Magic Squares (35g1)

Office Applications and Entertainment, Magic Squares
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14.0 Special Magic Squares, Prime Numbers

14.9 Magic Squares (11 x 11)

14.9.1 Concentric Magic Squares (11 x 11)

An 11^th order Prime Number Concentric Magic Square consists of a Prime Number Concentric Magic Square of the 9^th order with a border around it.

For Prime Number Concentric Magic Squares of order 11 with Magic Sum s11, it is convenient to split the supplementary rows and columns into parts summing to s4 = 4 * s11 / 11 and s3 = 3 * s11 / 11:

This results in following border equations:

which enable the development of a fast procedure to generate Prime Number Concentric Magic Squares of order 11 (ref. Priem11a).

Miscellaneous Prime Number Concentric Magic Squares of order 11, based on 9^th order Concentric Magic Squares as discussed in Section 14.7.4, are shown in Attachment 14.9.1.

Each square shown corresponds with numerous squares for the same Magic Sum.

A method to generate order 11 Concentric Mgic Squares with order 6 Diamond Inlays will be discussed in Section 20.1.4.

14.9.2 Bordered Magic Squares (11 x 11), Miscellaneous Inlays

Based on the collections of 9^th order Composed and miscellaneous Bordered Magic Squares, as discussed in Section 14.7.1, Section 14.7.5 and 14.7.12 also following 11^th order Bordered Magic Squares can be generated with routine Priem11a:

Center Squares with Overlapping Sub Squares (ref. Attachment 14.9.10)
Center Squares with Diamond Inlays (ref. Attachment 14.9.20)
Concentric Center Squares with Diamond Inlay    (ref. Attachment 14.9.11)
Concentric Center Squares with Diamond Inlay    (ref. Attachment 14.9.12)
Bordered   Center Squares with Diamond Inlays   (ref. Attachment 14.9.13)
Bordered Center Square, Embedded Ultra Magic Square with:

- Conc. Square and Square Inlay (a)             (ref. Attachment 14.9.14)
- Conc. Square and Square Inlay (b)             (ref. Attachment 14.9.15)
- Square Inlays                                 (ref. Attachment 14.9.16)
- Square and Diamond Inlay (a)                  (ref. Attachment 14.9.17)
- Square and Diamond Inlay (b)                  (ref. Attachment 14.9.18)
- Conc. Square and Diamond Inlay                (ref. Attachment 14.9.19)
Double Bordered Center Squares with:

- Embedded Ultra Magic Square (ref. Attachment 14.9.21)
- Embedded Square with Square Inlay (ref. Attachment 14.9.23)
- Embedded Square with Square and Diamond Inlay (ref. Attachment 14.9.24)
Composed Associated Magic Center Squares (ref. Attachment 14.9.42)

Each square shown corresponds with numerous squares for the same Magic Sum.

14.9.3 Bordered Magic Squares (11 x 11), Split Border

Alternatively an 11^th order Bordered Magic Square with Magic Sum s11 can be constructed based on:

a Symmetric Magic Center Square of order 7 with Magic Sum s7 = 7 * s11 / 11;
36 pairs, each summing to 2 * s11 / 11, surrounding the (Concentric) Magic Center Square;
a split of the supplementary rows and columns into three parts:
two summing to s3 = 3 * s11 / 11 and one to s5 = 5 * s11 / 11.

as illustrated below:

Based on the principles described in previous sections, a fast procedure (Priem11b) can be developed:

to read the previously generated Concentric Magic Squares (7 x 7);
to generate the four Magic Rectangles (2 x 5);
to generate, based on the remainder of the pairs, four Magic Squares of order 3;
to transform these Magic Squares into suitable Corner Squares, thus completing the border.

Attachment 14.9.3 shows based on 7^th order Concentric Magic Squares as discussed in Section 14.5.1, one Prime Number Bordered Magic Square for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum.

14.9.4 Eccentric Magic Squares (11 x 11)

For Prime Number Eccentric Magic Squares of order 11 it is convenient to split the supplementary rows and columns into: two parts summing to s4 = 4 * s11 / 11 and one part summing to s3 = 3 * s11 / 11.

This enables, based on the same principles, the development of a set of fast procedures (ref. Priem11c):

to read the previously generated Eccentric Magic Squares of order 9;
to complete the Main Diagonal and determine the related Border Pairs;
to generate, based on the remainder of the available pairs, a suitable Corner Square of order 3;
to complete the Eccentric Magic Square of order 11 with the two remaining 2 x 4 Magic Rectangles.

Attachment 14.9.4 shows, based on 9^th order Eccentric Magic Squares as discussed in Section 14.7.6, one Prime Number Eccentric Magic Square for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum.

14.9.5 Bordered Magic Squares (11 x 11), Composed Border (1)

Order 11 Composed Border Magic Squares with Magic Sum s11 can be constructed based on:

a Border composed out of:
- 4 Semi Magic Squares of order 3 with Magic Sum s3 = 3 * s11 / 11;
- 4 Magic Rectangles order 3 x 5 with s3 = 3 * s11 / 11 and s5 = 5 * s11 / 11.
a Symmetric Magic Center Square of order 5 with Magic Sum s5 = 5 * s11 / 11;

as illustrated below:

Based on the principles described in previous sections, a fast procedure (Priem11d) can be developed:

to read the previously generated Center Symmetric Magic Squares (5 x 5);
to generate, based on the remainder of the pairs, the four Semi Magic Squares of order 3;
to complete the Composed Border of order 11 with the four 3 x 5 Magic Rectangles.

Following Attachments show, for some of the occuring Magic Sums, one Bordered Magic Square per Magic Sum:

Attachment 14.9.5a, embedded Concentric Magic Center Squares
Attachment 14.9.5b, embedded Ultra Magic Center Squares

Each square shown corresponds with numerous squares for the same Magic Sum.

Note:

As a consequence of the applied properties (ref. Priem11d):

The opposite Semi Magic Corner Squares are Anti Symmetric and Complementary;
The Magic Center Square might be Associated;

In this case the 11^th order Composed Magic Square will be associated, if the opposite Magic Rectangles (3 x 5) are Anti Symmetric and Complementary as well.

Attachment 14.9.7 shows one Associated Composed Magic Square for some of the occurring Magic Sums (ref. Priem11d3).

14.9.6 Bordered Magic Squares (11 x 11), Composed Border (2)

Alternatively order 11 Composed Border Magic Squares with Magic Sum s11 can be constructed based on:

a Border composed out of:
- 4 (Semi) Magic Squares of order 4 with Magic Sum s4 = 4 * s11 / 11;
- 4 Associated Magic Rectangles order 3 x 4 with s3 = 3 * s11 / 11 and s4 = 4 * s11 / 11.
a Magic Center Square of order 3 with Magic Sum s3 = 3 * s11 / 11;

as illustrated below:

Based on the principles described in previous sections, a fast procedure (Priem11e) can be developed:

to generate the 3 x 3 Magic Center Square;
to generate, based on the remainder of the pairs, four 4 x 4 Associated Magic Squares;
to complete the Composed Border of order 11 with four 3 x 4 Associated Magic Rectangles.

Attachment 14.9.6 shows one Prime Number Composed Border Magic Square for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum.

Note:

If the applied properties are changed to:

the opposite Semi Magic Corner Squares (4 x 4) are Anti Symmetric and Complementary;
the opposite Magic Rectangles (3 x 4) are Anti Symmetric and Complementary;
the Magic Center Square (3 x 3) is Center Symmetric (per definition);

the 11^th order Composed Magic Square will be associated

Attachment 14.9.8 shows one Associated Composed Magic Square for some of the occurring Magic Sums (ref. Priem11f).

14.9.7 Composed Magic Squares (11 x 11)
Overlapping Sub Squares (1)

This 11^th order Composed Magic Square, with overlapping sub squares, is a sub square of a well known 13^th order Composed Magic Square (Andrews, 1909) which will be developed for Prime Numbers in Next Section.

The 11^th order Magic Square K, with Magic Sum s11, contains following sub squares:

One 9^th order Eccentric Magic Square H (right top K):
- with embedded 7^th order Eccentric Magic I (left bottom H)
- with embedded 5^th order Pan Magic Square C (left bottom I)
One 3^th order Semi Magic Square M, element a(91) common with C (left bottom K);
Four 2 x 4 Magic Rectangles: A and B (left), D and E (bottom);
Another 7^th order Magic Square L with C in the right top corner (overlapping I);

as illustrated below:

Attachment 14.8.5.07 shows, for the sake of completeness, one Prime Number Composed Magic Square of order 11 for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the same Magic Sum.

14.9.8 Composed Magic Squares (11 x 11)
Overlapping Sub Squares (2)

Following 11^th order Composed Magic Square, with overlapping sub squares, is also a sub square of another higher order Composed Magic Square as discussed by William Symes Andrews (ref. Magic Squares and Cubes, Fig. 352).

The 11^th order Magic Square E, with Magic Sum s11, contains following sub squares:

One 3^th order Magic Center Square C;
Two each other overlapping 5^th order Eccentric Magic Squares A1 and A2;
Two each other overlapping 7^th order Eccentric Magic Squares B1 and B2;
Two 4^th order Pan Magic Squares PM1 and PM2;

as illustrated below:

Attachment 14.9.8.1 shows, one Prime Number Composed Magic Square of order 11 for some of the occurring Magic Sums (ref. PriemE11).

The corresponding Composed Magic Squares of order 15 contain, in addition to the sub squares mentioned above, following Corner Squares:

Two 6^th order Eccentric Magic Squares F1 and F2 with embedded PM1 and PM2;
Two 9^th order Eccentric Magic Squares D1 and D2 with embedded B1 and B2;

Attachment 14.9.8.2 shows, for the sake of completeness, the Prime Number Composed Magic Squares of order 15 for some of the occurring Magic Sums (ref. PriemG15).

Each square shown corresponds with numerous squares for the same Magic Sum.

14.9.9 Inlaid Magic Squares (11 x 11)
Pan Magic Square Inlays Order 4 and 5 (Overlapping)

The 11^th order Inlaid Magic Square shown below:

contains following inlays:

two each 5^th order Pan Magic Squares - Magic Sums s(1) and s(4) - with the center element in common,
two each 4^th order Pan Magic Squares with Magic Sums s(2) and s(3).

The relation between the Magic Sums s(1), s(2), s(3) and s(4) is:

 s(1) = 10 * s11 / 11 - s(4)
 s(2) =  8 * s11 / 11 - s(3)

With s11 the Magic Sum of the 11^th order Inlaid Magic Square.

The Associated Border can be described by following linear equations:

a(115) =   -s11/11 + a(117) - s(3) + s(4)
a(114) =   -s11/11 + a(118) - s(3) + s(4)
a(113) =   -s11/11 + a(119) - s(3) + s(4)
a(112) =   -s11/11 + a(120) - s(3) + s(4)
a(111) = 15*s11/11 - a(116) - 2 * a(117) - 2 * a(118) - 2 * a(119) - 2 * a(120) - a(121) + 4 * s(3) - 4 * s(4)
a(100) =    s11    - a(110) - s(3) - s(4)
a( 89) =    s11    - a( 99) - s(3) - s(4)
a( 78) =    s11    - a( 88) - s(3) - s(4)
a( 67) =    s11    - a( 77) - s(3) - s(4)
a( 66) = 60*s11/11 - 2*a(77) - 2*a(88) - 2*a(99) - 2*a(110) - a(116) - 2*a(117) - 2*a(118) +
                                                                     - 2*a(119) - 2*a(120) - 2*a(121) - 8*s(4)

a(56) = 2 * s11/11 - a(66)
a(55) = 2 * s11/11 - a(67)
a(45) = 2 * s11/11 - a(77)
a(44) = 2 * s11/11 - a(78)
a(34) = 2 * s11/11 - a(88)
a(33) = 2 * s11/11 - a(89)

a(23) = 2 * s11/11 - a( 99)
a(22) = 2 * s11/11 - a(100)
a(12) = 2 * s11/11 - a(110)
a(11) = 2 * s11/11 - a(111)
a(10) = 2 * s11/11 - a(112)
a( 9) = 2 * s11/11 - a(113)
a( 8) = 2 * s11/11 - a(114)

a(7) = 2 * s11/11 - a(115)
a(6) = 2 * s11/11 - a(116)
a(5) = 2 * s11/11 - a(117)
a(4) = 2 * s11/11 - a(118)
a(3) = 2 * s11/11 - a(119)
a(2) = 2 * s11/11 - a(120)
a(1) = 2 * s11/11 - a(121)

Which can be incorporated in an optimised guessing routine MgcSqr11k1.

The Magic Center Squares can be constructed by means of:

A guessing routine, based on the defining linear equations as deducted in Section 14.2.2, resulting in the two 4^th order Pan Magic Sub Squares,
A guessing routine, based on the defining linear equations as deducted in Section 14.3.1, resulting in the two each other overlapping 5^th order Pan Magic Sub Squares.

Attachment 14.9.9 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 11.

Each square shown corresponds with numerous solutions, which can be obtained by variation of the four inlays and the border.

14.9.10 Associated Magic Squares (11 x 11)
Associated Square Inlays Order 5 and 6

Associated Magic Squares of order 11 with Square Inlays of order 5 and 6 can be obtained by means of transformation of order 11 Composed Magic Squares, as illustrated in Section 14.7.13 for order 9 Magic Squares.

The Associated Square shown above is composed out of:

One 5^th order Associated (Pan) Magic Square Inlay with Magic Sum s5 = 5 * s1 / 11,
One 6^th order Associated Magic Square Inlay with Magic Sum s6 = 6 * s1 / 11 and
Two Associated Magic Rectangle Inlays order 5 x 6 with s5 = 5 * s1 / 11 and s6 = 6 * s1 / 11

Based on this definition a routine can be developed to generate the required Composed Magic Squares (ref. Prime11c).

Attachment 14.8.15 shows for miscellaneous Magic Sums the first occurring Prime Number Composed Magic Square.

Attachment 14.8.16 shows the corresponding Associated Magic Squares with order 5 and 6 Square Inlays.

It should be noted that the reversed transformation is not necessarily possible because of the bottom-left / top-right Main Diagonal.

14.9.11 Associated Magic Squares (11 x 11)
Associated Center Square (5 x 5)

Associated Magic Squares of order 11 with Associated Center Square (5 x 5) can be obtained by means of transformation of order 11 Composed Magic Squares, as illustrated in Section 14.7.14 for order 9 Magic Squares.

Attachment 14.8.90 shows the Associated Magic Squares with order 5 Associated Center Squares, corresponding with the Composed Magic Squares as shown in Attachment 14.8.15.

It should be noted that the reversed transformation is not necessarily possible because of the bottom-left / top-right Main Diagonal.

14.9.12 Associated Magic Squares
Diamond Inlays Order 5 and 6

The 11^th order Associated Inlaid Magic Square shown below:

contains following Diamond Inlays:

one each 5^th order Associated Diamond Inlay with Magic Sum s5 = 5 * s1 / 11,
one each 6^th order Associated Diamond Inlay with Magic Sum s6 = 6 * s1 / 11.

The method to generate order 11 Associated Mgic Squares with order 5 and 6 Diamond Inlays will be discussed in Section 20.2.4.

14.9.13 Summary

The obtained results regarding the miscellaneous types of order 11 Prime Number Magic Squares as deducted and discussed in previous sections are summarized in following table:

Comparable routines as listed above, can be used to generate Prime Number Magic Squares of order 12, which will be described in following sections.

Index

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Type	Characteristics	Subroutine	Results
Concentric	-	Priem11a	Attachment 14.9.1
Bordered	Miscellaneous Inlays	-	Ref. Sect. 14.9.2
	Split Border Lines	Priem11b	Attachment 14.9.3
	Composed Border (1)	Priem11d	Attachment 14.9.5a Attachment 14.9.5b
	Composed Border (2)	Priem11e	Attachment 14.9.6
Eccentric	Split Border Lines	Priem11c	Attachment 14.9.4
Associated	Composed Border (1)	Priem11d3	Attachment 14.9.7
	Composed Border (2)	Priem11f	Attachment 14.9.8
	Associated Square Inlays Order 5 and 6	-	Attachment 14.8.16
	Associated Center Square Order 5	-	Attachment 14.8.90
Composed	Overlapping Sub Squares (1)	-	Attachment 14.8.5.7
	Overlapping Sub Squares (2), Order 11	PriemE11	Attachment 14.9.8.1
	Overlapping Sub Squares (2), Order 15	PriemG15	Attachment 14.9.8.2
	Associated Corner Squares and - Rectangles	Prime11c	Attachment 14.8.15
Inlaid	Pan Magic Square Inlays order 4 and 5	MgcSqr11k1	Attachment 14.9.9
-	-	-	-