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 4.7 Analytic Solution Semi Pan Magic Cubes One of the most magic cubes of the 5th order is the Semi Pan Magic Cube. An example is shown below.

Plane 11 (Top)

 78 109 15 41 72 45 71 77 108 14 107 13 44 75 76 74 80 106 12 43 11 42 73 79 110

Plane 12

 40 66 97 103 9 102 8 39 70 96 69 100 101 7 38 6 37 68 99 105 98 104 10 36 67

Plane 13

 122 3 34 65 91 64 95 121 2 33 1 32 63 94 125 93 124 5 31 62 35 61 92 123 4

Plane 14

 59 90 116 22 28 21 27 58 89 120 88 119 25 26 57 30 56 87 118 24 117 23 29 60 86

Plane 15

 16 47 53 84 115 83 114 20 46 52 50 51 82 113 19 112 18 49 55 81 54 85 111 17 48
 The equations for a Semi Pan Magic Cube of the fifth order can be summarised as follows: The Rows (25), Columns (25), Pillars (25) and Space Diagonals (4) sum to the Magic Sum (315); Main - and Pan Diagonals of 2 of the 3 sets of planes sum to the Magic Sum (315); The equations for Symmetrical Cubes are applicable In the example shown above the (pan) magic diagonals for the horizontal planes 1.1 thru 1.5 and the vertical planes 3.1 thru 3.5 sum to 315. This is, however, not the case for the vertical planes 2.1 thru 2.5. Although this depends from the orientation of the cube, this will be used as a guideline in the deduction below.

Plane 11 (Top)

 a101 a102 a103 a104 a105 a106 a107 a108 a109 a110 a111 a112 a113 a114 a115 a116 a117 a118 a119 a120 a121 a122 a123 a124 a125

Plane 12

 a76 a77 a78 a79 a80 a81 a82 a83 a84 a85 a86 a87 a88 a89 a90 a91 a92 a93 a94 a95 a96 a97 a98 a99 a100

Plane 13

 a51 a52 a53 a54 a55 a56 a57 a58 a59 a60 a61 a62 a63 a64 a65 a66 a67 a68 a69 a70 a71 a72 a73 a74 a75

Plane 14

 a26 a27 a28 a29 a30 a31 a32 a33 a34 a35 a36 a37 a38 a39 a40 a41 a42 a43 a44 a45 a46 a47 a48 a49 a50

Plane 15

 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21 a22 a23 a24 a25

The equations of section 4.2 are applicable with exception of the equations for the plane diagonals of the vertical planes 2.1 thru 2.5.

Further the equations for the pan diagonals of the horizontal planes and the vertical planes 3.1 thru 3.5 have to be added (total 10 * 8 = 80 equations).

If the equations for symmetric space diagonals are added:

 Outer Cube: a1  + a125 = 126 a5  + a121 = 126 a21 + a105 = 126 a25 + a101 = 126 Inner Cube: a32 + a94 = 126 a34 + a92 = 126 a42 + a84 = 126 a44 + a82 = 126

following equations, applicable for a Symmetrical Cube, are automatically fulfilled:

 Center Pillar: a13 + a113 = 126 Plane 13: a51 + a75 = 126 a52 + a74 = 126 a53 + a73 = 126 a54 + a72 = 126 a55 + a71 = 126 a56 + a70 = 126 a60 + a66 = 126 a61 + a65 = 126 Planes 12/14: a26 + a100 = 126 a27 + a99  = 126 a29 + a97  = 126 a30 + a96  = 126 a31 + a95  = 126 a35 + a91  = 126 a41 + a85  = 126 a45 + a81  = 126 a46 + a80  = 126 a47 + a79  = 126 a49 + a77  = 126 a50 + a76  = 126 Plane 23: a3   + a123 = 126 a23  + a103 = 126 a78  + a48  = 126 a98  + a28  = 126 Plane 33: a11  + a115 = 126 a15  + a111 = 126 a36  + a90  = 126 a40  + a86  = 126 Inner Cube: Center Pillar: a38 + a88 = 126 Plane 13: a57 + a69 = 126 a58 + a68 = 126 a59 + a67 = 126 a62 + a64 = 126 Plane 23: a33 + a93 = 126 a43 + a83 = 126 Plane 33: a37 + a89 = 126 a39 + a87 = 126

After deduction of the defining equations, the following set of linear equations - describing the Semi Pan Magic Cubes of the 5th order - can be obtained:

```a(121) =  315 - a(122) - a(123) - a(124) - a(125)
a(116) =  315 - a(117) - a(118) - a(119) - a(120)
a(115) =        a(117) + a(118) - a(125)
a(114) =  315 - a(118) - a(119) - a(120) - a(124)
a(113) =  315 - a(117) - a(118) - a(119) - a(123)
a(112) =        a(119) + a(120) - a(122)
a(111) =        a(118) + a(119) - a(121)
a(110) =  315 - a(117) - a(118) - a(122) - a(123)
a(109) = -315 + a(118) + a(119) + a(120) + a(123) + a(124) + a(125)
a(108) = -315 + a(117) + a(118) + a(119) + a(122) + a(123) + a(124)
a(107) =  315 - a(119) - a(120) - a(124) - a(125)
a(106) =  315 - a(118) - a(119) - a(123) - a(124)
a(105) =      - a(120) + a(122) + a(123)
a(104) =  315 - a(119) - a(123) - a(124) - a(125)
a(103) =  315 - a(118) - a(122) - a(123) - a(124)
a(102) =      - a(117) + a(124) + a(125)
a(101) = -315 + a(117) + a(118) + a(119) + a(120) + a(123) + a(124)
a(100) = -441 + a(117) + a(118) + a(119) + a(120) + a(122) + 2 * a(123) + a(124)
a( 99) =  189 - a(117) - a(123)
a( 98) =  504 - a(118) - 2 * a(122) - 2 * a(123) - 2 * a(124)
a( 97) =  189 - a(119) - a(123)
a( 96) = -126 - a(120) + a(122) + 2 * a(123) + a(124)
a( 95) =  189 + a(117) - a(119) - a(123) - a(124)
a( 94) =  504 - a(118) - 2 * a(119) - 2 * a(120) - a(124) - a(125)
a( 93) = -126 + a(122) + a(123) + a(124)
a( 92) = -441 + a(118) + 2 * a(119) + 2 * a(120) + a(123) + a(124) + a(125)
a( 91) =  189 - a(117) + a(119) - a(122) - a(123)
a( 90) = -126 - a(117) + a(119) + a(120) + a(124) + a(125)
a( 89) = -441 + a(118) + 3 * a(119) + 2 * a(120) - a(122) + a(123) + a(124) + a(125)
a( 88) = -126 + a(118) + a(122) + a(124)
a( 87) =  504 + a(117) - a(118) - 2 * a(119) - 2 * a(120) - 2 * a(124) - a(125)
a( 86) =  504 - a(118) - 2 * a(119) - a(120) - a(123) - a(124) - a(125)
a( 85) =  189 - a(119) - 2 * a(120) + a(122) + a(123) - a(125)
a( 84) =  504 + a(117) - a(118) - 2 * a(119) - a(120) - a(123) - 2 * a(124) - a(125)
a( 83) =  504 - a(117) - a(118) - a(119) - a(122) - 2 * a(123) - a(124)
a( 82) = -126 - a(117) + 2 * a(119) + a(120) - a(122) + a(124) + a(125)
a( 81) = -756 + a(117) + 2 * a(118) + 2 * a(119) + 2 * a(120) + a(122) + 2 * a(123) + 2 * a(124) + a(125)
a( 80) =  504 - a(117) - a(118) - 2 * a(122) - 2 * a(123) - a(124)
a( 79) = -441 + a(118) + a(119) + a(120) + a(122) + a(123) + 2 * a(124) + a(125)
a( 78) = -441 + a(117) + a(118) + a(119) + a(122) + 3 * a(123) + a(124)
a( 77) =  189 - a(119) - a(120) + a(122) - a(125)
a( 76) =  504 - a(118) - a(119) - a(122) - 2 * a(123) - 2 * a(124)
a( 75) =  693 - a(118) - a(119) - 2 * a(122) - 3 * a(123) - 2 * a(124) - a(125)
a( 74) =   63 - a(119) - a(120) + a(122) + a(123)
a( 73) = -567 + a(117) + a(118) + a(119) + 2 * a(122) + 3 * a(123) + 2 * a(124)
a( 72) = -252 + a(118) + a(119) + a(120) + a(123) + a(124)
a( 71) =  378 - a(117) - a(118) - a(122) - 2 * a(123) - a(124) + a(125)
a( 70) = -567 + a(118) + 2 * a(119) + a(120) + a(122) + 2 * a(123) + 2 * a(124) + a(125)
a( 69) = -252 - a(117) + a(118) + 2 * a(119) + 2 * a(120) - a(122) + a(124) + a(125)
a( 68) =  378 - a(118) - a(122) - 2 * a(123) - a(124)
a( 67) =  693 - a(118) - 3 * a(119) - 2 * a(120) - a(123) - 2 * a(124) - a(125)
a( 66) =   63 + a(117) - a(119) - a(120) + a(122) + a(123) - a(125)
a( 65) =  378 - a(118) - 2 * a(119) - 2 * a(120) + a(122) - a(124)
a( 64) =  693 + a(117) - a(118) - 3 * a(119) - 2 * a(120) - a(123) - 2 * a(124) - 2 * a(125)
```
 a( 63) =  63 a( 62) = 126 - a(64) a( 61) = 126 - a(65) a( 60) = 126 - a(66) a( 59) = 126 - a(67) a( 58) = 126 - a(68) a( 57) = 126 - a(69) a( 56) = 126 - a(70) a( 55) = 126 - a(71) a( 54) = 126 - a(72) a( 53) = 126 - a(73) a( 52) = 126 - a(74) a( 51) = 126 - a(75) a( 50) = 126 - a(76) a( 49) = 126 - a(77) a(48) = 126 - a(78) a(47) = 126 - a(79) a(46) = 126 - a(80) a(45) = 126 - a(81) a(44) = 126 - a(82) a(43) = 126 - a(83) a(42) = 126 - a(84) a(41) = 126 - a(85) a(40) = 126 - a(86) a(39) = 126 - a(87) a(38) = 126 - a(88) a(37) = 126 - a(89) a(36) = 126 - a(90) a(35) = 126 - a(91) a(34) = 126 - a(92) a(33) = 126 - a(93) a(32) = 126 - a(94) a(31) = 126 - a(95) a(30) = 126 - a(96) a(29) = 126 - a(97) a(28) = 126 - a(98) a(27) = 126 - a(99) a(26) = 126 - a(100) a(25) = 126 - a(101) a(24) = 126 - a(102) a(23) = 126 - a(103) a(22) = 126 - a(104) a(21) = 126 - a(105) a(20) = 126 - a(106) a(19) = 126 - a(107) a(18) = 126 - a(108) a(17) = 126 - a(109) a(16) = 126 - a(110) a(15) = 126 - a(111) a(14) = 126 - a(112) a(13) = 126 - a(113) a(12) = 126 - a(114) a(11) = 126 - a(115) a(10) = 126 - a(116) a( 9) = 126 - a(117) a( 8) = 126 - a(118) a( 7) = 126 - a(119) a( 6) = 126 - a(120) a( 5) = 126 - a(121) a( 4) = 126 - a(122) a( 3) = 126 - a(123) a( 2) = 126 - a(124) a( 1) = 126 - a(125)

The linear equations shown above, are ready to be solved, for the magic constant 315.

The solutions can be obtained by guessing a(117) ... a(120) and a(122) ... a(125) and filling out these guesses in the abovementioned equations.

For distinct integers also following relations should be applied:

0 < a(i) =< 125       for i = 1 ... 116, 121
a(i) ≠ a(j)           for i ≠ j

which have been incorporated in an optimized guessing routine (MgcCube5d).

Subject guessing routine produced - with careful selected sets of the variables a(125) and a(124) (note 47) - Semi Pan Magic Cubes, which are shown in Attachment 4.7.1.

The eight cubes which have been obtained with the 3 Grid Method are shown in Attachment 4.7.2.

A more efficient method to generate Semi Pan Magic Cubes, based on the equations deducted above, will be discussed in Section 5.3.