Office Applications and Entertainment, Magic Squares

15.0    Special Magic Squares, Bimagic Squares, Part 2

15.3    Bimagic Squares (8 x 8)

In his book Magische Quadrate (Version 19, 2017), Holger Danielsson provides a detailed description of classical construction methods of Bimagic Squares of order 8.

Following sections are adopted from a few of these descriptions and have been generalised were possible.

The results of these efforts have been compared with the collection Essential Different Bimagic Squares of order eight (136244), as published by Walter Trump and Francis Gaspalou (April 2014).

15.3.1  Achille Rilly
        (Half Generator Principle)

Rilly’s method (1901) is based on the application of Generators of 8 Bimagic Rows, built up out of two Half Generators of 4 Bimagic Rows.

The method was reconsidered by Francis Gaspalou in 2013, who showed that more squares then the original published ones could be found (refer summary below).

The Half Generators are based on Bimagic Series for which the four even numbers sum to 132 (2704 ea).

The upper Half Generators for the first 4 rows are based on:

• 16 even numbers selected from the ranges {2 ... 16} and {50 ... 64}
• 16 odd numbers selected from the range {17 ... 47}

The lower Half Generators for the last 4 rows are based on:

• 16 even numbers from the range {18 ... 48}
• 16 odd numbers from the ranges {1 ... 15} and {49 ... 63}

This reduces the applicable number of Bimagic Series to 136, which are shown in Attachment 15.3.01 (68 for the upper Half Generators and another 68 for the lower Half Generators).

Based on subject 136 Bimagic Series, 50 upper - and 50 lower Half Generators can be constructed (Ref. Attachment 15.3.02).

A portion of the resulting 2500 Generators can be transferred into Semi Bimagic Squares by permutating the numbers within the Bimagic Rows.

A portion of the resulting Semi Bimagic Squares can be transferred into Bimagic Squares by permutating rows and columns within the Semi Bimagic Squares.

With routine CnstrSqrs1501, 6400 Semi Bimagic Squares could be generated, which have been put into the Standard Position and compared with the "List of SM that lead to SQ" as made available by Walter Trump.

This resulted in 706 Semi Bimagic Squares able to produce Bimagic Squares, which are shown in Attachment 15.3.03, and 3312 related Essential Different Bimagic Squares.

Verification:

The 2500 Generators were used as input for routine CnstrSqrs1502, which returned the same 706 Semi Bimagic Squares and calculated the 3312 related Essential Different Bimagic Squares (feasible set of diagonals).

Alternative:

Francis Gaspalou defined the applicable integers for upper and lower Half Generators as follows:

The upper Half Generators are based on the ranges:

    {1 ... 8}, {25 ... 32}, {41 ... 48} and {49 ... 56}

The lower Half Generators are based on the ranges:

    {9 ... 16}, {17 ... 24}, {33 ... 40} and {57 ... 64}

This reduces the number of applicable Bimagic Series to 84, which are shown in Attachment 15.3.04 (42 for the upper Half Generators and another 42 for the lower Half Generators).

Based on subject 84 Bimagic Series, 28 upper - and 28 lower Half Generators can be constructed (ref. Attachment 15.3.05).

The 784 Generators were used as input for routine CnstrSqrs1502, which returned 717 Semi Bimagic Squares, which are shown in Attachment 15.3.06, and calculated 2851 related Essential Different Bimagic Squares.

Verification:

The 717 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.

The 717 Semi Bimagic Squares appeared to be Essential Different and the calculated number of related Essental Different Bimagic Squares matched the productivity indicated by Walter.

Summary:

The results of the Rilly method as applied by Achille Rilly, Francis Gaspalou and in the routines described above can be summarised as follows:

The differences are mainly due to the number of Bimagic Series which are considered for the columns while constructing the Semi Bimagic Squares and the diagonals while constructing the Bimagic Squares.

15.3.2  Victor Coccoz
        (Generator Principle)

The Coccoz method (1892) is based on the application of Generator Sets, each containing a Generator of 8 Bimagic Rows and a Generator of 8 Bimagic Columns.

The Generators are based on Bimagic Series, based on ordered integer pairs as shown in Attachment 15.3.21 (An example of a possible Bimagic Series is highlighted in grey).

Each of the 5 groups shown in Attachment 15.3.21 result in 48 Bimagic Series, which are shown in Attachment 15.3.22.

Each set of 48 Bimagic Series results in 576 Generators with Bimagic Rows (ref. CnstrGen01). The Generators based on Group 1 are shown in Attachment 15.3.23.

Generators with Bimagic Columns can be obtained by means of transposition.

The Coccoz method of construction can be summarised as follows:

• Select the first Generator with Bimagic Rows from a group;
• Select the second Generator with Bimagic Columns from another group, which can be 'conjugated' with the first Generator into a Semi Bimagic Square;
• The Semi Bimagic Square should be composed out of 16 Sub Squares (2 x 2) summing to 130 with the diagonals summing to p and (130 - p) with p = 33, 49, 57, 63, 64 or 65;
• Permutate the rows and columns within the Semi Bimagic Square in order to obtain a Bimagic Square (if possible).

The total number of Essential Different Bimag Squares is said to be 2188, based on the limited number of Bimagic Series mentioned above.

The method was reconsidered by Francis Gaspalou in 2013, who showed that 10317 Essential Different Bimagic Squares of the Coccoz type can be constructed, based on all possible Bimagic Series (38039 ea).

Alternative 1:

The 576 Generators of Group 1 have been used as input for routine CnstrSqrs1502, which returned 493 Semi Bimagic Squares, which are shown in Attachment 15.3.24, and calculated the 2128 related Essential Different Bimagic Squares.

For Group 2 thru 5 respectively 546, 590, 585 and 606 Semi Bimagic Squares could be found, resulting in respectively 2710, 2948, 3060 and 2517 related Essential Different Bimagic Squares.

The 2820 Semi Bimagic Squares of Group 1 thru 5 together are however not all (essential) different from each other.

Verification 1:

The 2820 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.

2601 Semi Bimagic Squares appeared to be Essential Different, resulting in 10806 related Essential Different Bimagic Squares based on the productivity indicated by Walter.

Alternative 2:

By conjugation of the Generators of different groups (ref. ConjSqrs1502), 360 Semi Bimagic Squares can be obtained (ref. Attachment 15.3.25), which might be converted to ‘Coccoz Squares’ as described above.

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.

The 360 Semi Bimagic Squares have been put into the Standard Position and compared with the "List of SM that lead to SQ" as made available by Walter Trump.

This resulted in 219 Semi Bimagic Squares able to produce Bimagic Squares, which are shown in Attachment 15.3.26, and 2557 related Essential Different Bimagic Squares.

Verification 2:

The 360 Semi Bimagic Squares were used as input for routine CnstrSqrs1503, which returned the same 219 Semi Bimagic Squares and calculated the same amount of related Essential Different Bimagic Squares.

Alternative 3:

Alternatively Semi Bimagic Squares can be based on Generators for which the Magic Series have been selected from two groups (4 series/group). This method is illustrated and discussed in Exhibit 15.2.

Summary:

The Semi Bimagic Squares (Sm) resulting in Essential Different Bimagic Squares (Sq) based on the variations of the Coccoz method(s) as applied under Alternative 1, 2 and 3 above can be summarised as follows:

All Sm and related Sq of Alternative 2  are included in Alternative 1.
All Sm and related Sq of Alternative 3a are included in Alternative 3b.

The 3497 different Semi Bimagic Squares (Sm) of Alternative 1 and Alternative 3b together result in 13265 Essential Different Bimagic Squares (Sq).

15.3.3  Victor Coccoz
        (Based on Sudoku Comparable Squares)

This method is based on the application of two Sudoku Comparable Squares (B1/B2) as shown below:

The resulting Square M1 = 8 * B1 + B2 + 1 will be Associated Magic if following conditions are fulfilled:

      A + a = B + b = C + c = D + d = P + p = Q + q = R + r = S + s = 7

      A + D + b + c = a + d + B + C = 14

      P + S + q + r = p + s + Q + R = 14

For the numbers {0, 1, ... , 7} this can only be realised with the sub sets {0, 3, 5, 6} and {1, 2, 4, 7} which ensures also that the Associated Square is Bimagic as:

      s1 = 0  + 3  + 5  + 6  = 1  + 2  + 4  + 7
      s2 = 0² + 3² + 5² + 6² = 1² + 2² + 4² + 7²

A self explanatory numerical example is shown below:

Based on these characteristics 48 Magic Lines can be constructed for both B1 and B2 (ref. Attachment 15.3.21).

With routine CnstrSqrs02 320 suitable sets of Sudoku Comparable Squares can be generated, resulting in 320 (80 unique) Bimagic Squares M1 (ref. Attachment 15.3.22).

The resulting squares are Bimagic, Associated with Trimagic Main Diagonals.

Verification:
With a Query (ReadDb8e), checking all 192 possible transformations of each available Essential Different Bimagic Square, a collection of 3840 Unique Associated Bimagic Squares based on Sudoku Comparable Squares could be found.

For each of the 80 Unique Bimagic Squares M1 an aspect could be found in this collection.

15.3.4  Gaston Tarry
        (Based on Diagonal Euler Squares)

This method (1903) is based on the application of two Diagonal Euler Squares (B1/B2) as shown below:

The resulting square M1a = B1 + 8 * (B2 - 1) will be Pandiagonal and Bimagic if following condition is fulfilled (Bouteloup):

      r * (a - b) = c * (p - q)

which is illustrated in following numerical example:

      a = 2, b = 3, c = 2, d = 4
      p = 3, q = 5, r = 4, s = 1

The resulting square M1a is Bimagic, Pandiagonal, Complete and 4 x 4 Compact with Trimagic Main Diagonals and Bimagic Semi Diagonals.

Based on the condition mentioned above 48 Magic Lines can be constructed for both B1 and B2 (ref. Attachment 15.3.41).

With routine CnstrSqrs04 320 suitable sets of Diagonal Euler Squares can be generated, resulting in 320 (80 unique) Bimagic Squares M1a, which are shown in Attachment 15.3.42, Page 1.

Another set of 320 (80 unique) Bimagic Squares M1b = B2 + 8 * (B1 - 1) is shown in Attachment 15.3.42, Page 2.

Alternatives:

Gaston Tarry developed 10 more mathematical models for Diagonal Euler Squares suitable for the construction of Pandiagonal Bimagic Squares, which are illustrated in Exhibit 15.3.

Each model results in 320 (80 unique) Complete Bimagic Squares, which are however not all different from each other.

All solutions together, including those of Attachment 15.3.42, result in 320 Unique Complete Bimagic Squares.

Verification:
For each of the 320 Unique Bimagic Squares mentioned above, an aspect could be found in the collection of “Unique Bimagic Squares, Complete with bimagic semi-diagonals” (10496 ea) as made available by Walter Trump.

15.3.6  André Gérardin
        (Based on Sudoku Comparable Squares)

As illustrated in Section 15.1.1, ten thousands of (Pan) Magic Squares can be found based on Sudoku Comparable Squares of which only a small portion Bimagic.

With following numerical example André Gérardin (1925) illustrated that Bimagic Squares can be constructed based on more strict defined Sudoku Comparable Squares:

For B1 can be noticed that:

• the numbers of each half row sum to 14
• the first  and third  number of each half row sum to 7
• the second and fourth number of each half row sum to 7

For B2 can be noticed that:

• the first and second pair of each row have the same pair sum
• the third and fourth pair of each row have the same pair sum

Based on these characteristics 96 Magic Lines can be constructed for both B1 and B2 (ref. Attachment 15.3.61).

With routine CnstrSqrs06 320 suitable sets of Sudoku Comparable Squares can be generated, resulting in 320 (80 unique) Bimagic Squares M1 (ref. Attachment 15.3.62).

The resulting squares are Bimagic, Pandiagonal, Complete and 4 x 4 Compact with Trimagic Main Diagonals and Bimagic Semi Diagonals.

In addition to this the numbers of each of the eight 2 x 4 rectangles return both the Magic - and the Bimagic Sum.

Verification:
For each of the 80 Unique Bimagic Squares M1 an aspect could be found in the collection of “Unique Bimagic Squares, Complete with bimagic semi-diagonals” (10496 ea) as made available by Walter Trump.

15.3.8  John Hendricks
        (Based on Octanary Squares)

This method is based on the application of two Octanary Squares (B1/B2) as shown below:

The resulting Panmagic Square M1 = 8 * B1 + B2 + 1 will be composed of four 4 x 4 corner squares if following condition is fulfilled:

      a + b + c + d = A + B + C + D

The numbers {0, 1, ... , 7} can be divided in two sub sets {0, 3, 5, 6} and {1, 2, 4, 7} with the same sums s1 and s2:

      s1 = 0  + 3  + 5  + 6  = 1  + 2  + 4  + 7
      s2 = 0² + 3² + 5² + 6² = 1² + 2² + 4² + 7²

Under these conditions the Panmagic Square M1 will be Semi Bimagic if following additional conditions are fulfilled:

      A + a = B + b = C + c = D + d = 7

The main diagonals of the square M1 can be made bimagic by exchanging either

• (row 2 and 6) and (row 3 and 7) or
• (column 2 and 6) and (column 3 and 7)

The resulting squares M2a and M2b will no longer be composed but Bimagic, Pandiagonal, Complete and 4 x 4 Compact with Trimagic Main Diagonals and Bimagic Semi Diagonals.

A self explanatory numerical example is shown below:

Based on the characteristics described above 48 suitable sets of Octanary Squares can be constructed, resulting in 96 Bimagic Squares (48 unique) with the properties described above (ref. Attachment 15.3.08).

Verification:
For each of the 48 Unique Bimagic Squares an aspect could be found in the collection of “Unique Bimagic Squares, Complete with bimagic semi-diagonals” (10496 ea) as made available by Walter Trump.

15.3.10 Aale de Winkel
        (Based on Sudoku Comparable Squares)

This method is based on the generation of Sudoku Comparable Squares by means of digital equations.

The row and column numbers (z, s) of the square are numbered from 0 to 7 and used as coordinates, which can be converted to binary numbers by means of following equations:

      s = 4 * s(1) + 2 * s(2) + s(3)
      z = 4 * z(1) + 2 * z(2) + z(3)

with s(i) and z(i) equal to 0 or 1 for i = 1 to 3 e.g. the coordinates (3, 5) can be converted to (011, 101).

The binary components d(1), d(2), d(3) of B(z,s) are determined by the formula's:

      d(3) = (a(1) * s(1) + a(2) * s(2) + a(3) * s(3) + a(4) * z(1) + a(5) * z(2) + a(6) * z(3)) Mod 2
      d(2) = (b(1) * s(1) + b(2) * s(2) + b(3) * s(3) + b(4) * z(1) + b(5) * z(2) + b(6) * z(3)) Mod 2
      d(1) = (c(1) * s(1) + c(2) * s(2) + c(3) * s(3) + c(4) * z(1) + c(5) * z(2) + c(6) * z(3)) Mod 2

with a(i), b(i) and c(i) equal to 0 or 1 for i = 1 ... 6. The resulting decimal value is:

      B(z,s) = 4 * d(3) + 2 * d(2) + d(1)

With routine SudSqrs8, 8064 Sudoku Comparable Squares could be generated based on this principle.

Bimagic Squares can be constructed based on two suitable selected Sudoku Comparable Squares (B1, B2) from this collection. Row and column permutations within square B2 are necessary as well as (occasionally) transposition.

A self explanatory numerical example (no transposition required) is shown below:

With routine CnstrSqrs10 and B2 = Constant 1152 Bimagic Squares (576 unique) could be generated which are shown in Attachment 15.3.10.

Verification:
The collection of "Essential Different Bimagic Squares Made of two diagonal Latin Squares" (472 ea) corresponds with a collection of 90624 Unique Bimagic Squares Made of two diagonal Latin squares .

For each of the 576 Unique Bimagic Squares M1 an aspect could be found in this collection.

15.3.11 Aale de Winkel, Pandiagonal
        (Based on Binary Squares)

This method is based on the construction of Pan Diagonal Bimagic Squares based on six Binary Pan Magic Squares A, B, C, D, E and F as shown below:

Every row, column and (pan) diagonal of these squares contains four numbers zero and four numbers one.

Pan Diagonal Bimagic Squares M1 can be found based on the formula:

      M1 = m(1) * A + m(2) * B + m(3) * C + m(4) * D + m(5) * E + m(6) * F + 1

with m(i) = {1, 2, 4, 8, 16, 32} for i = 1 ... 6 and m(i) ≠ m(j) for i,j = 1 ... 6.

Based on the six Binaries shown above 112 (unique) Pan Diagonal, Complete Bimagic Squares could be found with routine CnstrSqrs11a, which are shown in Attachment 15.3.11.

Alternatives:
It can be noticed that the 112 Unique Complete Bimagic Squares shown in Attachment 15.3.11 have 12 squares in common with (aspects of) the 48 Unique Complete Bimagic Squares of the Hendricks Collection (ref. Attachment 15.3.8).

Based on this observation, decompositions into Binary Squares have been made for careful selected Complete Bimagic Squares of the collections listed in the table below.

A generalised routine CnstrSqrs11b produced - based on these decompositions - collections of Complete Bimagic Squares as summarized in following table:

Verification:
For each of the Unique Bimagic Squares included in the above listed Results, an aspect could be found in the collection of “Unique Bimagic Squares, Complete with bimagic semi-diagonals” (10496 ea) as made available by Walter Trump.

15.3.12 Gil Lamb
        (Based on Octanary Squares)

This method is based on the application of two Octanary Squares (B1/B2), both based on the 8 numbers of the same Magic Rectangle (r1), as illustrated in following example:

For B1 can be noticed that:

• The 4 x 4 left top square rows are based on the upper row of r1 as highlighted
• The 4 x 4 right top square is identical to the left top square
• Both 4 x 4 bottom squares are the complement of the corresponding top squares

For B2 can be noticed that:

• The 4 x 4 left top square columns are based on the upper and lower row of r1 as highlighted
• The 4 x 4 left bottom square is identical to the left top square
• Both 4 x 4 right squares are the complement of the corresponding left squares

The resulting square M1 is Bimagic, Pandiagonal, Complete and 4 x 4 Compact with Trimagic Main Diagonals and Bimagic Semi Diagonals.

Based on the characteristics described above 48 suitable sets of Octanary Squares could be generated, resulting in 48 Bimagic Squares (24 unique) with the properties described above (ref. Attachment 15.3.12).

Verification:
For each of the 24 Unique Bimagic Squares M1 an aspect could be found in the collection of “Unique Bimagic Squares, Complete with bimagic semi-diagonals” (10496 ea) as made available by Walter Trump.

15.3.13 Summary

The obtained results regarding the miscellaneous types of order 8 Bimagic Squares as deducted and discussed in previous sections are summarized in following table:

Next section will provide an alternative construction method for Bimagic Squares of order 8.