Office Applications and Entertainment, Magic Squares

7.9   Bent Diagonals

7.9.1 Introduction

Harrey White studied in 2002 Magic Squares of order 7 for which the numbers of the four Main Bent Diagonals and the Bent Diagonals parallel to it sum to the Magic Sum and found 21446 unique results for a(1) = 1 thru 22.

7.9.2 Analysis, Magic Squares, Bent Diagonals

Magic Squares for which all Bent Diagonals sum to the Magic Sum are described by following set of linear equations:

a(44) =  100 - a(45) - a(47) - a(48)
a(43) =   75 - a(46) - a(49)
a(39) =   75 - a(45) - a(47)
a(37) =  100 - a(38) - a(40) - a(41)
a(36) =      - a(42) + a(45) + a(47)
a(33) =   75 - a(41) - a(49)
a(32) =   50 - a(38) - a(40) + a(45) - a(46) + a(47)
a(31) = -100 + a(38) + a(40) + a(41) + a(46) + a(49)
a(30) =  125 - a(34) - a(38) - a(40) - a(46)
a(29) =   25 - a(35) + a(38) + a(40) - a(45) + a(46) - a(47)
a(25) =   25
a(23) =  100 - a(24) - a(26) - a(27)
a(22) =   50 - a(28)
a(21) =   75 - a(27) - a(35)
a(20) =  125 - a(26) - a(27) - a(28) - a(34)
a(19) =   25 - a(26) - a(27) + a(41) + a(49)
a(18) =   50 - a(24) - a(26) + a(38) + a(40) - a(45) + a(46) - a(47)
a(17) =  100 + a(26) + a(27) - a(38) - a(40) - a(41) - a(46) - a(49)
a(16) = -150 + a(26) + a(27) + a(28) + a(34) + a(38) + a(40) + a(46)
a(15) = - 50 + a(24) + a(26) + a(27) + a(35) - a(38) - a(40) + a(45) - a(46) + a(47)
a(14) =   25 + a(27) - a(42)
a(13) = - 25 + a(26) + a(27) + a(28) - a(41)
a(12) =        a(26) + a(27) - a(40)
a(11) = - 75 + a(24) + a(26) + a(45) + a(47)
a(10) =  100 - a(26) - a(27) - a(38)
a( 9) =   25 - a(26) - a(27) - a(28) + a(38) + a(40) + a(41)
a( 8) =  125 - a(24) - a(26) - a(27) + a(42) - a(45) - a(47)
a( 7) =   75 - a(28) - a(49)
a( 6) =   75 - a(27) - a(48)
a( 5) =   75 - a(26) - a(47)
a( 4) =   50 - a(46)
a( 3) =   75 - a(24) - a(45)
a( 2) = -125 + a(24) + a(26) + a(27) + a(45) + a(47) + a(48)
a( 1) = - 50 + a(28) + a(46) + a(49)

An optimized guessing routine (MgcSqr7d1), might produce all 171568 (= 8 * 21446) possible solutions, however not within a reasonable time.

Attachment 7.8.1 contains 252 unique solutions for a(1) = 1 and is generated with an alternative program developed by Miguel Angel Amela in 2012 (ref. Amela7).

Following sections will deal with a few - more strict defined - sub collections of subject Magic Squares.

7.9.3 Analysis, Magic Squares, Bent Diagonals,
      Symmetrical Axes and Main Diagonals

When the symmetry conditions for the Axes and Main Diagonals are added to the equations describing a Magic Square for which all Bent Diagonals sum to the Magic Sum (ref. Section 7.9.2), the resulting square is described by following set of linear equations:

a(44) =  100 - a(45) - a(47) - a(48)
a(43) =   75 - a(46) - a(49)
a(39) =   75 - a(45) - a(47)
a(37) =  100 - a(38) - a(40) - a(41)
a(36) =      - a(42) + a(45) + a(47)
a(33) =   75 - a(41) - a(49)
a(32) =   50 - a(38) - a(40) + a(45) - a(46) + a(47)
a(31) = -100 + a(38) + a(40) + a(41) + a(46) + a(49)
a(30) =  125 - a(34) - a(38) - a(40) - a(46)
a(29) =   25 - a(35) + a(38) + a(40) - a(45) + a(46) - a(47)
a(28) =  100 - a(46) - 2 * a(49)
a(26) = -125 - a(27) + a(38) + a(40) + 2 * a(41) + a(46) + 2 * a(49)
a(25) =   25
a(21) =   75 - a(27) - a(35)
a(20) =  150 - a(34) - a(38) - a(40) - 2 * a(41)
a(16) = -175 + a(34) + 2 * a(38) + 2 * a(40) + 2 * a(41) + a(46)
a(15) =        a(27) + a(35) - a(38) - a(40) + a(45) - a(46) + a(47)
a(14) =   25 + a(27) - a(42)
a(12) = -125 + a(38) + 2 * a(41) + a(46) + 2 * a(49)
a(10) =  225 - 2 * a(38) - a(40) - 2 * a(41) - a(46) - 2 * a(49)
a( 8) =   75 - a(27) + a(42) - a(45) - a(47)
a( 6) =   75 - a(27) - a(48)
a( 5) =  200 + a(27) - a(38) - a(40) - 2 * a(41) - a(46) - a(47) - 2 * a(49)
a( 3) = -100 - a(27) + a(38) + a(40) + 2 * a(41) - a(45) + a(46) + 2 * a(49)
a( 2) =   25 + a(27) - a(44)

An optimized guessing routine (MgcSqr7d2), produced the 400 (= 8 * 50) possible solutions, which are shown in Attachment 7.8.2a.

7.9.4 Analysis, Pan Magic Squares, Bent Diagonals

Pan Magic Squares for which all Bent Diagonals sum to the Magic Sum are described by following set of linear equations:

a(44) =  100 - a(45) - a(47) - a(48)
a(43) =   75 - a(46) - a(49)
a(39) =   75 - a(45) - a(47)
a(37) =  100 - a(38) - a(40) - a(41)
a(36) =      - a(42) + a(45) + a(47)
a(35) =   50 - a(41) + a(42) - a(47) + a(48) - a(49)
a(34) =   75 - a(40) - a(42) - a(48) + a(49)
a(33) =   75 - a(41) - a(49)
a(32) =   50 - a(38) - a(40) + a(45) - a(46) + a(47)
a(31) = -100 + a(38) + a(40) + a(41) + a(46) + a(49)
a(30) =   50 - a(38) + a(42) - a(46) + a(48) - a(49)
a(29) = - 25 + a(38) + a(40) + a(41) - a(42) - a(45) + a(46) - a(48) + a(49)
a(25) =   25
a(23) =  100 - a(24) - a(26) - a(27)
a(22) =   50 - a(28)
a(21) =   25 - a(27) + a(41) - a(42) + a(47) - a(48) + a(49)
a(20) =   50 - a(26) - a(27) - a(28) + a(40) + a(42) + a(48) - a(49)
a(19) =   25 - a(26) - a(27) + a(41) + a(49)
a(18) =   50 - a(24) - a(26) + a(38) + a(40) - a(45) + a(46) - a(47)
a(17) =  100 + a(26) + a(27) - a(38) - a(40) - a(41) - a(46) - a(49)
a(16) = - 75 + a(26) + a(27) + a(28) + a(38) - a(42) + a(46) - a(48) + a(49)
a(15) =        a(24) + a(26) + a(27) - a(38) - a(40) - a(41) + a(42) + a(45) - a(46) + a(48) - a(49)
a(14) =   25 + a(27) - a(42)
a(13) = - 25 + a(26) + a(27) + a(28) - a(41)
a(12) =        a(26) + a(27) - a(40)
a(11) = - 75 + a(24) + a(26) + a(45) + a(47)
a(10) =  100 - a(26) - a(27) - a(38)
a( 9) =   25 - a(26) - a(27) - a(28) + a(38) + a(40) + a(41)
a( 8) =  125 - a(24) - a(26) - a(27) + a(42) - a(45) - a(47)
a( 7) =   75 - a(28) - a(49)
a( 6) =   75 - a(27) - a(48)
a( 5) =   75 - a(26) - a(47)
a( 4) =   50 - a(46)
a( 3) =   75 - a(24) - a(45)
a( 2) = -125 + a(24) + a(26) + a(27) + a(45) + a(47) + a(48)
a( 1) = - 50 + a(28) + a(46) + a(49)

An optimized guessing routine (MgcSqr7d3), produced the 112 (= 8 * 14) possible solutions, which are shown in Attachment 7.8.3.

7.9.5 Analysis, Pan Magic Squares, Bent Diagonals,
      Symmetrical Axes and Main Diagonals

When the symmetry conditions for the Axes and Main Diagonals are added to the equations describing a Pan Magic Square for which all Bent Diagonals sum to the Magic Sum (ref. Section 7.9.4), the resulting square is described by following set of linear equations:

a(44) =  100 - a(45) - a(47) - a(48)
a(43) =   75 - a(46) - a(49)
a(39) =   75 - a(45) - a(47)
a(37) =  100 - a(38) - a(40) - a(41)
a(36) =      - a(42) + a(45) + a(47)
a(35) =   50 - a(41) + a(42) - a(47) + a(48) - a(49)
a(34) =   75 - a(40) - a(42) - a(48) + a(49)
a(32) =   50 - a(38) - a(40) + a(45) - a(46) + a(47)
a(31) = -100 + a(38) + a(40) + a(41) + a(46) + a(49)
a(30) =   50 - a(38) + a(42) - a(46) + a(48) - a(49)
a(29) = - 25 + a(38) + a(40) + a(41) - a(42) - a(45) + a(46) - a(48) + a(49)
a(28) =  100 - a(46) - 2 * a(49)
a(26) = -125 - a(27) + a(38) + a(40) + 2 * a(41) + a(46) + 2 * a(49)
a(21) =   25 - a(27) + a(41) - a(42) + a(47) - a(48) + a(49)
a(20) =   75 - a(38) - 2 * a(41) + a(42) + a(48) - a(49)
a(16) = -100 + 2 * a(38) + a(40) + 2 * a(41) - a(42) + a(46) - a(48) + a(49)
a(15) =   50 + a(27) - a(38) - a(40) - a(41) + a(42) + a(45) - a(46) + a(48) - a(49)
a(14) =   25 + a(27) - a(42)
a(12) = -125 + a(38) + 2 * a(41) + a(46) + 2 * a(49)
a(10) =  225 - 2 * a(38) - a(40) - 2 * a(41) - a(46) - 2 * a(49)
a( 8) =   75 - a(27) + a(42) - a(45) - a(47)
a( 6) =   75 - a(27) - a(48)
a( 5) =  200 + a(27) - a(38) - a(40) - 2 * a(41) - a(46) - a(47) - 2 * a(49)
a( 3) = -100 - a(27) + a(38) + a(40) + 2 * a(41) - a(45) + a(46) + 2 * a(49)
a( 2) = - 75 + a(27) + a(45) + a(47) + a(48)

An optimized guessing routine (MgcSqr7d4), produced the 16 (= 8 * 2) possible solutions, which are shown in Attachment 7.8.4.

7.9.6 Analysis, Magic Squares, Bent Diagonals,
      Almost Associated

It can be proven that Magic Squares for which all Bent Diagonals sum to the Magic Sum can't be Associated.

However when the associated pairs:

   {a(2), a(48)}, {a(6), a(44)}, {a(8), a(42)} and {a(14), a(36)}

are replaced by:

   {a(2), a(8)}, {a(6), a(14)}, {a(36), a(44)} and {a(42), a(48)}

and subject conditions are added to the equations describing a Magic Square for which all Bent Diagonals sum to the Magic Sum (ref. Section 7.9.2), the resulting square is described by following set of linear equations:

a(44) =  100 - a(45) - a(47) - a(48)
a(43) =   75 - a(46) - a(49)
a(39) =   75 - a(45) - a(47)
a(38) = (175 - 2 * a(41) -a(46) - 2 * a(49))/2
a(37) =  100 - a(38) - a(40) - a(41)
a(35) =  100 - a(41) - a(47) - a(49)
a(34) =   25 - a(40) + a(49)
a(33) =   75 - a(41) - a(49)
a(32) =   50 - a(38) - a(40) + a(45) - a(46) + a(47)
a(31) =   75 - a(38) + a(40) - a(41) - a(49)
a(30) =   25 - a(37) - a(40) + a(41) + a(49)
a(29) =  100 - a(38) + a(40) - a(41) - a(45) - a(49)
a(28) =  100 - a(46) - 2 * a(49)
a(27) =   25 - a(38) + a(40) - a(45) + a(47)
a(26) =   25 + a(45) - a(47)
a(14) =      - a(38) + a(40) - a(45) + a(47) + a(48)
a( 8) =  100 - a(14) - a(45) - a(47)

An optimized guessing routine (MgcSqr7d5), produced the 16 (= 8 * 2) possible solutions, which are shown in Attachment 7.8.2b.

7.9.7 Additional Properties

Consequential properties resulting from the defining properties of 7^th order Magic Squares for which the Bent Diagonals sum to the Magic Sum are summarised and illustrated in Attachment 7.8.5.

7.9.8 Spreadsheet Solutions

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

• CnstrSngl7g1, Magic Squares of order 7, Bent Diagonals
  
  
• CnstrSngl7g2, Magic Squares of order 7, Bent Diagonals,
                Symmetrical Axes and Main Diagonals
  
  
• CnstrSngl7g3, Pan Magic Squares of order 7, Bent Diagonals
  
  
• CnstrSngl7g4, Pan Magic Squares of order 7, Bent Diagonals,
                Symmetrical Axes and Main Diagonals

Only the red figures have to be “guessed” to construct one of the applicable Magic Squares of the 7^th order (wrong solutions are obvious).