Office Applications and Entertaiment, Magic Squares

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8.6   Further Analysis, Miscellaneous Properties

8.6.1 Associated versus Complete


In previous sections the properties Associated (Center Symmetric) and Complete have been introduced and applied several times.

For the sake of completeness, the definitions of subject properties are repeated below for Magic Squares of order 8:

  • Associated: All pairs of integers, which can be connected with a straight line through the center,
    are equal to n2 + 1 = 65.
  • Complete:   All pairs of integers distant n/2 = 4 along a (main) diagonal are equal to n2 + 1 = 65.

In Section 8.6.2 and Section 8.6.3 below, the properties described above will be applied on Pan Magic Squares.

8.6.2 Pan Magic, Associated

For Associated Pan Magic Squares, following equations should be added to the equations defining a Pan Magic Square:

a(1) + a(64) = s1/4
a(2) + a(63) = s1/4
a(3) + a(62) = s1/4
a(4) + a(61) = s1/4
a(5) + a(60) = s1/4
a(6) + a(59) = s1/4
a(7) + a(58) = s1/4
a(8) + a(57) = s1/4

a(9 ) + a(56) = s1/4
a(10) + a(55) = s1/4
a(11) + a(54) = s1/4
a(12) + a(53) = s1/4
a(13) + a(52) = s1/4
a(14) + a(51) = s1/4
a(15) + a(50) = s1/4
a(16) + a(49) = s1/4

a(17) + a(48) = s1/4
a(18) + a(47) = s1/4
a(19) + a(46) = s1/4
a(20) + a(45) = s1/4
a(21) + a(44) = s1/4
a(22) + a(43) = s1/4
a(23) + a(42) = s1/4
a(24) + a(41) = s1/4

a(25) + a(40) = s1/4
a(26) + a(39) = s1/4
a(27) + a(38) = s1/4
a(28) + a(37) = s1/4
a(29) + a(36) = s1/4
a(30) + a(35) = s1/4
a(31) + a(34) = s1/4
a(32) + a(33) = s1/4

which results - after deduction - in following linear equations describing Associated Pan Magic or Ultra Magic Squares:

a(57) =    s1 - a(58) - a(59) - a(60) - a(61) - a(62) - a(63) - a(64)
a(49) =    s1 - a(50) - a(51) - a(52) - a(53) - a(54) - a(55) - a(56)
a(45) =  2*s1 - a(46) - a(47) - a(48) + a(50) + a(51) - a(53) - 2*a(54) - 2*a(55) - a(56) + a(58) + a(59) +
                                                                        - 2*a(61) - 3*a(62) - 3*a(63) - 2*a(64)
a(43) =    s1 - a(44) - a(47) - a(48) - a(51) - a(52) - a(55) - a(56)
a(42) =         a(44) - a(45) + a(47) - a(49) - a(50) + a(55) + a(56) - a(58) - a(59) + a(62) + a(63)
a(41) =    s1 - a(42) - a(45) - a(46) - a(49) - a(50) - a(53) - a(54)
a(40) =  2*s1 - a(47) - 2*a(48) - a(54) - 2*a(55) - 2*a(56) - a(61) - 2*a(62) - 2*a(63) - 2*a(64)
a(39) =         a(44) - a(49) + a(62)
a(38) =       - a(44) - a(46) - a(47) + a(49) + a(50) + a(51) + a(58) + a(59) - a(62)
a(37) = -2*s1 + a(46) + 2*a(47) + 2*a(48) - 2*a(50) - 2*a(51) + 2*a(54) + 3*a(55) + 2*a(56) - 2*a(58) +
                                                              - 2*a(59) + 2*a(61) + 4*a(62) + 4*a(63) + 3*a(64)
a(36) =    s1 - a(44) + a(47) + a(48) - a(50) - a(51) - a(52) + a(55) + a(56) - a(57) - 2*a(58) - 2*a(59) - 2*a(60)
a(35) =         a(48) - a(53) + a(58)
a(34) =         a(45) - a(56) + a(59)
a(33) = -  s1 + a(44) + a(46) - a(48) + a(50) + a(51) + a(52) + 2*a(53) + a(54) + a(56) + a(60)

a(32) = s1/4 - a(33)
a(31) = s1/4 - a(34)
a(30) = s1/4 - a(35)
a(29) = s1/4 - a(36)
a(28) = s1/4 - a(37)
a(27) = s1/4 - a(38)
a(26) = s1/4 - a(39)
a(25) = s1/4 - a(40)

a(24) = s1/4 - a(41)
a(23) = s1/4 - a(42)
a(22) = s1/4 - a(43)
a(21) = s1/4 - a(44)
a(20) = s1/4 - a(45)
a(19) = s1/4 - a(46)
a(18) = s1/4 - a(47)
a(17) = s1/4 - a(48)

a(16) = s1/4 - a(49)
a(15) = s1/4 - a(50)
a(14) = s1/4 - a(51)
a(13) = s1/4 - a(52)
a(12) = s1/4 - a(53)
a(11) = s1/4 - a(54)
a(10) = s1/4 - a(55)
a( 9) = s1/4 - a(56)

a(8) = s1/4 - a(57)
a(7) = s1/4 - a(58)
a(6) = s1/4 - a(59)
a(5) = s1/4 - a(60)
a(4) = s1/4 - a(61)
a(3) = s1/4 - a(62)
a(2) = s1/4 - a(63)
a(1) = s1/4 - a(64)

The solutions can be obtained by guessing a(64) ... a(58), a(56) ... a(50), a(48) ... a(46), a(44) and filling out these guesses in the equations shown above.

However rather than trying to find solutions based on the equations deducted above, subject equations will be combined with properties described in Section 8.6.4, into more strict defined Associated Pan Magic Squares of order 8.

8.6.3 Pan Magic, Complete

The combination of the properties Pan Magic, Compact and Complete (Most Perfect Magic Squares) has been mentioned before, amongst others as a consequence of the construction method discussed in Section 8.1.2.

When the defining properties are moderated till Pan Magic Complete, only following equations have to be added to the equations defining a Pan Magic Square:

a(1) + a(37) = s1/4
a(2) + a(38) = s1/4
a(3) + a(39) = s1/4
a(4) + a(40) = s1/4
a(5) + a(33) = s1/4
a(6) + a(34) = s1/4
a(7) + a(35) = s1/4
a(8) + a(36) = s1/4

a( 9) + a(45) = s1/4
a(10) + a(46) = s1/4
a(11) + a(47) = s1/4
a(12) + a(48) = s1/4
a(13) + a(41) = s1/4
a(14) + a(42) = s1/4
a(15) + a(43) = s1/4
a(16) + a(44) = s1/4

a(17) + a(53) = s1/4
a(18) + a(54) = s1/4
a(19) + a(55) = s1/4
a(20) + a(56) = s1/4
a(21) + a(49) = s1/4
a(22) + a(50) = s1/4
a(23) + a(51) = s1/4
a(24) + a(52) = s1/4

a(25) + a(61) = s1/4
a(26) + a(62) = s1/4
a(27) + a(63) = s1/4
a(28) + a(64) = s1/4
a(29) + a(57) = s1/4
a(30) + a(58) = s1/4
a(31) + a(59) = s1/4
a(32) + a(60) = s1/4

which results - after deduction - in following linear equations describing Complete Pan Magic Squares:

a(57) =   s1 - a(58) - a(59) - a(60) - a(61) - a(62) - a(63) - a(64)
a(49) =   s1 - a(50) - a(51) - a(52) - a(53) - a(54) - a(55) - a(56)
a(41) =   s1 - a(42) - a(43) - a(44) - a(45) - a(46) - a(47) - a(48)
a(37) = 2*s1 - a(38) - a(39) - a(40) - a(45) - a(46) - a(47) - a(48) - a(53) - a(54) - a(55) - a(56) +
                                                                             - a(61) - a(62) - a(63) - a(64)
a(36) =        a(40) - a(44) + a(48) - a(52) + a(56) - a(60) + a(64)
a(35) =        a(39) - a(43) + a(47) - a(51) + a(55) - a(59) + a(63)
a(34) =        a(38) - a(42) + a(46) - a(50) + a(54) - a(58) + a(62)
a(33) =      - a(38) - a(39) - a(40) + a(42) + a(43) + a(44) + a(45) + a(50) + a(51) + a(52) + a(53) + 
                                                                             - a(57) - a(62) - a(63) - a(64)

a(32) = s1/4 - a(60)
a(31) = s1/4 - a(59)
a(30) = s1/4 - a(58)
a(29) = s1/4 - a(57)
a(28) = s1/4 - a(64)
a(27) = s1/4 - a(63)
a(26) = s1/4 - a(62)
a(25) = s1/4 - a(61)

a(24) = s1/4 - a(52)
a(23) = s1/4 - a(51)
a(22) = s1/4 - a(50)
a(21) = s1/4 - a(49)
a(20) = s1/4 - a(56)
a(19) = s1/4 - a(55)
a(18) = s1/4 - a(54)
a(17) = s1/4 - a(53)

a(16) = s1/4 - a(44)
a(15) = s1/4 - a(43)
a(14) = s1/4 - a(42)
a(13) = s1/4 - a(41)
a(12) = s1/4 - a(48)
a(11) = s1/4 - a(47)
a(10) = s1/4 - a(46)
a( 9) = s1/4 - a(45)

a(8) = s1/4 - a(36)
a(7) = s1/4 - a(35)
a(6) = s1/4 - a(34)
a(5) = s1/4 - a(33)
a(4) = s1/4 - a(40)
a(3) = s1/4 - a(39)
a(2) = s1/4 - a(38)
a(1) = s1/4 - a(37)

The solutions can be obtained by guessing a(64) ... a(58), a(56) ... a(50), a(48) ... a(42), a(40) ... a(38) and filling out these guesses in the equations shown above.

However rather than trying to find solutions based on the equations deducted above, subject equations will be combined with properties described in Section 8.6.4, into more strict defined Complete Pan Magic Squares of order 8.

8.6.4 Compact versus Partly Compact

In previous sections the property Compact has been introduced and several times applied (e.g. Franklin Squares).

In following sections the concept will be elaborated, as certain Magic Squares (e.g. Bimagic) require less restricting properties.

The following possible grids will be considered:

  • Squared Grids, consisting of 2 x 2 squares for which the elements sum to half the Magic Sum;
  • Rectangular Grids, consisting of 2 x 4 (4 x 2) rectangles for which the elements sum to the Magic Sum;
  • Combined Rectangular Grids, which combine 2 x 4 and 4 x 2 rectangles.

Notes: Rectangles are described by width (columns) x height (rows).
       4 x 2 Grids can be considered as a rotation of 2 x 4 Grids and will not be considered separately.

For each grid the following (squares) rectangles might sum to (half the) Magic Sum:

  • All Overlapping squares/rectangles (Compact), resulting in 8 x 8 = 64 defining equations;
  • The Non Overlapping squares/rectangles, resulting in 16/8 defining equations;
  • A fraction of the Overlapping squares/rectangles (Partly Compact).
    The number of defining equations depends of the fraction.

In following sections the properties described above will be applied on Associated (ref. Section 8.6.2) and Complete (ref. Section 8.6.3) Pan Magic Squares.

8.6.5 Pan Magic, Compact, Associated

For Compact Associated Pan Magic Squares, following equations should be added to the equations defining an Associated Pan Magic Square:

a(i) + a(i+1) + a(i+ 8) + a(i+ 9) = s1/2 with 1 =< i < 56 and i ≠ 8 * n for n = 1, 2 ... 7

a(i) + a(i+1) + a(i+ 8) + a(i- 7) = s1/2 with i = 8 * n for n = 1, 2 ... 7

a(i) + a(i+1) + a(i+56) + a(i+57) = s1/2 with i = 1, 2 ... 7

a(1) + a(8) + a(57) + a(64) = s1/2

which results - after deduction - in following linear equations describing Associated Compact Pan Magic:

a(61) =  s1/2 - a(62) - a(63) - a(64)
a(59) =  s1/2 - a(60) - a(63) - a(64)
a(58) = -s1/2 + a(60) + a(62) + 2 * a(63) + a(64)
a(57) =  s1/2 - a(60) - a(62) - a(63)
a(55) =  s1/2 - a(56) - a(63) - a(64)
a(54) =         a(56) - a(62) + a(64)
a(53) =       - a(56) + a(62) + a(63)
a(52) =         a(56) - a(60) + a(64)
a(51) =       - a(56) + a(60) + a(63)
a(50) =  s1/2 + a(56) - a(60) - a(62) - 2 * a(63)
a(49) =       - a(56) + a(60) + a(62) + a(63) - a(64)
a(47) =       - a(48) + a(63) + a(64)
a(46) =         a(48) + a(62) - a(64)
a(45) =  s1/2 - a(48) - a(62) - a(63)
a(44) =         a(48) + a(60) - a(64)
a(43) =  s1/2 - a(48) - a(60) - a(63)
a(42) = -s1/2 + a(48) + a(60) + a(62) + 2 * a(63)
a(41) =  s1/2 - a(48) - a(60) - a(62) - a(63) + a(64)
a(40) =  s1/2 - a(48) - a(56) - a(64)
a(39) =         a(48) + a(56) - a(63)
a(38) =  s1/2 - a(48) - a(56) - a(62)
a(37) = -s1/2 + a(48) + a(56) + a(62) + a(63) + a(64)
a(36) =  s1/2 - a(48) - a(56) - a(60)
a(35) = -s1/2 + a(48) + a(56) + a(60) + a(63) + a(64)
a(34) =  s1   - a(48) - a(56) - a(60) - a(62) - 2 * a(63) - a(64)
a(33) = -s1/2 + a(48) + a(56) + a(60) + a(62) + a(63)

a(32) = s1/4 - a(33)
a(31) = s1/4 - a(34)
a(30) = s1/4 - a(35)
a(29) = s1/4 - a(36)
a(28) = s1/4 - a(37)
a(27) = s1/4 - a(38)
a(26) = s1/4 - a(39)
a(25) = s1/4 - a(40)

a(24) = s1/4 - a(41)
a(23) = s1/4 - a(42)
a(22) = s1/4 - a(43)
a(21) = s1/4 - a(44)
a(20) = s1/4 - a(45)
a(19) = s1/4 - a(46)
a(18) = s1/4 - a(47)
a(17) = s1/4 - a(48)

a(16) = s1/4 - a(49)
a(15) = s1/4 - a(50)
a(14) = s1/4 - a(51)
a(13) = s1/4 - a(52)
a(12) = s1/4 - a(53)
a(11) = s1/4 - a(54)
a(10) = s1/4 - a(55)
a( 9) = s1/4 - a(56)

a(8) = s1/4 - a(57)
a(7) = s1/4 - a(58)
a(6) = s1/4 - a(59)
a(5) = s1/4 - a(60)
a(4) = s1/4 - a(61)
a(3) = s1/4 - a(62)
a(2) = s1/4 - a(63)
a(1) = s1/4 - a(64)

The solutions can be obtained by guessing a(64) ... a(62), a(60), a(56), a(48) and filling out these guesses in the equations shown above.

Based on the equations above, it can be proven that both the half rows and half columns sum to half the Magic Sum.

An optimized guessing routine (Priem8e), produced 64 * 720 = 46080 Compact Associated Pan Magic Squares within 1,5 hours, of which the first 720 are shown in Attachment 8.6.5.

8.6.6 Pan Magic, Compact, Complete

Pan Magic, Compact and Complete or Most Perfect Magic Squares have been discussed in detail in Section 8.5.5.

The total number of Most Perfect Pan Magic Squares is 64 * 46080 = 2949120, of which 384 are shown in Attachment 8.5.5.

8.6.7 Pan Magic, Non Overlapping Sub Squares, Associated (1)

For Associated Pan Magic Squares with Non Overlapping Sub Squares, following equations should be added to the equations defining an Associated Pan Magic Square:

a(i) + a(i + 1) + a(i + 8) + a(i + 9) = s1/2 with i = m + 16 * n`and m = 1, 3, 5, 7 ; n = 0, 1, 2, 3

which results - after deduction - in following linear equations describing subject Associated Pan Magic Squares:

a(57) =    s1   - a(58) - a(59) - a(60) - a(61) - a(62) - a(63) - a(64)
a(55) =    s1/2 - a(56) - a(63) - a(64)
a(53) =    s1/2 - a(54) - a(61) - a(62)
a(51) =    s1/2 - a(52) - a(59) - a(60)
a(49) =    s1/2 - a(50) - a(57) - a(58)
a(45) =    s1   - a(46) - a(47) - a(48) + a(50) - a(52) - a(54) + a(56) + a(58) - a(60) - a(61) - 2 * a(62) - a(63)
a(44) = -  s1   + a(48) - a(50) + a(54) + a(59) + a(60) + 2 * a(61) + 2 * a(62) + a(63) + a(64)
a(43) =    s1   - a(47) - 2 * a(48) + a(50) - a(54) - 2 * a(61) - 2 * a(62)
a(42) = -  s1   + a(46) + 2 * a(47) + 2 * a(48) - 2 * a(50) + a(52) + 2 * a(54) - a(56) - 2 * a(58) - a(59) + 
                                                                    + a(60) + 2 * a(61) + 4 * a(62) + a(63) - a(64)
a(41) =    s1   - a(46) - a(47) - a(48) + a(50) - a(54) + a(58) - a(60) - a(61) - 2 * a(62) - a(63)
a(40) =    s1   - a(47) - 2 * a(48) - a(54) - a(61) - 2 * a(62)
a(39) = -  s1/2 + a(48) + a(54) + a(61) + 2 * a(62)
a(38) =    s1   - a(46) - a(47) - a(48) + a(50) - a(52) - a(54) + a(58) - a(60) - a(61) - 2 * a(62)
a(37) = -3*s1/2 + a(46) + 2 * a(47) + 2 * a(48) - 2 * a(50) + 2 * a(52) + 2 * a(54) - a(56) - 2 * a(58) + 
                                                                        + 2 * a(60) + 2 * a(61) + 4 * a(62) + a(63)
a(36) =            a(47) - a(54) + a(57)
a(35) = -  s1/2 + a(48) + a(54) + a(58) + a(61) + a(62)
a(34) =    s1   - a(46) - a(47) - a(48) + a(50) - a(52) - a(54) + a(58) + a(59) - a(60) - a(61) - 2 * a(62) - a(63)
a(33) = -  s1/2 + a(46) + a(56) + a(60) + a(63) + a(64)

a(32) = s1/4 - a(33)
a(31) = s1/4 - a(34)
a(30) = s1/4 - a(35)
a(29) = s1/4 - a(36)
a(28) = s1/4 - a(37)
a(27) = s1/4 - a(38)
a(26) = s1/4 - a(39)
a(25) = s1/4 - a(40)

a(24) = s1/4 - a(41)
a(23) = s1/4 - a(42)
a(22) = s1/4 - a(43)
a(21) = s1/4 - a(44)
a(20) = s1/4 - a(45)
a(19) = s1/4 - a(46)
a(18) = s1/4 - a(47)
a(17) = s1/4 - a(48)

a(16) = s1/4 - a(49)
a(15) = s1/4 - a(50)
a(14) = s1/4 - a(51)
a(13) = s1/4 - a(52)
a(12) = s1/4 - a(53)
a(11) = s1/4 - a(54)
a(10) = s1/4 - a(55)
a( 9) = s1/4 - a(56)

a(8) = s1/4 - a(57)
a(7) = s1/4 - a(58)
a(6) = s1/4 - a(59)
a(5) = s1/4 - a(60)
a(4) = s1/4 - a(61)
a(3) = s1/4 - a(62)
a(2) = s1/4 - a(63)
a(1) = s1/4 - a(64)

The solutions can be obtained by guessing a(64) ... a(58), a(56), a(54), a(52), a(50), a(48) ... a(46) and filling out these guesses in the equations shown above.

Subject equations can be used for the generation of Sudoku Comparable Squares as discussed in Section 8.7.2 or Quaternary Squares as discussed in Section 8.7.4.

Examples of Associated Pan Magic Squares with Non Overlapping Sub Squares based on Sudoku Comparable Squares are shown in Attachment 8.6.7.

8.6.8 Pan Magic, Non Overlapping Sub Squares, Associated (2)

For Associated Pan Magic Squares, composed out of 16 Non Overlapping Sub Squares, with also the half rows and half columns summing to half the Magic Sum, following set of linear equations is applicable:

a(61) =  s1/2 - a(62) - a(63) - a(64)
a(57) =  s1/2 - a(58) - a(59) - a(60)
a(55) =  s1/2 - a(56) - a(63) - a(64)
a(54) =         a(56) - a(62) + a(64)
a(53) =       - a(56) + a(62) + a(63)
a(51) =  s1/2 - a(52) - a(59) - a(60)
a(50) =         a(52) - a(58) + a(60)
a(49) =       - a(52) + a(58) + a(59)
a(47) =       - a(48) + a(63) + a(64)
a(45) =  s1/2 - a(46) - a(63) - a(64)
a(44) =         a(48) - a(52) + a(56) + a(58) + a(59) - a(62) - a(63)
a(43) =       - a(48) + a(52) - a(56) - a(58) + a(60) + a(62) + a(63)
a(42) =         a(46) - a(52) + a(56) - a(59) - a(60) + a(63) + a(64)
a(41) =  s1/2 - a(46) + a(52) - a(56) - a(63) - a(64)
a(40) =  s1/2 - a(48) - a(56) - a(64)
a(39) =         a(48) + a(56) - a(63)
a(38) =  s1/2 - a(46) - a(56) - a(64)
a(37) = -s1/2 + a(46) + a(56) + a(63) + 2 * a(64)
a(36) =  s1/2 - a(48) - a(56) - a(58) - a(59) - a(60) + a(62) + a(63)
a(35) =         a(48) + a(56) + a(58) - a(62) - a(63)
a(34) =  s1/2 - a(46) - a(56) + a(59) - a(63) - a(64)
a(33) = -s1/2 + a(46) + a(56) + a(60) + a(63) + a(64)

a(32) = s1/4 - a(33)
a(31) = s1/4 - a(34)
a(30) = s1/4 - a(35)
a(29) = s1/4 - a(36)
a(28) = s1/4 - a(37)
a(27) = s1/4 - a(38)
a(26) = s1/4 - a(39)
a(25) = s1/4 - a(40)

a(24) = s1/4 - a(41)
a(23) = s1/4 - a(42)
a(22) = s1/4 - a(43)
a(21) = s1/4 - a(44)
a(20) = s1/4 - a(45)
a(19) = s1/4 - a(46)
a(18) = s1/4 - a(47)
a(17) = s1/4 - a(48)

a(16) = s1/4 - a(49)
a(15) = s1/4 - a(50)
a(14) = s1/4 - a(51)
a(13) = s1/4 - a(52)
a(12) = s1/4 - a(53)
a(11) = s1/4 - a(54)
a(10) = s1/4 - a(55)
a( 9) = s1/4 - a(56)

a(8) = s1/4 - a(57)
a(7) = s1/4 - a(58)
a(6) = s1/4 - a(59)
a(5) = s1/4 - a(60)
a(4) = s1/4 - a(61)
a(3) = s1/4 - a(62)
a(2) = s1/4 - a(63)
a(1) = s1/4 - a(64)

The solutions can be obtained by guessing a(64) ... a(62), a(60) ... a(58), a(56), a(52), a(48), (46) and filling out these guesses in the equations shown above.

An optimized guessing routine (ref. Priem8d) produced - with a(64), a(63), a(62) and a(61) constant - 16320 Associated Pan Magic Squares within 55 minutes, of which the first 620 are shown in Attachment 8.6.8.

8.6.9 Pan Magic, Non Overlapping Sub Squares, Complete

For Complete Pan Magic Squares, composed out of 16 Non Overlapping Sub Squares, with also the half rows and half columns summing to half the Magic Sum, following set of linear equations is applicable:

a(61) =  s1/2 - a(62) - a(63) - a(64)
a(57) =  s1/2 - a(58) - a(59) - a(60)
a(55) =  s1/2 - a(56) - a(63) - a(64)
a(53) =       - a(54) + a(63) + a(64)
a(51) =  s1/2 - a(52) - a(59) - a(60)
a(49) =       - a(50) + a(59) + a(60)
a(45) =  s1/2 - a(46) - a(47) - a(48)
a(41) =  s1/2 - a(42) - a(43) - a(44)
a(40) =  s1/2 - a(48) - a(56) - a(64)
a(39) =       - a(47) + a(56) + a(64)
a(38) =  s1/2 - a(46) - a(54) - a(62)
a(37) = -s1/2 + a(46) + a(47) + a(48) + a(54) + a(62)
a(36) =  s1/2 - a(44) - a(52) - a(60)
a(35) =       - a(43) + a(52) + a(60)
a(34) =  s1/2 - a(42) - a(50) - a(58)
a(33) = -s1/2 + a(42) + a(43) + a(44) + a(50) + a(58)

a(32) = s1/4 - a(60)
a(31) = s1/4 - a(59)
a(30) = s1/4 - a(58)
a(29) = s1/4 - a(57)
a(28) = s1/4 - a(64)
a(27) = s1/4 - a(63)
a(26) = s1/4 - a(62)
a(25) = s1/4 - a(61)

a(24) = s1/4 - a(52)
a(23) = s1/4 - a(51)
a(22) = s1/4 - a(50)
a(21) = s1/4 - a(49)
a(20) = s1/4 - a(56)
a(19) = s1/4 - a(55)
a(18) = s1/4 - a(54)
a(17) = s1/4 - a(53)

a(16) = s1/4 - a(44)
a(15) = s1/4 - a(43)
a(14) = s1/4 - a(42)
a(13) = s1/4 - a(41)
a(12) = s1/4 - a(48)
a(11) = s1/4 - a(47)
a(10) = s1/4 - a(46)
a( 9) = s1/4 - a(45)

a(8) = s1/4 - a(36)
a(7) = s1/4 - a(35)
a(6) = s1/4 - a(34)
a(5) = s1/4 - a(33)
a(4) = s1/4 - a(40)
a(3) = s1/4 - a(39)
a(2) = s1/4 - a(38)
a(1) = s1/4 - a(37)

The solutions can be obtained by guessing:

   a(64) ... a(61), a(60) ... a(58), a(56), a(54), a(52), a(50), a(48) ... a(46) and a(44) ... a(42)

and filling out these guesses in the equations shown above.

Subject equations can be used for the generation of Sudoku Comparable Squares as discussed in Section 8.7.2 or Quaternary Squares as discussed in Section 8.7.4.

Examples of Pan Magic and Complete Squares with Non Overlapping Sub Squares based on Sudoku Comparable Squares are shown in Attachment 8.6.9.

8.6.10 Pan Magic, Associated, Rectangular Compact

Rectangular Compact (2 x 4) Ultra Magic Squares are described by following linear equations:

a(59) =  s1/2 - a(60) - a(63) - a(64)
a(57) =  s1/2 - a(58) - a(61) - a(62)
a(54) =  s1/2 - a(55) - a(62) - a(63)
a(53) =  s1/2 - a(56) - a(61) - a(64)
a(51) =  s1/2 - a(52) - a(55) - a(56)
a(50) = -s1/2 + a(52) + a(55) + a(56) - a(58) + a(60) + a(63) + a(64)
a(49) =       - a(52) + a(58) - a(60) + a(61) + a(62)
a(47) =  s1   - a(48) - a(55) - a(56) - a(61) - a(62) - a(63) - a(64)
a(46) = -s1/2 + a(48) + a(55) + a(56) + a(58) - a(60) + a(61) + a(62)
a(45) =  s1/2 - a(48) - a(58) + a(60) - a(61) - a(62)
a(44) = -s1/2 + a(48) - a(52) + a(56) + a(58) - a(60) + 2 * a(61) + a(62) + a(64)
a(43) =       - a(48) + a(52) + a(55) - a(58) + a(60) - a(61) + a(63)
a(42) =         a(48) - a(52) - a(55) + a(58) - a(60) + a(61) + a(62)
a(41) =  s1/2 - a(48) + a(52) - a(56) - a(58) + a(60) - a(61) - a(62)
a(40) =  s1/2 - a(48) - a(56) - a(64)
a(39) = -s1/2 + a(48) + a(56) + a(61) + a(62) + a(64)
a(38) =  s1/2 - a(48) - a(56) - a(58) + a(60) - a(61) - a(62) + a(63)
a(37) = -s1/2 + a(48) + a(56) + a(58) - a(60) + a(61) + a(62) + a(64)
a(36) =  s1   - a(48) - a(56) - a(58) - 2 * a(61) - a(62) - a(64)
a(35) = -s1/2 + a(48) + a(56) + a(58) + a(61) + a(64)
a(34) =  s1   - a(48) - a(56) - a(58) - a(61) - a(62) - a(63) - a(64)
a(33) = -s1/2 + a(48) + a(56) + a(58) + a(61) + a(62)

a(32) = s1/4 - a(33)
a(31) = s1/4 - a(34)
a(30) = s1/4 - a(35)
a(29) = s1/4 - a(36)
a(28) = s1/4 - a(37)
a(27) = s1/4 - a(38)
a(26) = s1/4 - a(39)
a(25) = s1/4 - a(40)

a(24) = s1/4 - a(41)
a(23) = s1/4 - a(42)
a(22) = s1/4 - a(43)
a(21) = s1/4 - a(44)
a(20) = s1/4 - a(45)
a(19) = s1/4 - a(46)
a(18) = s1/4 - a(47)
a(17) = s1/4 - a(48)

a(16) = s1/4 - a(49)
a(15) = s1/4 - a(50)
a(14) = s1/4 - a(51)
a(13) = s1/4 - a(52)
a(12) = s1/4 - a(53)
a(11) = s1/4 - a(54)
a(10) = s1/4 - a(55)
a( 9) = s1/4 - a(56)

a(8) = s1/4 - a(57)
a(7) = s1/4 - a(58)
a(6) = s1/4 - a(59)
a(5) = s1/4 - a(60)
a(4) = s1/4 - a(61)
a(3) = s1/4 - a(62)
a(2) = s1/4 - a(63)
a(1) = s1/4 - a(64)

The solutions can be obtained by guessing a(64) ... a(60), a(58), a(56), a(55), a(52), a(48) and filling out these guesses in the equations shown above.

An optimized guessing routine (ref. Priem8f) produced - with a(64), a(63), a(62) and a(61) constant - 2928 Rectangular Compact Ultra Magic Squares within 18,5 minutes, of which the first 168 are shown in Attachment 8.6.10.

The occuring 2 x 2 Compact Associated Pan Magic Squares as deducted in Section 8.6.5 are highlighted in red.

8.6.11 Pan Magic, Complete, Rectangular Compact

Rectangular Compact (2 x 4) Complete Pan Magic Squares are described by following linear equations:

a(59) =  s1/2 - a(60) - a(63) - a(64)
a(58) =         a(60) - a(62) + a(64)
a(57) =  s1/2 - a(60) - a(61) - a(64)
a(51) =  s1/2 - a(52) - a(55) - a(56)
a(50) =         a(52) - a(54) + a(56)
a(49) =  s1/2 - a(52) - a(53) - a(56)
a(43) =  s1/2 - a(44) - a(47) - a(48)
a(42) =         a(44) - a(46) + a(48)
a(41) =  s1/2 - a(44) - a(45) - a(48)
a(39) =  s1   - a(40) - a(47) - a(48) - a(55) - a(56) - a(63) - a(64)
a(38) =         a(40) - a(46) + a(48) - a(54) + a(56) - a(62) + a(64)
a(37) =  s1   - a(40) - a(45) - a(48) - a(53) - a(56) - a(61) - a(64)
a(36) =         a(40) - a(44) + a(48) - a(52) + a(56) - a(60) + a(64)
a(35) = -s1/2 - a(40) + a(44) + a(47) + a(52) + a(55) + a(60) + a(63)
a(34) =         a(40) - a(44) + a(46) - a(52) + a(54) - a(60) + a(62)
a(33) = -s1/2 - a(40) + a(44) + a(45) + a(52) + a(53) + a(60) + a(61)

a(32) = s1/4 - a(60)
a(31) = s1/4 - a(59)
a(30) = s1/4 - a(58)
a(29) = s1/4 - a(57)
a(28) = s1/4 - a(64)
a(27) = s1/4 - a(63)
a(26) = s1/4 - a(62)
a(25) = s1/4 - a(61)

a(24) = s1/4 - a(52)
a(23) = s1/4 - a(51)
a(22) = s1/4 - a(50)
a(21) = s1/4 - a(49)
a(20) = s1/4 - a(56)
a(19) = s1/4 - a(55)
a(18) = s1/4 - a(54)
a(17) = s1/4 - a(53)

a(16) = s1/4 - a(44)
a(15) = s1/4 - a(43)
a(14) = s1/4 - a(42)
a(13) = s1/4 - a(41)
a(12) = s1/4 - a(48)
a(11) = s1/4 - a(47)
a(10) = s1/4 - a(46)
a( 9) = s1/4 - a(45)

a(8) = s1/4 - a(36)
a(7) = s1/4 - a(35)
a(6) = s1/4 - a(34)
a(5) = s1/4 - a(33)
a(4) = s1/4 - a(40)
a(3) = s1/4 - a(39)
a(2) = s1/4 - a(38)
a(1) = s1/4 - a(37)

The solutions can be obtained by guessing a(64) ... a(60), a(56) ... a(52), a(48) ... a(44), a(40) and filling out these guesses in the equations shown above.

Subject equations can be used for the generation of Sudoku Comparable Squares as discussed in Section 8.7.2 or Quaternary Squares as discussed in Section 8.7.4.

Examples of Rectangular Compact Pan Magic and Complete Squares based on Sudoku Comparable Squares are shown in Attachment 8.6.11.

8.6.12 Magic, Associated, Partly Rectangular Compact

The highlighted elements in enclosed square are the left-top corner points of the 4 x 2 rectangles (44 ea) summing to the Magic Sum.

In addition to this:
- The Semi Diagonals sum to the Magic Sum and
- The half columns sum to half the Magic Sum.

a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8)
a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16)
a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24)
a(25) a(26) a(27) a(28) a(29) a(30) a(31) a(32)
a(33) a(34) a(35) a(36) a(37) a(38) a(39) a(40)
a(41) a(42) a(43) a(44) a(45) a(46) a(47) a(48)
a(49) a(50) a(51) a(52) a(53) a(54) a(55) a(56)
a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64)


Partly Rectangular Compact (4 x 2) Associated Magic Squares, as defined above, are described by following linear equations:

a(57) =  s1 -   a(58) - a(59) - a(60) - a(61) - a(62) - a(63) - a(64)
a(53) =  s1 -   a(54) - a(55) - a(56) - a(61) - a(62) - a(63) - a(64)
a(52) =         a(56) - a(60) + a(64)
a(51) =         a(55) - a(59) + a(63)
a(50) =         a(54) - a(58) + a(62)
a(49) =         a(53) - a(57) + a(61)
a(45) =       - a(46) - a(47) - a(48) + a(61) + a(62) + a(63) + a(64)
a(43) =       - a(44) + a(47) + a(48) + a(59) + a(60) - a(63) - a(64)
a(41) =  s1   - a(42) - a(47) - a(48) - a(59) - a(60) - a(61) - a(62)
a(40) =  s1/2 - a(48) - a(56) - a(64)
a(39) =  s1/2 - a(47) - a(55) - a(63)
a(38) =  s1/2 - a(46) - a(54) - a(62)
a(37) = -s1/2 + a(46) + a(47) + a(48) + a(54) + a(55) + a(56) - a(61)
a(36) =  s1/2 - a(44) - a(56) - a(64)
a(35) =  s1/2 + a(44) - a(47) - a(48) - a(55) - a(59) - a(60) + a(64)
a(34) =  s1/2 - a(42) - a(54) - a(62)
a(33) =  s1/2 + a(42) + a(47) + a(48) - a(53) - a(57) - a(58) - a(61) - a(63) - a(64)

a(32) = s1/4 - a(33)
a(31) = s1/4 - a(34)
a(30) = s1/4 - a(35)
a(29) = s1/4 - a(36)
a(28) = s1/4 - a(37)
a(27) = s1/4 - a(38)
a(26) = s1/4 - a(39)
a(25) = s1/4 - a(40)

a(24) = s1/4 - a(41)
a(23) = s1/4 - a(42)
a(22) = s1/4 - a(43)
a(21) = s1/4 - a(44)
a(20) = s1/4 - a(45)
a(19) = s1/4 - a(46)
a(18) = s1/4 - a(47)
a(17) = s1/4 - a(48)

a(16) = s1/4 - a(49)
a(15) = s1/4 - a(50)
a(14) = s1/4 - a(51)
a(13) = s1/4 - a(52)
a(12) = s1/4 - a(53)
a(11) = s1/4 - a(54)
a(10) = s1/4 - a(55)
a( 9) = s1/4 - a(56)

a(8) = s1/4 - a(57)
a(7) = s1/4 - a(58)
a(6) = s1/4 - a(59)
a(5) = s1/4 - a(60)
a(4) = s1/4 - a(61)
a(3) = s1/4 - a(62)
a(2) = s1/4 - a(63)
a(1) = s1/4 - a(64)

The solutions can be obtained by guessing:

   a(64) ... a(58), a(56) ... a(54), a(48) ... a(46), a(44), a(42)

and filling out these guesses in the equations shown above.

Subject equations can be used for the generation of Sudoku Comparable Squares as discussed in Section 8.7.2 or Quaternary Squares as discussed in Section 8.7.4.

Examples of Partly Rectangular Compact Associated Magic Squares based on Sudoku Comparable Squares are shown in Attachment 8.6.12 and Attachment 15.1.3 (Bimagic).

8.6.13 Pan Magic, Complete, Parttly Rectangular Compact

The highlighted elements in enclosed square are the left-top corner points of the 2 x 4 rectangles (48 ea) summing to the Magic Sum.

In addition to this the half rows sum to half the Magic Sum.

a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8)
a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16)
a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24)
a(25) a(26) a(27) a(28) a(29) a(30) a(31) a(32)
a(33) a(34) a(35) a(36) a(37) a(38) a(39) a(40)
a(41) a(42) a(43) a(44) a(45) a(46) a(47) a(48)
a(49) a(50) a(51) a(52) a(53) a(54) a(55) a(56)
a(57) a(58) a(59) a(60) a(61) a(62) a(63) a(64)


Partly Rectangular Compact (2 x 4) Complete Pan Magic Squares, as defined above, are described by following linear equations:

a(61) =  s1/2 - a(62) - a(63) - a(64)
a(59) =  s1/2 - a(60) - a(63) - a(64)
a(57) =       - a(58) + a(63) + a(64)
a(53) =  s1/2 - a(54) - a(55) - a(56)
a(51) =  s1/2 - a(52) - a(55) - a(56)
a(50) =         a(52) - a(54) + a(56) - a(58) + a(60) - a(62) + a(64)
a(49) =       - a(52) + a(54) + a(55) + a(58) - a(60) + a(62) - a(64)
a(45) =  s1/2 - a(46) - a(47) - a(48)
a(43) =  s1/2 - a(44) - a(47) - a(48)
a(41) =       - a(42) + a(47) + a(48)
a(39) =  s1   - a(40) - a(47) - a(48) - a(55) - a(56) - a(63) - a(64)
a(38) =         a(40) - a(46) + a(48) - a(54) + a(56) - a(62) + a(64)
a(37) = -s1/2 - a(40) + a(46) + a(47) + a(54) + a(55) + a(62) + a(63)
a(36) =         a(40) - a(44) + a(48) - a(52) + a(56) - a(60) + a(64)
a(35) = -s1/2 - a(40) + a(44) + a(47) + a(52) + a(55) + a(60) + a(63)
a(34) =         a(40) - a(42) + a(48) - a(52) + a(54) - a(60) + a(62)
a(33) =  s1   - a(40) + a(42) - a(47) - 2 * a(48) + a(52) - a(54) - a(55) - a(56) + a(60) - a(62) - a(63) - a(64)

a(32) = s1/4 - a(60)
a(31) = s1/4 - a(59)
a(30) = s1/4 - a(58)
a(29) = s1/4 - a(57)
a(28) = s1/4 - a(64)
a(27) = s1/4 - a(63)
a(26) = s1/4 - a(62)
a(25) = s1/4 - a(61)

a(24) = s1/4 - a(52)
a(23) = s1/4 - a(51)
a(22) = s1/4 - a(50)
a(21) = s1/4 - a(49)
a(20) = s1/4 - a(56)
a(19) = s1/4 - a(55)
a(18) = s1/4 - a(54)
a(17) = s1/4 - a(53)

a(16) = s1/4 - a(44)
a(15) = s1/4 - a(43)
a(14) = s1/4 - a(42)
a(13) = s1/4 - a(41)
a(12) = s1/4 - a(48)
a(11) = s1/4 - a(47)
a(10) = s1/4 - a(46)
a( 9) = s1/4 - a(45)

a(8) = s1/4 - a(36)
a(7) = s1/4 - a(35)
a(6) = s1/4 - a(34)
a(5) = s1/4 - a(33)
a(4) = s1/4 - a(40)
a(3) = s1/4 - a(39)
a(2) = s1/4 - a(38)
a(1) = s1/4 - a(37)

The solutions can be obtained by guessing:

   a(64) ... a(62), a(60), a(58), a(56) ... a(54), a(52), a(48) ... a(46), a(44), a(42), a(40)

and filling out these guesses in the equations shown above.

Subject equations can be used for the generation of Sudoku Comparable Squares as discussed in Section 8.7.2 or Quaternary Squares as discussed in Section 8.7.4.

Examples of Partly Rectangular Compact Pan Magic and Complete Squares based on Sudoku Comparable Squares are shown in Attachment 8.6.13 and Attachment 15.1.2 (Bimagic).

8.6.14 Spreadsheet Solutions

The linear equations deducted above, have been applied in following Excel Spread Sheets:

  • CnstrSngl865,  Pan Magic, Compact, Associated
  • CnstrSngl8g5,  Pan Magic, Compact, Complete (Most Perfect)
  • CnstrSngl867,  Pan Magic, Non Overlapping Sub Squares, Associated (1)
  • CnstrSngl868,  Pan Magic, Non Overlapping Sub Squares, Associated (2)
  • CnstrSngl869,  Pan Magic, Non Overlapping Sub Squares, Complete
  • CnstrSngl8610, Pan Magic, Rectangular Compact, Associated
  • CnstrSngl8611, Pan Magic, Rectangular Compact, Complete
  • CnstrSngl8612,     Magic, Partly Rectangular Compact, Associated
  • CnstrSngl8613, Pan Magic, Partly Rectangular Compact, Complete

Only the red figures have to be “guessed” to construct one of the applicable 8th order Magic Squares (wrong solutions are obvious).


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