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 4.6 Analytic Solution Almost Perfect Center Symmetric Magic Cubes About a century before the 5th order Perfect Magic Cube was published, following Almost Perfect Center Symmetric Magic Cube was published by Andrews (1908):

Plane 11 (Top)

 67 98 104 10 36 110 11 42 73 79 48 54 85 111 17 86 117 23 29 60 4 35 61 92 123

Plane 12

 106 12 43 74 80 49 55 81 112 18 87 118 24 30 56 5 31 62 93 124 68 99 105 6 37

Plane 13

 50 51 82 113 19 88 119 25 26 57 1 32 63 94 125 69 100 101 7 38 107 13 44 75 76

Plane 14

 89 120 21 27 58 2 33 64 95 121 70 96 102 8 39 108 14 45 71 77 46 52 83 114 20

Plane 15

 3 34 65 91 122 66 97 103 9 40 109 15 41 72 78 47 53 84 115 16 90 116 22 28 59
 The defining properties of an Almost Perfect Center Symmetric 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); The Main Diagonals of the horizontal planes sum to the Magic Sum (315); For both sets vertical planes 6 out of 10 Main Diagonals sum to the Magic Sum (315); The equations for Center Symmetrical Cubes are applicable Although this definition depends from the orientation of the cube, this will be used as a guideline in the equations 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

After deduction of the defining equations, the following set of linear equations - describing the Almost Perfect Center Symmetric Cubes of the 5th order - can be obtained:

```a121 =     s1 - a122 - a123 - a124 - a125
a116 =     s1 - a117 - a118 - a119 - a120
a111 =     s1 - a112 - a113 - a114 - a115
a109 =          a110 - a113 + a115 - a117 + a120 - a121 + a125
a107 =(  3*s1 - a108-2*a109 - a112-3*a113 - a114-2*a117 - a118 - 2*a119)/2
a106 =     s1 - a107 - a108 - a109 - a110
a105 =     s1 - a109 - a113 - a117 - a121
a104 =     s1 - a109 - a114 - a119 - a124
a103 =     s1 - a108 - a113 - a118 - a123
a102 =     s1 - a107 - a112 - a117 - a122
a101 =     s1 - a102 - a103 - a104 - a105
a96  =     s1 - a97  - a98  - a99  - a100
a91  =     s1 - a92  - a93  - a94  - a95
a88  =          a89 -2*a93+ 2*a94- 2*a98+2*a99-  a113+ a115-2*a118+2*a120-2*a123+2*a125
a87  =(  5*s1 - a88 -2*a89- 4*a90+   a93-4*a94-2*a95-2*a97- 4*a99- 4*a100-3*a112+  a114-3*a115-2*a117+2*a118+
+2*a119-2*a120+3*a123+2*a124)/3
a86  =     s1 - a87 -  a88 -  a89 -  a90
a85  =(- 4*s1+6*a89-   a90 +3*a92 -9*a93+9*a94-3*a95-3*a97-11*a98 +5*a99- 6*a100+ 2*a108+3*a110-4*a113+4*a114+10*a115+
- 2*a117-8*a118+2*a119+13*a120+2*a122-5*a123+4*a124+14*a125)/5
a84  =    -s1  +  a85 - a89 + a90-a92 +2*a93 -2*a94 +a95 +  a97+3*a98-a99+2*a100+a113-a115+2*a118-2*a120+2*a123-2*a125
a83  =            a84 + a93 - a94-a108 + a110+  a118-a120
a82  =(  7*s1  +2*a84+2*a87-2*a89-a108-4*a110+3*a112+a113-5*a114-4*a115+4*a117-a118-4*a119-4*a120-2*a122-4*a123+
-6*a124-8*a125)/2
a81  =     s1  -  a82 - a83 - a84 -  a85
a80  =     s1  -  a84 - a88 - a92 -  a96
a79  =     s1  -  a84 - a89 - a94 -  a99
a78  =     s1  -  a83 - a88 - a93 -  a98
a77  =     s1  -  a82 - a87 - a92 -  a97
a76  =     s1  -  a77 - a78 - a79 -  a80
a75  =(-13*s1/5+2*a76-2*a100+ a108+2*a110+a112-a113+a114+2*a115+  a118+2*a120+2*a122+2*a123+2*a124)/2
a74  =(- 3*s1/5+2*a77-2*a99 + a108+2*a110-a112+a113+a114+2*a115-2*a116-4*a117-  a118+2*a123+4*a125)/2
a73  =   6*s1/5+  a78 - a98 - a108-  a113-a118-   2*a123
a72  =     s1  +  a73 - a97 + a99 +  a113-a114+a117-a119-2*a122-a123-2*a124
a71  =     s1  -  a72 - a73 - a74 -  a75
a70  =(- 3*s1/5+2*a81-2*a95 - a108-2*a110+a112+3*a113+a114+2*a117+a118+2*a119-2*a120)/2
a69  =( 12*s1/5+2*a82-2*a94 - a108-2*a110+a111-a115-a118-4*a119-2*a120+2*a121-2*a125)/2
a68  =     s1/5+  a83 - a93 + a108 - a118
a67  =     s1/5+  a84 - a92 + a110 - a113+a115-2*a117+a120-a121+a125
a66  =     s1  -  a67 - a68 - a69  - a70
a65  =  11*s1/5-  a87-2*a89-2*a90 +2*a93-2*a94+2*a98-2*a99-a112-a114-3*a115+2*a118-2*a120+2*a123-2*a125
a64  =     s1/5+  a87 - a89 + a112 - a114
```
 a63 =   s1/5 a62 = 2*s1/5 - a64 a61 = 2*s1/5 - a65 a60 = 2*s1/5 - a66 a59 = 2*s1/5 - a67 a58 = 2*s1/5 - a68 a57 = 2*s1/5 - a69 a56 = 2*s1/5 - a70 a55 = 2*s1/5 - a71 a54 = 2*s1/5 - a72 a53 = 2*s1/5 - a73 a52 = 2*s1/5 - a74 a51 = 2*s1/5 - a75 a50 = 2*s1/5 - a76 a49 = 2*s1/5 - a77 a48 = 2*s1/5 - a78 a47 = 2*s1/5 - a79 a46 = 2*s1/5 - a80 a45 = 2*s1/5 - a81 a44 = 2*s1/5 - a82 a43 = 2*s1/5 - a83 a42 = 2*s1/5 - a84 a41 = 2*s1/5 - a85 a40 = 2*s1/5 - a86 a39 = 2*s1/5 - a87 a38 = 2*s1/5 - a88 a37 = 2*s1/5 - a89 a36 = 2*s1/5 - a90 a35 = 2*s1/5 - a91 a34 = 2*s1/5 - a92 a33 = 2*s1/5 - a93 a32 = 2*s1/5 - a94 a31 = 2*s1/5 - a95 a30 = 2*s1/5 - a96 a29 = 2*s1/5 - a97 a28 = 2*s1/5 - a98 a27 = 2*s1/5 - a99 a26 = 2*s1/5 - a100 a25 = 2*s1/5 - a101 a24 = 2*s1/5 - a102 a23 = 2*s1/5 - a103 a22 = 2*s1/5 - a104 a21 = 2*s1/5 - a105 a20 = 2*s1/5 - a106 a19 = 2*s1/5 - a107 a18 = 2*s1/5 - a108 a17 = 2*s1/5 - a109 a16 = 2*s1/5 - a110 a15 = 2*s1/5 - a111 a14 = 2*s1/5 - a112 a13 = 2*s1/5 - a113 a12 = 2*s1/5 - a114 a11 = 2*s1/5 - a115 a10 = 2*s1/5 - a116 a9  = 2*s1/5 - a117 a8  = 2*s1/5 - a118 a7  = 2*s1/5 - a119 a6  = 2*s1/5 - a120 a5  = 2*s1/5 - a121 a4  = 2*s1/5 - a122 a3  = 2*s1/5 - a123 a2  = 2*s1/5 - a124 a1  = 2*s1/5 - a125

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

The solutions can be obtained by guessing:

a( 89), a( 90), a( 92) ... a( 95), a( 97) ... a(100),
a(108), a(110), a(112) ... a(115), 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 ... 88, 91, 96, 101 ... 107, 109, 111, 116, 121
a(i) ≠ a(j)           for i ≠ j
Cint(a(i)) = a(i)     for i = 69, 70, 74, 75, 82, 85, 87 and 107

which can be incorporated in a guessing routine which might be used to find other 5th order Almost Perfect Center Symmetric Cubes.

However, the equations deducted above can be applied in a more efficient way to generate Almost Perfect Center Symmetric Cubes, which will be discussed in Section 5.8.