Office Applications and Entertainment, Magic Squares

14.0    Special Magic Squares, Prime Numbers

14.22   Magic Squares, Higher Order, Inlaid (2)

14.22.1 Introduction

Following sections will describe how Prime Number Inlaid Magic Squares with Concentric Main and/or Sub Squares can be generated with comparable routines as described in previous sections.

The advantage of applying Concentric Magic Squares either as Main Square or as Inlays is the relative high generation speed compared with other Magic Squares.

14.22.2 Magic Squares, Composed (15 x 15)

Examples of 15 x 15 Prime Number Inlaid Magic Squares with Concentric Sub Squares have been provided in Attachment 14.6.53.

The 15^th order Inlaid Magic Square shown below (s15 = 110535), is composed out of twenty five each 3^th order Simple Magic Squares with different Magic Sums s(1) ... s(25).

The Magic Sums s(1) thru s(25) of the Sub Squares comply with the equations defining a Concentric Magic Square of order 5 (ref. Section 14.3.2).

The Prime Number Inlaid Magic Square shown above can be constructed by selecting:

• An order 5 Concentric Magic Main Square containing possible Center Elements for the Sub Squares
• Twenty five order 3 Simple Magic Squares based on the corresponding Magic Sums and the equations deducted in Section 14.1.1

Attachment 14.6.58 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 15.

Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or rotation/reflection of the twenty five inlays.

14.22.3 Magic Squares, Composed (18 x 18)

The 18^th order Inlaid Magic Square shown below (s18 = 126234), is composed out of thirty six each 3^th order Simple Magic Squares with different Magic Sums s(1) ... s(36).

The Magic Sums s(1) thru s(36) of the Simple Magic Sub Squares comply with the equations defining a Concentric Magic Square of order 6 (ref. Section 14.4.3).

The Prime Number Inlaid Magic Squares described above can be constructed by selecting:

• An order 6 Concentric Magic Main Square containing possible Center Elements for the Sub Squares
• Thirty six order 3 Simple Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.1.1

Attachment 14.6.59 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 18, based on Concentric Magic Main Squares.

Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or rotation/reflection of the thirty six inlays.

Alternatively Order 18 Prime Number Inlaid Magic Squares can be constructed by selecting:

• An order 3 Simple Magic Main Square containing possible Order 6 Magic Sums
• Nine order 6 Concentric Magic Sub Squares based on these Magic Sums and the equations deducted in Section 14.4.4

Attachment 14.6.60 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 18.

Each square shown corresponds with numerous solutions, which can be obtained by rotation/reflection of the main square or selecting other aspects of the nine inlays.

14.22.4 Magic Squares, Composed (20 x 20)

Examples of 20^th order Inlaid Magic Squares, composed out of twenty five each 4^th order Pan Magic Squares with different Magic Sums s(1) ... s(25), are shown in Attachment 14.6.83.

The Magic Sums s(1) thru s(25) of the Pan Magic Sub Squares comply with the equations defining a Concentric Magic Square of order 5 (ref. Section 14.3.2).

The Prime Number Inlaid Magic Squares described above can be constructed by selecting:

• An order 5 Concentric Magic Main Square containing possible Order 4 Magic Sums
• Twenty five order 4 Pan Magic Sub Squares based on these Magic Sums and the equations deducted in Section 14.2.2

Attachment 14.6.83 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 20, based on Concentric Magic Main Squares.

Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the twenty five inlays.

Alternatively Order 20 Prime Number Inlaid Magic Squares can be constructed by selecting:

• An order 4 Pan Magic Main Square containing possible Center Elements for the Sub Squares
• Sixteen order 5 Concentric Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.3.2

Attachment 14.6.84 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 20.

Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the sixteen inlays.

14.22.5 Magic Squares, Composed (21 x 21)

Examples of 21^th order Inlaid Magic Squares, composed out of forty nine each 3^th order Simple Magic Squares with different Magic Sums s(1) ... s(49), are shown in Attachment 14.6.85.

The Magic Sums s(1) thru s(49) of the Simple Magic Sub Squares comply with the equations defining a Concentric Magic Square of order 7 (ref. Section 14.5.1).

The Prime Number Inlaid Magic Squares described above can be constructed by selecting:

• An order 7 Concentric Magic Main Square containing possible Center Elements for the Sub Squares
• Forty nine order 3 Simple Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.1.1

Attachment 14.6.85 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 21, based on Concentric Magic Main Squares.

Each square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or rotation/reflection of the forty nine inlays.

Alternatively Order 21 Prime Number Inlaid Magic Squares can be constructed by selecting:

• An order 3 Simple Magic Main Square containing possible Center Elements for the Sub Squares
• Nine order 7 Concentric Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.5.1

Attachment 14.6.86 shows for a few Magic Sums the first occurring Prime Number Inlaid Magic Square of order 21.

Each square shown corresponds with numerous solutions, which can be obtained by rotation/reflection of the main square or selecting other aspects of the nine inlays.

14.22.6 Magic Squares, Composed (24 x 24)

Attachment 14.6.87 shows an example of a 24^th order Inlaid Magic Square, composed out of thirty six each 4^th order Pan Magic Squares with different Magic Sums s(1) ... s(36).

The Magic Sums s(1) thru s(36) of the Pan Magic Sub Squares comply with the equations defining a Concentric Magic Square of order 6 (ref. Section 14.4.3).

The Prime Number Inlaid Magic Square described above can be constructed by selecting:

• An order 6 Concentric Magic Main Square containing possible Order 4 Magic Sums
• Thirty six order 4 Pan Magic Sub Squares based on these Magic Sums and the equations deducted in Section 14.2.2

The square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the thirty six inlays.

Attachment 14.6.88 shows an example of a 24^th order Inlaid Magic Square, composed out of sixteen each 6^th order Concentric Magic Squares with different Magic Sums s(1) ... s(16).

The Magic Sums s(1) thru s(16) of the Concentric Magic Sub Squares comply with the equations defining a Pan Magic Square of order 4 (ref. Section 14.2.2).

The Prime Number Inlaid Magic Square described above can be constructed by selecting:

• An order 4 Pan Magic Main Square containing possible Order 6 Magic Sums
• Sixteen order 6 Concentric Magic Sub Squares based on these Magic Sums and the equations deducted in Section 14.4.3

The square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the sixteen inlays.

14.22.7 Magic Squares, Composed (25 x 25)

Attachment 14.6.92 shows an example of a 25^th order Inlaid Magic Square, composed out of twenty five each 5^th order Magic Squares with different Magic Sums s(1) ... s(25).

The Magic Sums s(1) thru s(25) of the Magic Sub Squares comply with the equations defining a Concentric Magic Square of order 5 (ref. Section 14.3.2).

The Prime Number Inlaid Magic Square described above can be constructed by selecting:

• An order 5 Concentric Magic Main Square containing possible Center Elements for the Sub Squares
• Twenty five order 5 Concentric Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.3.2

The square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the twenty five inlays.

14.22.8 Magic Squares, Composed (28 x 28)

Attachment 14.6.93 shows an example of a 28^th order Inlaid Magic Square, composed out of forty nine each 4^th order Pan Magic Squares with different Magic Sums s(1) ... s(49).

The Magic Sums s(1) thru s(49) of the Pan Magic Sub Squares comply with the equations defining a Concentric Magic Square of order 7 (ref. Section 14.5.1).

The Prime Number Inlaid Magic Square described above can be constructed by selecting:

• An order 7 Concentric Magic Main Square containing possible Order 4 Magic Sums
• Forty nine order 4 Pan Magic Sub Squares based on these Magic Sums and the equations deducted in Section 14.2.2

The square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the forty nine inlays.

Attachment 14.6.94 shows an example of a 28^th order Inlaid Magic Square, composed out of sixteen each 7^th order Concentric Magic Squares with different Magic Sums s(1) ... s(16).

The Magic Sums s(1) thru s(16) of the Concentric Magic Sub Squares comply with the equations defining a Pan Magic Square of order 4 (ref. Section 14.2.2).

The Prime Number Inlaid Magic Square described above can be constructed by selecting:

• An order 4 Pan Magic Main Square containing possible Center Elements for the Sub Squares
• Sixteen order 7 Concentric Magic Sub Squares based on the corresponding Magic Sums and the equations deducted in Section 14.5.1

The square shown corresponds with numerous solutions, which can be obtained by selecting other aspects of the main square or the sixteen inlays.

14.22.9 Summary

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

Following sections will describe how Prime Number Magic Squares with Consecutive Primes can be found with comparable routines as described in previous chapters.