Office Applications and Entertainment, Magic Squares

7.5   Concentric and Eccentric Magic Squares

7.5.6 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 Concentric Magic Squares (ref. Priem7e18).

The solutions can be obtained by guessing the 16 parameters:

    a(i) for i = 11, 12, 14, 23, 28, 30, 32 ... 35, 44 ... 49

and filling out these guesses in the abovementioned equations.

Attachment 7.5.6 shows for a(33) = 2 ... 24, 26 ... 48 the first occurring Concentric Magic Square with Crosswise Symmetric Border.

7.5.7 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 Concentric Pan Magic Squares (ref. Priem7e19).

The solutions can be obtained by guessing the 14 parameters:

    a(i) for i = 32 ... 35, 39 ... 42, 44 ... 49

and filling out these guesses in the abovementioned equations.

Attachment 7.5.7 shows for a(49) = 1 ... 24, 26 ... 49 the first occurring Concentric Pan Magic Squares.

7.5.8 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 Concentric Pan Magic Squares (ref. Priem7e20).

The solutions can be obtained by guessing the 10 parameters:

    a(i) for i = 39 ... 42, 44 ... 49

and filling out these guesses in the abovementioned equations.

Attachment 7.5.8 shows for a(49) = 1 ... 24, 26 ... 49 the first occurring Concentric Pan Magic Squares.

7.5.9 Summary

The obtained results regarding the miscellaneous types of order 7 Magic Squares as deducted and discussed in previous sections are summarized in following table:

Comparable routines as listed above, can be used to generate Inlaid Magic Squares of order 7, which will be described in following sections.