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9.8   Quadrant Magic Squares

The concept of Quadrant Magic Squares, as discussed in following sections for order 9 Magic Squares, was introduced by Harvey Heinz (2001/2002).

9.8.1 Definition and Terminology

An order 9 magic square can be divided into four overlapping quadrants of 5 x 5 cells.

Each quadrant might contain symmetric patterns of 9 cells (8 cells + the centre cell) which sum to the Magic Sum s1.

For order 9 magic squares seven patterns can be recognised, as illustrated below:

P1 (Plus Magic)
o o o o o
o o o o o
o o o o o
o o o o o
o o o o o
P2
o o o o o
o o o o o
o o o o o
o o o o o
o o o o o
P3 (Diamond Magic)
o o o o o
o o o o o
o o o o o
o o o o o
o o o o o
P4
o o o o o
o o o o o
o o o o o
o o o o o
o o o o o
P5
o o o o o
o o o o o
o o o o o
o o o o o
o o o o o
P6 (Cross Magic)
o o o o o
o o o o o
o o o o o
o o o o o
o o o o o
P7
o o o o o
o o o o o
o o o o o
o o o o o
o o o o o

further referred to as Magic Pattern P1 thru P7.

An order 9 Magic Square is called Quadrant Magic if the same pattern(s) occur in all four quadrants.

9.8.2 Equations Quadrant Magic Patterns

The quadrant properties defined in Section 9.8.1 above can be described by following linear equations:

a( 3) + a(12) + a(21) + a(30) + a(39) + a(19) + a(20) + a(22) + a(23) = s1 P1 Magic
a( 7) + a(16) + a(25) + a(34) + a(43) + a(23) + a(24) + a(26) + a(27) = s1
a(39) + a(48) + a(57) + a(66) + a(75) + a(55) + a(56) + a(58) + a(59) = s1
a(43) + a(52) + a(61) + a(70) + a(79) + a(59) + a(60) + a(62) + a(63) = s1

a(11) + a(12) + a(13) + a(20) + a(21) + a(22) + a(29) + a(30) + a(31) = s1 P2 Magic
a(15) + a(16) + a(17) + a(24) + a(25) + a(26) + a(33) + a(34) + a(35) = s1
a(47) + a(48) + a(49) + a(56) + a(57) + a(58) + a(65) + a(66) + a(67) = s1
a(51) + a(52) + a(53) + a(60) + a(61) + a(62) + a(69) + a(70) + a(71) = s1

a( 3) + a(11) + a(13) + a(19) + a(21) + a(23) + a(29) + a(31) + a(39) = s1 P3 Magic
a( 7) + a(15) + a(17) + a(23) + a(25) + a(27) + a(33) + a(35) + a(43) = s1
a(39) + a(47) + a(49) + a(55) + a(57) + a(59) + a(65) + a(67) + a(75) = s1
a(43) + a(51) + a(53) + a(59) + a(61) + a(63) + a(69) + a(71) + a(79) = s1

a( 1) + a(12) + a( 5) + a(20) + a(21) + a(22) + a(37) + a(30) + a(41) = s1 P4 Magic
a( 5) + a(16) + a( 9) + a(24) + a(25) + a(26) + a(41) + a(34) + a(45) = s1
a(37) + a(48) + a(41) + a(56) + a(57) + a(58) + a(73) + a(66) + a(77) = s1
a(41) + a(52) + a(45) + a(60) + a(61) + a(62) + a(77) + a(70) + a(81) = s1

a( 1) + a( 3) + a( 5) + a(19) + a(21) + a(23) + a(37) + a(39) + a(41) = s1 P5 Magic
a( 5) + a( 7) + a( 9) + a(23) + a(25) + a(27) + a(41) + a(43) + a(45) = s1
a(37) + a(39) + a(41) + a(55) + a(57) + a(59) + a(73) + a(75) + a(77) = s1
a(41) + a(43) + a(45) + a(59) + a(61) + a(63) + a(77) + a(79) + a(81) = s1

a( 1) + a( 5) + a(11) + a(13) + a(21) + a(29) + a(31) + a(37) + a(41) = s1 P6 Magic
a( 5) + a( 9) + a(15) + a(17) + a(25) + a(33) + a(35) + a(41) + a(45) = s1
a(37) + a(41) + a(47) + a(49) + a(57) + a(65) + a(67) + a(73) + a(77) = s1
a(41) + a(45) + a(51) + a(53) + a(61) + a(69) + a(71) + a(77) + a(81) = s1

a( 2) + a( 4) + a(10) + a(14) + a(21) + a(28) + a(32) + a(38) + a(40) = s1 P7 Magic
a( 6) + a( 8) + a(14) + a(18) + a(25) + a(32) + a(36) + a(42) + a(44) = s1
a(38) + a(40) + a(46) + a(50) + a(57) + a(64) + a(68) + a(74) + a(76) = s1
a(42) + a(44) + a(50) + a(54) + a(61) + a(68) + a(72) + a(78) + a(80) = s1

which can be combined with the equations describing miscellaneous types of order 9 Magic Squares.

9.8.3 P1 Magic

Quadrant P1 Magic Squares, with order 4 and order 5 Square Inlays, can be obtained by means of a transformation as described in Section 9.7.4 and illustrated below:

Composed
a1 a2 a3 a4 b1 b2 b3 b4 b5
a5 a6 a7 a8 b6 b7 b8 b9 b10
a9 a10 a11 a12 b11 b12 b13 b14 b15
a13 a14 a15 a16 b16 b17 b18 b19 b20
c1 c2 c3 c4 d1 d2 d3 d4 d5
c5 c6 c7 c8 d6 d7 d8 d9 d10
c9 c10 c11 c12 d11 d12 d13 d14 d15
c13 c14 c15 c16 d16 d17 d18 d19 d20
c17 c18 c19 c20 d21 d22 d23 d24 d25
= > Quadrant P1 Magic
d1 c1 d2 c2 d3 c3 d4 c4 d5
b1 a1 b2 a2 b3 a3 b4 a4 b5
d6 c5 d7 c6 d8 c7 d9 c8 d10
b6 a5 b7 a6 b8 a7 b9 a8 b10
d11 c9 d12 c10 d13 c11 d14 c12 d15
b11 a9 b12 a10 b13 a11 b14 a12 b15
d16 c13 d17 c14 d18 c15 d19 c16 d20
b16 a13 b17 a14 b18 a15 b19 a16 b20
d21 c17 d22 c18 d23 c19 d24 c20 d25

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

  • One 4th order Associated Magic Corner Square A with Magic Sum s4 = 164

  • One 5th order Associated Quadrant Plus Magic Corner Square D - as discussed in Section 5.6.4 - with Magic Sum s5 = 205 for which:

    d2  + d6  + d7  + d8  + d12 = s5
    d4  + d8  + d9  + d10 + d14 = s5
    d12 + d16 + d17 + d18 + d22 = s5
    d14 + d18 + d19 + d20 + d24 = s5

  • One Row Symmetric Magic Rectangle B order 4 x 5 with s4 = 164 and s5 = 205, for which:

    b1  + b6  = b2  + b7  = b3  + b8  = b4  + b9  = b5  + b10 = 82 and
    b11 + b16 = b12 + b17 = b13 + b18 = b14 + b19 = b15 + b20 = 82

  • One Column Symmetric Magic Rectangle C order 5 x 4 with s4 = 164 and s5 = 205, for which:

    c1 + c2 = c5 + c6 = c9  + c10 = c13 + c14 = c17 + c18 = 82 and
    c3 + c4 = c7 + c8 = c11 + c12 = c15 + c16 = c19 + c20 = 82

The transformed square shown at the right side above is a Quadrant P1 Magic Square as:

d2  + d6  + d7  + d8  + d12 + b2  + b7  + c5  + c6  = s9
d4  + d8  + d9  + d10 + d14 + b4  + b9  + c7  + c8  = s9
d12 + d16 + d17 + d18 + d22 + b12 + b17 + c13 + c14 = s9
d14 + d18 + d19 + d20 + d24 + b14 + b19 + c15 + c16 = s9

Attachment 9.8.33 shows a few Quadrant P1 Magic Squares, with symmetrical diagonals, which could be constructed based on the method described above.

9.8.4 P2, P5 Magic

All 3 x 3 Compact Magic Squares, of which a few sub collections have been discussed in Section 9.4 and Section 9.5 are Quadrant P2 Magic.

From the linear equations describing 3 x 3 Compact Magc Squares can be deducted that:

a( 1) + a( 3) + a( 5) + a(19) + a(21) + a(23) + a(37) + a(39) + a(41) = s1
a( 5) + a( 7) + a( 9) + a(23) + a(25) + a(27) + a(41) + a(43) + a(45) = s1
a(37) + a(39) + a(41) + a(55) + a(57) + a(59) + a(73) + a(75) + a(77) = s1
a(41) + a(43) + a(45) + a(59) + a(61) + a(63) + a(77) + a(79) + a(81) = s1

Which are the defining equations for Quadrant P5 Magic Squares. Consequently all (3 x 3) Compact Magic Squares are Quadrant (P2, P5) Magic.

Attachment 9.8.41 shows a few Quadrant (P2, P5) Magic Square selected from the collection of Associated Compact Pan Magic Squares as discussed in Section 9.4.2.

9.8.5 P3, P4 Magic

Based on the equations defining the Quadrant P3 and P4 properties:

a( 3) + a(11) + a(13) + a(19) + a(21) + a(23) + a(29) + a(31) + a(39) = s1 P3 Magic
a( 7) + a(15) + a(17) + a(23) + a(25) + a(27) + a(33) + a(35) + a(43) = s1
a(39) + a(47) + a(49) + a(55) + a(57) + a(59) + a(65) + a(67) + a(75) = s1
a(43) + a(51) + a(53) + a(59) + a(61) + a(63) + a(69) + a(71) + a(79) = s1

a( 1) + a(12) + a( 5) + a(20) + a(21) + a(22) + a(37) + a(30) + a(41) = s1 P4 Magic
a( 5) + a(16) + a( 9) + a(24) + a(25) + a(26) + a(41) + a(34) + a(45) = s1
a(37) + a(48) + a(41) + a(56) + a(57) + a(58) + a(73) + a(66) + a(77) = s1
a(41) + a(52) + a(45) + a(60) + a(61) + a(62) + a(77) + a(70) + a(81) = s1

Sudoku Comparable Quadrant (P2, P3, P4, P5) Pan Magic Squares can be filtered from the 1152 Sudoku Comparable Compact Associated Pan Magic Squares as deducted in Section 9.5.3.

Attachment 9.8.51 shows the 64 Sudoku Comparable Quadrant (P2, P3, P4, P5) Pan Magic Square as filtered from subject collection (ref. Attachment 9.6.11).

Routine CnstrSqrs94, in which the relations ensuring unique magic squares:

    a(81) < a(73), a(9), a(1)
    a(73) < a(9)

have been incorporated, generated 64 unique Quadrant (P2, P3, P4, P5) Pan Magic Squares within 6,25 seconds, which are shown in Attachment 9.8.52.

Note: The patterns highlighted in the first four squares, occur all four in every square shown.

9.8.6 P6 Magic

Associated Quadrant P6 Magic Squares, with order 4 and order 5 Square Inlays, can be obtained by means of a transformation as described in Section 9.7.4 and illustrated below:

Composed
a1 a2 a3 a4 b1 b2 b3 b4 b5
a5 a6 a7 a8 b6 b7 b8 b9 b10
a9 a10 a11 a12 b11 b12 b13 b14 b15
a13 a14 a15 a16 b16 b17 b18 b19 b20
c1 c2 c3 c4 d1 d2 d3 d4 d5
c5 c6 c7 c8 d6 d7 d8 d9 d10
c9 c10 c11 c12 d11 d12 d13 d14 d15
c13 c14 c15 c16 d16 d17 d18 d19 d20
c17 c18 c19 c20 d21 d22 d23 d24 d25
= > Quadrant P6 Magic
d1 c1 d2 c2 d3 c3 d4 c4 d5
b1 a1 b2 a2 b3 a3 b4 a4 b5
d6 c5 d7 c6 d8 c7 d9 c8 d10
b6 a5 b7 a6 b8 a7 b9 a8 b10
d11 c9 d12 c10 d13 c11 d14 c12 d15
b11 a9 b12 a10 b13 a11 b14 a12 b15
d16 c13 d17 c14 d18 c15 d19 c16 d20
b16 a13 b17 a14 b18 a15 b19 a16 b20
d21 c17 d22 c18 d23 c19 d24 c20 d25

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

  • One 4th order Associated Magic Corner Square A with Magic Sum s4 = 164, for which:

    a1  + a2  + a5  + a6  = s4
    a3  + a4  + a7  + a8  = s4
    a9  + a10 + a13 + a14 = s4
    a11 + a12 + a15 + a16 = s4

  • One 5th order Associated Quadrant Cross Magic Corner Square D - as discussed in Section 5.6.4 - with Magic Sum s5 = 205 for which:

    d1  + d3  + d7  + d11 + d13 = s5
    d3  + d5  + d9  + d13 + d15 = s5
    d11 + d13 + d17 + d21 + d23 = s5
    d13 + d15 + d19 + d23 + d25 = s5

  • Two Associated Magic Rectangles B/C order 4 x 5 with s4 = 164 and s5 = 205

The transformed square shown at the right side above is an Associated Quadrant P6 Magic Square as:

a1  + a2  + a5  + a6  + d1  + d3  + d7  + d11 + d13 = s9
a3  + a4  + a7  + a8  + d3  + d5  + d9  + d13 + d15 = s9
a9  + a10 + a13 + a14 + d11 + d13 + d17 + d21 + d23 = s9
a11 + a12 + a15 + a16 + d13 + d15 + d19 + d23 + d25 = s9

Attachment 9.8.61 shows a few Associated Quadrant P6 Magic Squares which could be constructed based on the method described above.

9.8.7 P1, P6 Magic

Also Quadrant (P1, P6) Magic Squares, with order 4 and order 5 Square Inlays, can be obtained with the method described above:

Composed
a1 a2 a3 a4 b1 b2 b3 b4 b5
a5 a6 a7 a8 b6 b7 b8 b9 b10
a9 a10 a11 a12 b11 b12 b13 b14 b15
a13 a14 a15 a16 b16 b17 b18 b19 b20
c1 c2 c3 c4 d1 d2 d3 d4 d5
c5 c6 c7 c8 d6 d7 d8 d9 d10
c9 c10 c11 c12 d11 d12 d13 d14 d15
c13 c14 c15 c16 d16 d17 d18 d19 d20
c17 c18 c19 c20 d21 d22 d23 d24 d25
= > Quadrant (P1, P6) Magic
d1 c1 d2 c2 d3 c3 d4 c4 d5
b1 a1 b2 a2 b3 a3 b4 a4 b5
d6 c5 d7 c6 d8 c7 d9 c8 d10
b6 a5 b7 a6 b8 a7 b9 a8 b10
d11 c9 d12 c10 d13 c11 d14 c12 d15
b11 a9 b12 a10 b13 a11 b14 a12 b15
d16 c13 d17 c14 d18 c15 d19 c16 d20
b16 a13 b17 a14 b18 a15 b19 a16 b20
d21 c17 d22 c18 d23 c19 d24 c20 d25

The (Semi) Magic Square shown at the left side above should for this case be composed out of:

  • One 4th order Associated Magic Corner Square A with Magic Sum s4 = 164, for which:

    a1  + a2  + a5  + a6  = s4
    a3  + a4  + a7  + a8  = s4
    a9  + a10 + a13 + a14 = s4
    a11 + a12 + a15 + a16 = s4

  • One 5th order Ultra Magic Magic Corner Square D, Plus and Cross Magic as discussed in Section 5.6.4, with Magic Sum s5 = 205 for which:

    d2  + d6  + d7  + d8  + d12 = s5 Plus Magic
    d4  + d8  + d9  + d10 + d14 = s5
    d12 + d16 + d17 + d18 + d22 = s5
    d14 + d18 + d19 + d20 + d24 = s5

    d1  + d3  + d7  + d11 + d13 = s5 Cross Magic
    d3  + d5  + d9  + d13 + d15 = s5
    d11 + d13 + d17 + d21 + d23 = s5
    d13 + d15 + d19 + d23 + d25 = s5

  • One (Partly) Row Symmetric Magic Rectangle B order 4 x 5 with s4 = 164 and s5 = 205, for which at least:

    b2 + b7 = b4  + b9  = b12 + b17 = b14 + b19 = 82

  • One (Partly) Column Symmetric Magic Rectangle C order 5 x 4 with s4 = 164 and s5 = 205, for which at least:

    c5 + c6 = c13 + c14 = c7  + c8  = c15 + c16 = 82

The transformed square shown at the right side above is a Quadrant (P1, P6) Magic Square as:

d2  + d6  + d7  + d8  + d12 + b2  + b7  + c5  + c6  = s9 P1 Magic
d4  + d8  + d9  + d10 + d14 + b4  + b9  + c7  + c8  = s9
d12 + d16 + d17 + d18 + d22 + b12 + b17 + c13 + c14 = s9
d14 + d18 + d19 + d20 + d24 + b14 + b19 + c15 + c16 = s9

a1  + a2  + a5  + a6  + d1  + d3  + d7  + d11 + d13 = s9 P6 Magic
a3  + a4  + a7  + a8  + d3  + d5  + d9  + d13 + d15 = s9
a9  + a10 + a13 + a14 + d11 + d13 + d17 + d21 + d23 =s9
a11 + a12 + a15 + a16 + d13 + d15 + d19 + d23 + d25 = s9

Attachment 9.8.62 shows a few Quadrant (P1, P6) Magic Squares, with symmetrical diagonals, which could be constructed based on the method described above.

Note: The patterns highlighted in the first two squares, occur both in every square shown.

9.8.8 P1, P2, P5, P7 Magic

When the equations defining the Quadrant P7 property:

a( 2) + a( 4) + a(10) + a(14) + a(21) + a(28) + a(32) + a(38) + a(40) = s1 P7 Magic
a( 6) + a( 8) + a(14) + a(18) + a(25) + a(32) + a(36) + a(42) + a(44) = s1
a(38) + a(40) + a(46) + a(50) + a(57) + a(64) + a(68) + a(74) + a(76) = s1
a(42) + a(44) + a(50) + a(54) + a(61) + a(68) + a(72) + a(78) + a(80) = s1

are added to the defining equations of order 9 Associated Compact Magic Squares, the resulting Quadrant P7 Magic Square is described by following equations: 

a(73) =     s1   - a(74) - a(75) - a(76) - a(77) - a(78) - a(79) - a(80) - a(81)
a(44) =     s1/9 + a(74) - a(80)
a(69) = -   s1   + a(72) + a(74) + a(75) + 2 * a(76) + a(77) + a(79) + a(80) + 2 * a(81)
a(66) = -   s1   + a(72) + a(74) + a(76) + a(77) + a(78) + 2 * a(79) + a(80) + 2 * a(81)
a(70) =  12*s1/9 - a(71) - a(72) - a(76) - a(77) - a(78) - 2 * a(79) - 2 * a(80) - 2 * a(81)
a(68) =            a(71) - a(74) + a(80)
a(67) =  12*s1/9 - a(71) - a(72) - a(75) - 2 * a(76) - a(77) - a(79) - 2 * a(80) - 2 * a(81)
a(65) =            a(71) - 2 * a(74) + 2 * a(80)
a(64) =   3*s1/9 - a(71) - a(72) + a(74) - a(80)
a(60) =     s1   + a(63) - a(74) - a(75) - 2 * a(76) - a(77) - a(78) - a(79) - a(80) - a(81)
a(57) =     s1   + a(63) - a(74) - a(75) - a(76) - a(77) - a(78) - 2 * a(79) - a(80) - a(81)
a(61) = - 3*s1/9 - a(62) - a(63) + a(76) + a(77) + a(78) + a(79) + a(80) + a(81)
a(59) =            a(62) + a(74) - a(77)
a(58) = - 3*s1/9 - a(62) - a(63) + a(75) + a(76) + a(77) + a(79) + a(80) + a(81)
a(56) =            a(62) + a(74) - a(80)
a(55) = - 3*s1/9 - a(62) - a(63) + a(75) + a(76) + a(77) + a(78) + a(79) + a(80)
a(54) = -   s1/9 - 2 * a(62) - 3 * a(63) - a(72) + 2 * a(76) + 2 * a(77) + a(78) + a(79) + a(80) + a(81)
a(53) = - 4*s1/9 + a(62) - a(71) + 2 * a(74) + 2 * a(75) - a(77) + 2 * a(79)
a(52) =   5*s1/9 + a(62) + 3*a(63) + a(71) + a(72) - 2*a(74) - 2*a(75) - 2*a(76) - a(77) - a(78) - 2*a(79)
a(51) = -   s1/9 - 2 * a(62) - 3 * a(63) - a(72) + 2 * a(76) + 2*a(77) + 2*a(78) + a(79) + a(80)
a(50) = - 4*s1/9 + a(62) - a(71) + 2 * a(74) + 2 * a(75) + 2 * a(79) - a(80)
a(49) =   5*s1/9 + a(62) + 3*a(63) + a(71) + a(72) - 2*a(74) - 2*a(75) - a(76) - a(77) - a(78) - 3*a(79)
a(48) = -   s1/9 - 2 * a(62) - 3 * a(63) - a(72) + a(75) + 2 * a(76) + 2 * a(77) + a(78) + a(79) + a(80)
a(47) = - 4*s1/9 + a(62) - a(71) + 3 * a(74) + 2 * a(75) - a(77) + 2 * a(79) - a(80)
a(46) =  14*s1/9 + a(62) + 3*a(63) + a(71) + a(72) - 3*a(74) - 3*a(75) - 3*a(76) +
                                                             - 2*a(77) - 2*a(78) - 4*a(79) - a(80) - a(81)
a(45) =  25*s1/9 + 2*a(62) + 4*a(63) - 3*a(74) - 3*a(75) - 5*a(76) - 4*a(77) +
                                                                   - 3*a(78) - 5*a(79) - 3*a(80) - 4*a(81)
a(43) = -14*s1/9 - 2*a(62) - 4*a(63) + 2*a(74) + 3*a(75) + 4*a(76) + 3*a(77) +
                                                                   + 2*a(78) + 3*a(79) + 2*a(80) + 2*a(81)
a(42) =  16*s1/9 + 2*a(62) + 4*a(63) - 2*a(74) - 2*a(75) - 3*a(76) - 3*a(77) +
                                                                   - 3*a(78) - 4*a(79) - 2*a(80) - 2*a(81)
a(41) =     s1/9

a(40) = 2*s1/9 - a(42)
a(39) = 2*s1/9 - a(43)
a(38) = 2*s1/9 - a(44)
a(37) = 2*s1/9 - a(45)
a(36) = 2*s1/9 - a(46)
a(35) = 2*s1/9 - a(47)
a(34) = 2*s1/9 - a(48)
a(33) = 2*s1/9 - a(49)
a(32) = 2*s1/9 - a(50)
a(31) = 2*s1/9 - a(51)

a(30) = 2*s1/9 - a(52)
a(29) = 2*s1/9 - a(53)
a(28) = 2*s1/9 - a(54)
a(27) = 2*s1/9 - a(55)
a(26) = 2*s1/9 - a(56)
a(25) = 2*s1/9 - a(57)
a(24) = 2*s1/9 - a(58)
a(23) = 2*s1/9 - a(59)
a(22) = 2*s1/9 - a(60)
a(21) = 2*s1/9 - a(61)

a(20) = 2*s1/9 - a(62)
a(19) = 2*s1/9 - a(63)
a(18) = 2*s1/9 - a(64)
a(17) = 2*s1/9 - a(65)
a(16) = 2*s1/9 - a(66)
a(15) = 2*s1/9 - a(67)
a(14) = 2*s1/9 - a(68)
a(13) = 2*s1/9 - a(69)
a(12) = 2*s1/9 - a(70)
a(11) = 2*s1/9 - a(71)

a(10) = 2*s1/9 - a(72)
a( 9) = 2*s1/9 - a(73)
a( 8) = 2*s1/9 - a(74)
a( 7) = 2*s1/9 - a(75)
a( 6) = 2*s1/9 - a(76)
a( 5) = 2*s1/9 - a(77)
a( 4) = 2*s1/9 - a(78)
a( 3) = 2*s1/9 - a(79)
a( 2) = 2*s1/9 - a(80)
a( 1) = 2*s1/9 - a(81)

From the equations shown above it can be deducted that:

a( 2) + a( 4) + a(10) + a(14) + a(21) + a(28) + a(32) + a(38) + a(40) = s1
a( 6) + a( 8) + a(14) + a(18) + a(25) + a(32) + a(36) + a(42) + a(44) = s1
a(38) + a(40) + a(46) + a(50) + a(57) + a(64) + a(68) + a(74) + a(76) = s1
a(42) + a(44) + a(50) + a(54) + a(61) + a(68) + a(72) + a(78) + a(80) = s1

Which are the defining equations for Quadrant P1 Magic Squares. Consequently all Quadrant P7 Associated Compact Magic Squares are Quadrant (P1, P7) Magic.

Attachment 9.8.72 shows a few Associated Compact Quadrant (P1, P2, P5, P7) Magic Squares (ref. MgcSqr97a).

Note: The patterns highlighted in the first four squares, occur all four in every square shown.

9.8.9 Multi Magic

(Non Consecutive Integers)

When the equations defining the seven quadrant magic patterns (P1 ... P7), as listed in Section 9.8.2 above, are added to the equations defining order 9 Associated Magic Squares, the resulting Quadrant Multi Magic Square is described by following equations: 

a(73) =       s1 - a(74) - a(75) - a(76) - a(77) - a(78) - a(79) - a(80) - a(81)
a(64) =       s1 - a(65) - a(66) - a(67) - a(68) - a(69) - a(70) - a(71) - a(72)
a(61) = (- 37*s1/9 +  6*a(74) +  6*a(75) +  6*a(76) -  6*a(77) + 10*a(78) + 10*a(79) + 10*a(80) + 2*a(81))/7
a(60) = (  76*s1/9 - 14*a(62) -  7*a(63) - 11*a(74) - 11*a(75) - 11*a(76) + 11*a(77) +
                                                               -  9*a(78) -  2*a(79) -  9*a(80) +   a(81))/14
a(57) = (- 19*s1/9 +  8*a(74) +  8*a(75) +  8*a(76) -  8*a(77) +  4*a(78) +  4*a(79) +  4*a(80) - 2*a(81))/7
a(56) = (   8*s1/9 - 14*a(58) + 14*a(59) +  7*a(63) -  3*a(74) + 11*a(75) -  3*a(76) +
                                                    +  3*a(77) -  5*a(78) +  2*a(79) -  5*a(80) -   a(81)) /14
a(55) = (  77*s1/9 - 14*a(59) -  7*a(63) -  7*a(74) - 14*a(75) -  7*a(76) +  7*a(77) +
                                                                          -  7*a(78) - 14*a(79) - 7*a(80))/7
a(54) = (- 39*s1/9 -  7*a(63) -  7*a(72) + 12*a(74) + 12*a(75) + 12*a(76) - 12*a(77) +
                                                               + 13*a(78) + 13*a(79) + 13*a(80) - 3*a(81))/7
a(52) = (  24*s1/9 +  7*a(63) - 14*a(70) +  5*a(74) +  5*a(75) +  5*a(76) -  5*a(77) +
                                                                      - a(78) - 8*a(79) - a(80) - 3*a(81))/14
a(51) = (  50*s1/9 -  7*a(53) -  7*a(69) -  7*a(71) -  3*a(74) -  3*a(75) -  3*a(76) +
                                                    +  3*a(77) -  5*a(78) -  5*a(79) -  5*a(80) -   a(81))/7
a(50) = (- 84*s1/9 -  7*a(59) -  7*a(68) + 28*a(73) + 49*a(77) + 28*a(81))/7
a(48) = (  74*s1/9 - 14*a(59) -  7*a(63) - 14*a(66) -  5*a(74) - 19*a(75) -  5*a(76) +
                                                    +  5*a(77) +    a(78) -  6*a(79) +    a(80) + 3*a(81))/14
a(47) = (  41*s1/9 -  7*a(49) -  7*a(65) -  7*a(67) -  4*a(74) -  4*a(75) -  4*a(76) +
                                                    +  4*a(77) -  2*a(78) -  2*a(79) -  2*a(80) +   a(81))/7
a(46) = (-143*s1/9 + 14*a(59) +  7*a(63) -  7*a(64) + 23*a(74) + 30*a(75) + 23*a(76) +
                                                    - 16*a(77) + 22*a(78) + 29*a(79) + 22*a(80) + 3*a(81))/7
a(45) = (  43*s1/9 -  3*a(74) -  3*a(75) -  3*a(76) -  4*a(77) -  5*a(78) -  5*a(79) +
                                                                                     -  5*a(80) - 8*a(81))/7
a(44) = ( 104*s1/9 - 14*a(49) - 14*a(53) - 14*a(58) + 14*a(59) - 14*a(62) +  7*a(63) - 14*a(67) +
              - 14*a(71) + 3*a(74) + 3*a(75) - 11*a(76) + 11*a(77) - 9*a(78) - 2*a(79) - 23*a(80) + a(81))/14
a(43) = (  50*s1/9 -  7*a(59) -  7*a(63) -  3*a(74) -  3*a(75) -  3*a(76) +  3*a(77) +
                                                               -  5*a(78) - 12*a(79) -  5*a(80) -   a(81))/7
a(42) = (-162*s1/9 + 14*a(49) + 14*a(53) + 14*a(58) + 14*a(62) +  7*a(63) + 14*a(67) +
           + 14*a(71) + 17*a(74) + 17*a(75) + 31*a(76) - 17*a(77) + 5*a(78) + 12*a(79) + 19*a(80) + a(81))/14

a(41) =     s1/9

a(40) = 2*s1/9 - a(42)
a(39) = 2*s1/9 - a(43)
a(38) = 2*s1/9 - a(44)
a(37) = 2*s1/9 - a(45)
a(36) = 2*s1/9 - a(46)
a(35) = 2*s1/9 - a(47)
a(34) = 2*s1/9 - a(48)
a(33) = 2*s1/9 - a(49)
a(32) = 2*s1/9 - a(50)
a(31) = 2*s1/9 - a(51)

a(30) = 2*s1/9 - a(52)
a(29) = 2*s1/9 - a(53)
a(28) = 2*s1/9 - a(54)
a(27) = 2*s1/9 - a(55)
a(26) = 2*s1/9 - a(56)
a(25) = 2*s1/9 - a(57)
a(24) = 2*s1/9 - a(58)
a(23) = 2*s1/9 - a(59)
a(22) = 2*s1/9 - a(60)
a(21) = 2*s1/9 - a(61)

a(20) = 2*s1/9 - a(62)
a(19) = 2*s1/9 - a(63)
a(18) = 2*s1/9 - a(64)
a(17) = 2*s1/9 - a(65)
a(16) = 2*s1/9 - a(66)
a(15) = 2*s1/9 - a(67)
a(14) = 2*s1/9 - a(68)
a(13) = 2*s1/9 - a(69)
a(12) = 2*s1/9 - a(70)
a(11) = 2*s1/9 - a(71)

a(10) = 2*s1/9 - a(72)
a( 9) = 2*s1/9 - a(73)
a( 8) = 2*s1/9 - a(74)
a( 7) = 2*s1/9 - a(75)
a( 6) = 2*s1/9 - a(76)
a( 5) = 2*s1/9 - a(77)
a( 4) = 2*s1/9 - a(78)
a( 3) = 2*s1/9 - a(79)
a( 2) = 2*s1/9 - a(80)
a( 1) = 2*s1/9 - a(81)

which can be solved for Magic Sums s1 whch are a multiple of 63 (= 7 * 9). The minimum possible Magic Sum s1 = 378 occurs for the range {i} = {1 ... 83}.

Attachment 9.8.81 shows a few (2 unique) Associated Quadrant Multi Magic Squares (ref. MgcSqr98a).

Note: The patterns highlighted in the first seven squares, occur all seven in every square shown.

9.8.10 Summary

The obtained results regarding the miscellaneous types of order 9 Quadrant Magic Squares as deducted and discussed in previous sections are summarized in following table:

Type

Characteristics

Subroutine

Results

P1

Order 4 and 5 Square Inlays

-

Attachment 9.8.33

P2, P5

Associated, Compact Pan Magic

MgcSqr9g

Attachment 9.8.41

P2, P3, P4, P5

Associated, Compact Pan Magic

CnstrSqrs94

Attachment 9.8.52

P6

Associated, Square Inlays

-

Attachment 9.8.61

P1, P6

Order 4 and 5 Square Inlays

-

Attachment 9.8.62

P1, P2, P5, P7

Associated, Compact

MgcSqr97a

Attachment 9.8.72

P1 ... P7 (multi)

Associated

MgcSqr98a

Attachment 9.8.81

Comparable routines as listed above, can be used to generate alternative types of order 9 Magic Squares.

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