Office Applications and Entertaiment, Magic Squares

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6.4 Further Analysis, Symmetrical Main Diagonals

For a magic square with symmetrical main diagoanls, following equations should be added to the equations describing a simple magic square of order 6 (Section 6.2).

   a(1) + a(36) = a(8 ) + a(29) = a(15) + a(22) = s1/3

   a(6) + a(31) = a(11) + a(26) = a(16) + a(21) = s1/3

which results, after deduction, in following linear equations:

a(31) = s1 - a(32) - a(33) - a(34) - a(35) - a(36)
a(25) = s1 - a(26) - a(27) - a(28) - a(29) - a(30)
a(19) = s1 - a(20) - a(21) - a(22) - a(23) - a(24)
a(16) = {s1 - 3 * a(21)}/3
a(15) = {s1 - 3 * a(22)}/3
a(13) = {s1 - 3 * a(14) - 3 * a(17) - 3 * a(18) + 3 * a(21) + 3 * a(22)}/3
a(12) = {2 * s1 - 3 * a(18) - 3 * a(24) - 3 * a(30) + 3 * a(31) - 3 * a(36)}/3
a(11) = {s1 - 3 * a(26)}/3
a( 9) = {4 * s1 - 3 * a(10) - 3 * a(14) - 3 * a(17) - 3 * a(20) - 3 * a(23) - 3 * a(27) - 3 * a(28)}/3
a( 8) = {s1 - 3 * a(29)}/3
a( 7) = {s1 + 3 * a(14) + 3*a(17) + 3*a(18) - 3*a(19) - 3*a(21) - 3 * a(22) - 3 * a(25) - 3 * a(31) + 3 * a(36)}/3
a( 6) = {s1 - 3 * a(31)}/3
a( 5) = {2 * s1 - 3 * a(17) - 3 * a(23) + 3 * a(26) - 3 * a(29) - 3 * a(35)}/3
a( 4) = {2 * s1 - 3 * a(10) + 3 * a(21) - 3 * a(22) - 3 * a(28) - 3 * a(34)}/3
a( 3) = {s1 + 3 * a(10) + 3 * a(14) + 3 * a(17) - 3 * a(19) - 6 * a(21) - 3 * a(24) + 3 * a(28) - 3 * a(33)}/3
a( 2) = {2 * s1 - 3 * a(14) - 3 * a(20) - 3 * a(26) + 3 * a(29) - 3 * a(32)}/3
a( 1) = {s1 - 3 * a(36)}/3

The number of magic squares - with symmetrical diagonals - is still very huge because the squares are determined by 19 independent variables (red).

6 32 3 34 35 1
7 11 27 28 8 30
19 14 16 15 23 24
18 20 22 21 17 13
25 29 10 9 26 12
36 5 33 4 2 31

With the highlighted variables constant, an optimized guessing routine (MgcSqr6c), produced 416 magic squares within 32 minutes, which are shown in Attachment 6.6.2.

This is about half the number of squares found in section 6.2 under less restrictive conditions (ref. Attachment 6.5.1).

As described in next section, magic squares with symmetrical diagonals will allow for another Class definition.

Note:

An alternative routine - in which after the bottom row, first the main diagonals and afterward the remaining rows and columns are calculated - appeared to be much faster (MgcSqr6c2).

Subject routine generated within the same 32 minutes 38333 magic squares with symmetrical diagonals, of which the first 416 are shown in Attachment 6.6.3.

6.5 Class Definition, Symmetrical Main Diagonals

For a magic square with symmetrical main diagonals:

6 32 3 34 35 1
7 11 27 28 8 30
19 14 16 15 23 24
18 20 22 21 17 13
25 29 10 9 26 12
36 5 33 4 2 31

with:

   a(1) + a(36) = a(8 ) + a(29) = a(15) + a(22) = s1/3

   a(6) + a(31) = a(11) + a(26) = a(16) + a(21) = s1/3

we can define following operations, which will result also in a magic square with symmetrical main diagonals.

Swap row 1 and 6, Swap row 2 and 5, Swap row 3 and 4 or a combination of these operations.

Swap column 1 and 6, Swap column 2 and 5, Swap column 3 and 4 or a combination of these operations.

Any combination of above mentioned row and column operations will also result in a magic square with symmetrical main diagonals.

This can be formalised to a set of operators:

  Rij(A) with (i = 0, 1, ... 7; j = 0, 1, ... 7)

for which the definitions are summarised in following table:

Swap none 1;6 2;5 3;4 1;6 and 2;5 1;6 and 3;4 2;5 and 3;4 1;6, 2;5 and 3;4
Row i = 0 i = 1 i = 2 i = 3 i = 4 i = 5 i = 6 i = 7
Column j = 0 j = 1 j = 2 j = 3 j = 4 j = 5 j = 6 j = 7

If above defined operators are applied on the magic square A1 at the top of this page, a collection of 64 squares will result.

It should be noted that the squares which can be obtained by means of horizontal reflection (A1* Is), vertical reflection (Is* A1) and 180o rotation (Is* A1* Is) are included in this collection (ref. Attachment 6.7.1).

If above defined operators are applied on the transposed A1T of the magic square A1 at the top of this page, another collection of 64 squares will result.

It should be noted that the squares which can be obtained by means of 90o rotation (A1T* I1), 270o rotation(Is* A1T) and vertical reflection on 90o rotation (Is* A1T* Is) are included in this collection (ref. Attachment 6.7.1).

As a consequence of the fact that all magic squares, which can be obtained by means of rotation and/or reflection, were found as a result of applying the operator Rij(A) on A1 and A1T, we can conclude that, when we write A1T as A2, any magic square with symmetrical diagonals, will result in a sub collection or Class {Aijk} with:

  Aijk = Rij(Ak) for (i = 0, 1, ... 7; j = 0, 1, ... 7 and k = 1, 2)

with:

  A00k = R00(Ak) = Ak for (k = 1, 2)

Attachment 6.7.1 shows the 128 magic squares constructed based on the above, applied on the Base Square A1.


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