Office Applications and Entertainment, Magic Squares  
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16.0 Special Magic Squares, Squares of Squares
In contradiction to Bimagic Squares, as discussed in previous section, Magic Squares of Squares are not magic when its entries are not squared.
16.1 Squares of Squares (3 x 3)
Up to now (Jan. 2016), nobody has been able to construct a 3^{th} order Magic Square of Squares, nor proven that it is impossible to construct such a square.
16.1.1
Semi Magic, Edouard Lucas, Six Magic Lines
The first example of a Semi Magic Square of the 3^{th} order, as published by Eduoard Lucas in 1876, was based on following formula’s:
as described in an article published by Christian Boyer in The Mathematical Intelligencer (Vol. 27, N. 2, 2005, pages 5264).
16.1.2
Semi Magic, Edouard Lucas, Seven Magic Lines
Semi Magic Squares with seven Magic Lines of the Lucas Family will occur for higher values of p, q, r and s.
16.1.3
Semi Magic, Linear Equations, Six Magic Lines
Semi Magic Squares of Squares can also be generated with comparable routines as used for Prime Numbers but now based on distinct squared variable values {a_{i}^{2}} with i = 1 ... n.
The equations defining 3^{th} order Semi Magic Squares with six Magic Lines, are: a(7) = s1  a(8)  a(9) a(4) = s1  a(5)  a(6) a(3) =  a(6) + a(7) + a(8) a(2) = s1  a(5)  a(8) a(1) = a(5) + a(6)  a(7)
with a(9), a(8), a(6) and a(5) the independent variables and a(j) = a_{i}^{2} for j = 1 ... 9, i = 1 ... n.
The majority of the Magic Sums found for Squares not belonging to the Lucas Family are not squared numbers, although occasionally remarkably close (e.g. 10011).
16.1.4
Semi Magic, Linear Equations, Seven Magic Lines
The equations defining 3^{th} order Semi Magic Squares with seven Magic Lines, are: a(7) = s1  a(8)  a(9) a(5) =  s1 + a(6) + a(8) + 2 * a(9) a(4) = 2 * s1  2 * a(6)  a(8)  2 * a(9) a(3) = s1  a(6)  a(9) a(2) = 2 * s1  a(6)  2 * a(8)  2 * a(9) a(1) = 2 * s1 + 2 * a(6) + 2 * a(8) + 3 * a(9)
with a(9), a(8) and a(6) the independent variables and a(j) = a_{i}^{2} for j = 1 ... 9, i = 1 ... n.
16.1.5
Magic, Eight Magic Lines, Squared Entries >= 7
As mentioned in Section 16.1 above, up to now nobody has been able to construct a 3^{th} order Magic Square of Squares.
a(7) = s1  a(8)  a(9) a(6) = 4 * s1/3  a(8)  2 * a(9) a(5) = s1/3 a(4) = 2 * s1/3 + a(8) + 2 * a(9) a(3) =  s1/3 + a(8) + a(9) a(2) = 2 * s1/3  a(8) a(1) = 2 * s1/3  a(9)
with a(9) and a(8) the independent variables and a(j) = a_{i}^{2} for j = 1 ... 9, i = 1 ... n.
16.1.6
Semi Magic, Center Element a(5) = s1/3
It should be noticed that none of the Semi Magic Squares shown in the attachments listed in Section 16.1.3 above contain an element a(j) = s1/3.
The obtained results regarding the miscellaneous types of order 3 Semi Magic Squares of Squares as deducted and discussed in previous sections are summarized in following table: 
Main Characteristics
Original Author
Subroutine
Results
Semi Magic, 6 Magic Lines
Eduoard Lucas
Semi Magic, 6 Magic Lines

Semi Magic, 7 Magic Lines
Eduoard Lucas
Semi Magic, 8 Magic Lines, 7 Squared Entries
Andrew Brenner
Semi Magic, 6 Magic Lines, Center = s1/3
Lee Morgenstern
16.2 Squares of Squares (4 x 4)
16.2.1
Magic, Parametric Solution, Leonhard Euler
The first four Magic Squares of the 4^{th} order, as send by Leonhard Euler in 1770 to Joseph Lagrange, were based on following formula’s (according to the source mentioned above):
The 4 rows and 4 columns sum to the Magic Sum s1, and two supplemental conditions:
ensure that also the two magic diagonals sum to s1 = (a^{2} + b^{2} + c^{2} + d^{2})(p^{2} + q^{2} + r^{2} + s^{2}).
Attachment 16.2.1 shows the occurring Magic Sums and corresponding distinct variable values {a_{i}^{2}} while varying the parameters a, b, c, d, p, q, r over the range {1 ... 9} for s = 4.
For (a, b, c, d, p, q, r, s) = (2, k, 5, 0, 1, 2, 8, –4) the formula's can be simplified to:
and will result in Magic Squares with s1 = 85 * (k^{2} + 29) and distinct integers for k = 3, 5, 8, ...
(ref. Attachment 16.2.2)
and will result in Magic Squares with s1 = 65 * (k^{2} + 106) and distinct integers for k = 2, 3, 5, ...
(ref. Attachment 16.2.2)
16.2.2
Magic, Parametric Solution, Seiji Tomita
More recently comparable Parametric Solutions have been published by Seiji Tomita (November 2018) and are shown in
Attachment 16.2.4.
16.2.3
Magic, Linear Equations
Magic Squares of Squares of order 4 can also be generated with comparable routines as used above for Semi Magic Squares of Squares of order 3.
a(13) = s1  a(14)  a(15)  a(16) a( 9) = s1  a(10)  a(11)  a(12) a( 7) = a( 8)  a(10) + a(12)  a(13) + a(16) a( 6) = s1  a( 8)  a(11)  a(12) + a(13)  a(16) a( 5) =  a( 8) + a(10) + a(11) a( 4) = s1  a( 7)  a(10)  a(13) a( 3) = s1  a( 8) + a( 9) + 2*a(10) + 2*a(13) + a(14) a( 2) = a( 8)  a( 9)  2*a(10) + a(15) + 2*a(16) a( 1) = a( 8) + a(12)  a(13)
with a(16), a(15), a(14), a(12), a(11), a(10) and a(8) the independent variables and a(j) = a_{i}^{2} for j = 1 ... 16, i = 1 ... n.
The obtained results regarding the miscellaneous types of order 4 Magic Squares of Squares as deducted and discussed in previous sections are summarized in following table: 
Main Characteristics
Original Author
Subroutine
Results
Magic Squares of Squares
Leonhard Euler

Magic Squares of Squares
Seiji Tomita
Magic Squares of Squares

16.3 Squares of Squares (5 x 5)
The first Magic Square of Squares of the 5^{th},
6^{th} and
7^{th}
order have been published by Christian Boyer in 2004/2005.
16.3.1
Magic, Generator Method
Magic Squares of Squares of order 5 can be constructed by means of the Generator Method as applied for the construction of Bimagic Squares.
An example of a Magic Square obtained by permutation of the rows and columns within a Semi Magic Square, is shown below:
Each 5^{th} order Magic Square of Squares corresponds with 32 (= 8 * 4) transformations as described in Section 3.6.
16.3.2
Magic, Linear Equations
Magic Squares of Squares of order 5 can also be generated with a comparable routine as used above for Magic Squares of Squares of order 4.
a( 1) = s2  a( 7)  a(13)  a(19)  a(25) a( 5) = s2  a( 9)  a(13)  a(17)  a(21) a( 6) = s2  a( 7)  a( 8)  a( 9)  a(10) a(16) = s2  a(21)  a(11)  a( 6)  a( 1) a(20) = s2  a(25)  a(15)  a(10)  a( 5) a(18) = s2  a(16)  a(17)  a(19)  a(20) a(12) = s2  a(11)  a(13)  a(14)  a(15) a(23) = s2  a( 3)  a( 8)  a(13)  a(18) a(22) = s2  a( 2)  a( 7)  a(12)  a(17) a( 4) = s2  a( 1)  a( 2)  a( 3)  a( 5) a(24) = s2  a(21)  a(22)  a(23)  a(25)
with a(1), a(4), a(5), a(6), a(12), a(16), a(18), a(20), a(22), a(23) and a(24) the independent variables and a(j) = a_{i}^{2} for j = 1 ... 25, i = 1 ... n.
16.4 Squares of Squares (6 x 6)
Magic Square of Squares of order 6 can be constructed with the Generator Method, as discussed in
Section 16.3.1 above.
Each 6^{th} order Magic Square of Squares corresponds with 192 (= 8 * 24) transformations as described in Section 6.3.
The first Bimagic Squares of order 6 were constructed by Jaroslaw Wroblewski
and Lee Morgenstern in 2006 and have been discussed in
Section 6.13.4 and
6.13.5.
16.5 Squares of Squares (7 x 7)
Magic Square of Squares of order 7 can be constructed with the Generator Method, as discussed in
Section 16.3.1 above.
which ensures that subject generators contain the consecutive integers 1 ... 49 and speeds up the construction process.
The comparable Boyer Square (bimagic rows) is based on the consecutive integers 0 ... 48 with related Magic Sums s1 = 168 and s2 = 5432 .
Although a mathematical proof has not yet been provided (2020), it has been demonstrated by Christian Boyer and others that Bimagic Squares of order 7 can't exist for consecutive integers.
The obtained results regarding the miscellaneous types of order 5, 6 and 7 Magic Squares of Squares as deducted and discussed in previous sections are summarized in following table: 
Main Characteristics
Property
Subroutine
Results
Magic Squares of Squares Order 5

Magic Squares of Squares Order 6

Magic Squares of Squares Order 7
Bimagic Columns
Next section will provide miscellaneous construction methods for Two and Four Way, V type Zig Zag Magic Squares.

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