Office Applications and Entertainment, Magic Squares

14.0    Special Magic Squares, Prime Numbers

14.5    Magic Squares (7 x 7), Part II

In next sections a few more solutions will be found for more strict defined Prime Number Magic Squares of the 7^th order.

14.5.17 Non Overlapping Sub Squares

The order 7 Associated Magic Squares as described Section 14.5.7 can also be obtained by means of a transformation of order 7 Composed Magic Squares as illustrated below:

The Magic Square shown at the left side above is composed out of:

• One 3^th order Simple Magic Corner Square with Magic Sum s3 = 3 * s1 / 7 (top/left)
• One 4^th order Associated Magic Corner Square with Magic Sum s4 = 4 * s1 / 7 (bottom/right)
• Two Associated Magic Rectangles order 3 x 4 with s3 = 3 * s1 / 7 and s4 = 4 * s1 / 7

Based on this definition a routine can be developed to generate subject Composed Magic Squares (ref. Priem7e3).

Attachment 14.6.30 shows for miscellaneous Magic Sums the first occurring Prime Number Composed Magic Square.

Also (Associated) Bordered Magic Squares can be obtained by means of transformation of order 7 Composed Magic Squares as illustrated below:

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

14.5.18 Concentric Magic Squares (7 x 7)
        Crosswise Symmetric Border

Based on the equations defining order 7 Concentric Magic Squares with Crosswise Symmetric Border:

a(31) = 3 * s1/7 - a(32) - a(33)
a(26) =     s1/7 + a(31) - a(33)
a(25) =     s1/7

a(43) =     s1   - a(44) - a(45) - a(46) - a(47) - a(48) - a(49)
a(29) =     s1   - a(30) - a(34) - a(35) - a(31) - a(32) - a(33)
a(13) =    -s1/7 + a(14) + a(28) - a(34) + a(35) - a(48) + a(49)
a(10) =    -s1/7 + a(12) - a(45) + a(47) + a(26) - a(31) + a(33)
a( 9) = (24*s1/7 - a(11) - 2 * a(12) - 2 * a(14) - a(23) - a(28) - 2 * a(30) +
                         - 2 * a(35) - 2 * a(44) - a(46) - 2 * a(47) - 2*a(49) - a(26) - a(32) - 2*a(33))/2
a( 8) =     s1 - a(9) - a(10) - a(11) - a(12) - a(13) - a(14)

a routine can be written to generate subject Prime Number Concentric Magic Squares (ref. Priem7e18).

Attachment 14.5.18 shows Prime Number Concentric Magic Squares with Crosswise Symmetric Border for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

14.5.19 Concentric Pan Magic Squares (7 x 7)
        Simple Magic Center Square

Based on the equations defining order 7 Concentric Pan Magic Squares with order 5 Simple Magic Center Square:

a(43) =      s1   - a(44) - a(45) - a(46) - a(47) - a(48) - a(49)
a(38) =  4 * s1/7 - a(39) - a(40) - a(46)
a(37) =      s1/7 - a(41) + a(46)
a(31) =  6 * s1/7 - a(32) - a(33) - a(39) - a(45) - a(47)
a(30) =  3 * s1/7 - a(34) - a(38) - a(40) + a(45) - a(46) + a(47)
a(28) =      s1   - a(34) - a(40) - a(42) - a(46) - a(48) - a(49)
a(27) =  6 * s1/7 - a(33) - a(34) - a(35) + a(38) - 2 * a(41) + a(46) - a(47) - a(49)
a(26) =  8 * s1/7 - a(32) - a(33) - a(34) - a(40) - a(41) - a(42) - a(48)
a(24) = -4 * s1/7 + a(33) + a(34) + a(39) + a(40) + a(41) + a(42) - a(44)
a(23) =           - a(27) + a(32) - a(39) + a(44) + a(48)
a(21) =     -s1/7 + a(34) + a(40) + a(41) - a(45)
a(20) = -3 * s1/7 - a(27) + a(40) + a(42) + a(46) + a(48) + a(49)
a(19) =     -s1/7 - a(31) + a(34) - a(38) - a(39) + 2 * a(41) + a(42) - a(46) + a(48) + a(49)
a(18) =  3 * s1/7 + a(38) + a(40) - a(43) - a(45) - a(47) - a(49)
a(17) =  9 * s1/7 - a(33) - a(34) - a(40) - 2 * a(41) - a(42) - a(48) - a(49)
a(16) =  5 * s1/7 + a(26) - a(35) - a(39) - a(40) - a(41) + a(43) - a(47) - a(49)
a(14) =  6 * s1/7 - a(35) - a(41) - a(44) - a(47) - a(49)
a(13) =  8 * s1/7 - a(34) - a(40) - a(41) - a(42) - a(46) - a(48) - a(49)
a(12) =  4 * s1/7 - a(33) + a(38) - a(41) - a(45) + a(46) - a(47) - a(49)
a(11) =  4 * s1/7 - a(32) - a(44) - a(48)
a(10) =      s1/7 + a(32) + a(33) - a(38) + a(41) - a(43) - a(46)
a( 9) = -5 * s1/7 + a(34) + a(40) + a(41) + a(42) + a(48) + a(49)
a(25) =      s1/7

a routine can be written to generate subject Prime Number Concentric Pan Magic Squares (ref. Priem7e19).

Attachment 14.5.19 shows Prime Number Concentric Pan Magic Squares for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

14.5.20 Concentric Pan Magic Squares (7 x 7)
        Associated Center Square

Based on the equations defining order 7 Concentric Pan Magic Squares with order 5 Associated Center Square:

a(43) =       s1   - a(44) - a(45) - a(46) - a(47) - a(48) - a(49)
a(38) =   4 * s1/7 - a(39) - a(40) - a(46)
a(37) =       s1/7 - a(41) + a(46)
a(35) =   6 * s1/7 - a(41) - a(42) - a(47) - a(48) - a(49)
a(34) =       s1   - a(40) - 2 * a(41) - a(42) - a(48) - a(49)
a(33) =  10 * s1/7 - 2 * a(39) - 2 * a(40) - a(41) - a(45) - a(46) - a(47) - a(49)
a(32) =   2 * s1/7 + a(39) - a(44) - a(48)
a(31) =  -6 * s1/7 + 2 * a(40) + a(41) + a(44) + a(46) + a(48) + a(49)
a(30) =  -8 * s1/7 + a(39) + a(40) + 2 * a(41) + a(42) + a(45) + a(47) + a(48) + a(49)
a(28) =              2 * a(41) - a(46)
a(27) = -13 * s1/7 + a(39) + 2 * a(40) + 2 * a(41) + 2 * a(42) + a(45) + a(46) + a(47) + 2*a(48) + 2*a(49)
a(26) = -11 * s1/7 + a(39) + 2 * a(40) + 2 * a(41) + a(44) + a(45) + a(46) + a(47) + a(48) + 2*a(49)
a(21) =   6 * s1/7 - a(41) - a(42) - a(45) - a(48) - a(49)
a(14) =              a(42) - a(44) + a(48)
a(25) =       s1/7

a routine can be written to generate subject Prime Number Concentric Pan Magic Squares (ref. Priem7e20).

Attachment 14.5.20 shows Prime Number Concentric Pan Magic Squares for some of the occurring Magic Sums.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

14.5.21 Summary

The obtained results regarding the miscellaneous types of order 7 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 8, which will be described in following sections.