17
99 80 26 30 22 92 58 51 19 28
56 57 96 23 38 25 100 4 54 52
41 48 95 83 33 27 37 11 32 98
61 29 31 13 12 77 36 91 93 62
66 87 86 7 20 16 46 59 84 34
71 73 50 79 82 9 2 75 21 43
Simple Bimagic Square
63 47 78 5 45 49 1 44 76 97
60 69 74 6 68 90 3 18 53 64
89 88 8 39 72 65 40 24 70 10
58 22 19 80 30 92 99 26 51 28
100 38 54 57 23 25 56 96 4 52
2 82 21 73 79 9 71 50 75 43
36 12 93 29 13 77 61 31 91 62
14 94 42 81 85 55 67 35 15 17
46 20 84 87 7 16 66 86 59 34
37 33 32 48 83 27 41 95 11 98

Each (Essential Different) Simple Bimagic Square corresponds with 1920 transformations, as described below.

  • Any line n can be interchanged with line (11 - n). The possible number of transformations is 25 = 32.
    It should be noted that for each square the 180o rotated aspect is included in this collection.

  • Any permutation can be applied to the lines 1, 2, 3, 4, 5, provided that the same permutation is applied to the lines 10, 9, 8, 7, 6. The possible number of transformations is 5! = 120.

The resulting number of transformations, excluding the 180o rotated aspects, is 32/2 * 120 = 1920.

Because of the extremely high number of possible permutations within the rows of order 10 Generators, a non-iterative procedure has been used for the construction of Semi Bimagic Squares (ref. SemiSqrs10a).

Unfortunately this results (generally) in Semi Bimagic Squares with only six thru eight Bimagic Columns.

This can however be corrected with a procedure, which recalculates the last seven columns (ref. SemiSqrs10b), and returns occasionally Semi Bimagic Squares with 10 Bimagic Columns:

Generator, 7 Bimagic Columns
1 5 44 45 47 49 63 76 78 97
3 6 18 53 60 64 68 69 74 90
40 39 24 10 88 70 65 72 8 89
67 81 35 55 17 15 85 42 14 94
99 80 26 19 22 92 58 30 51 28
56 57 96 4 100 38 25 52 23 54
41 48 95 83 33 98 32 37 27 11
61 29 31 77 36 12 93 91 13 62
66 87 86 84 20 46 7 59 16 34
71 73 50 75 82 21 9 79 43 2
Recalculated Columns
45 68 72 85 30 23 83 13 7 79
47 69 88 94 22 38 33 12 20 82
49 90 65 55 92 25 27 77 16 9
63 60 89 14 58 100 37 36 46 2
76 53 70 15 51 4 11 91 59 75
78 74 8 42 19 54 32 93 84 21
97 64 10 17 28 52 98 62 34 43
Semi Bimagic Square
1 5 44 45 47 49 63 76 78 97
3 6 18 68 69 90 60 53 74 64
40 39 24 72 88 65 89 70 8 10
67 81 35 85 94 55 14 15 42 17
99 80 26 30 22 92 58 51 19 28
56 57 96 23 38 25 100 4 54 52
41 48 95 83 33 27 37 11 32 98
61 29 31 13 12 77 36 91 93 62
66 87 86 7 20 16 46 59 84 34
71 73 50 79 82 9 2 75 21 43

The resulting Semi Bimagic Square shown above, has been used in the construction example at the start of this section.

Notes

  1. The sub collection of 16603 magic series described above, contains 2848 series starting with the number one (1).

  2. One Generator was constructed for each magic series containing the number one, which resulted all in Semi Bimagic Squares with 6, 7 or 8 Bimagic Columns.

  3. While using the method described above 111 Semi Bimagic Squares with 10 Bimagic Columns could be constructed.

  4. Only Semi Bimagic Square 12 - based on Generator 361 - resulted in a Simple Bimagic Square.

Attachment 15,5.2 shows a few related Simple Bimagic Squares which could be obtained by transformation (complementary and quarter exchange).

15.5.6  Summary

The obtained results regarding the miscellaneous types of order 10 Bimagic Squares as deducted and discussed in previous sections are summarized in following table:

Type

Characteristics

Subroutine

Results

-

-

-

-

Bimagic

Historical Squares

-

Attachment 15.5.1

Bimagic

Half Generator Based

CnstrSqrs10

Attachment 15.5.2

-

-

-

-

Next section will provide methods for the construction of Bimagic Squares of order 16.

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