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 4.8   Pantriagonal Magic Cubes (5 x 5 x 5) 4.8.1 Introduction The cube shown below is a Pantriagonal Magic Cube of the 5th order, meaning a Simple Magic Cube for which all main and broken triagonals sum to the Magic Constant.

Plane 11 (Top)

 42 56 100 114 3 8 47 61 80 119 124 13 27 66 85 90 104 18 32 71 51 95 109 23 37

Plane 12

 58 97 111 5 44 49 63 77 116 10 15 29 68 82 121 101 20 34 73 87 92 106 25 39 53

Plane 13

 99 113 2 41 60 65 79 118 7 46 26 70 84 123 12 17 31 75 89 103 108 22 36 55 94

Plane 14

 115 4 43 57 96 76 120 9 48 62 67 81 125 14 28 33 72 86 105 19 24 38 52 91 110

Plane 15

 1 45 59 98 112 117 6 50 64 78 83 122 11 30 69 74 88 102 16 35 40 54 93 107 21
 A Pantriagonal Magic Cube can be transformed into another Pantriagonal Magic Cube by moving an orthogonal plane from one side of the cube to the other. (Comparable with the row and column movements for Pandiagonal Magic Squares as discussed in 'Magic Squares' Section 3.3). Consequently the cube belongs to a collection {Aijkm} of 53 * 48 = 6000 elements which can be found by means of rotation, reflection or plane movements. The Class of 48 elements which can be obtained by rotation / reflection of this Pantriagonal Cube is shown in Attachment 4.8.1. The Class of 125 elements which can be obtained by planar shifts of this Pantriagonal Cube is shown in Attachment 4.8.2. Each cube of Attachment 4.8.1 can be used as a Base for Attachment 4.8.2. It should be noted that the planar shifts are from left to right (L1 ... L4), from back to front (B1 ... B4) and from bottom to top (T1 ... T4). 4.8.2 Analytic Solution In general Magic Cubes of order 5 can be represented as follows:

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 for a Pantriagonal Magic Cube of the fifth order can be summarised as follows: The Rows (25), Columns (25) and Pillars (25) sum to the Magic Sum (315); All pantriagonals (100) sum to the Magic Sum (315). After deduction of the defining equations (175), the following set of linear equations - describing the Pantriagonal 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) =  315 - a(117) - a(118) - a(120) - a(125)
a(114) =        a(118) + a(120) - a(124)
a(113) =        a(117) + a(119) - a(123)
a(112) =  315 - a(117) - a(119) - a(120) - a(122)
a(111) = -315 + a(117) + a(120) + a(122) + a(123) + a(124) + a(125)
a(110) = -315 + a(117) + a(118) + a(120) + a(122) + a(123) + a(125)
a(109) =  315 - a(118) - a(120) - a(123) - a(125)
a(108) =  315 - a(117) - a(119) - a(122) - a(124)
a(107) = -315 + a(117) + a(119) + a(120) + a(122) + a(124) + a(125)
a(106) =  315 - a(117) - a(120) - a(122) - a(125)
a(105) =  315 - a(120) - a(122) - a(123) - a(125)
a(104) =      - a(119) + a(123) + a(125)
a(103) =      - a(118) + a(122) + a(124)
a(102) =  315 - a(117) - a(122) - a(124) - a(125)
a(101) = -315 + a(117) + a(118) + a(119) + a(120) + a(122) + a(125)
a( 96) =  315 - a(97) - a(98) - a(99) - a(100)
a( 91) =  315 - a(92) - a(93) - a(94) - a(95)
a( 90) =  315 - a(92) - a(93) - a(95) - a(100)
a( 89) =        a(93) + a(95) - a(99)
a( 88) =        a(92) + a(94) - a(98)
a( 87) =  315 - a(92) - a(94) - a(95) - a(97)
a( 86) = -315 + a(92) + a(95) + a(97) + a(98) + a(99) + a(100)
a( 85) = -315 + a(92) + a(93) + a(95) + a(97) + a(98) + a(100)
a( 84) =  315 - a(93) - a(95) - a(98) - a(100)
a( 83) =  315 - a(92) - a(94) - a(97) - a(99)
a( 82) = -315 + a(92) + a(94) + a(95) + a(97) + a(99) + a(100)
a( 81) =  315 - a(92) - a(95) - a(97) - a(100)
a( 80) =  315 - a(95) - a(97) - a(98) - a(100)
a( 79) =      - a(94) + a(98) + a(100)
a( 78) =      - a(93) + a(97) + a(99)
a( 77) =  315 - a(92) - a(97) - a(99) - a(100)
a( 76) = -315 + a(92) + a(93) + a(94) + a(95) + a(97) + a(100)
a( 75) =  315 - a(97) - a(98) - a(100) - a(125)
a( 74) =        a(98) + a(100) - a(124)
a( 73) =        a(97) + a(99) - a(123)
a( 72) =  315 - a(97) - a(99) - a(100) - a(122)
a( 71) = -315 + a(97) + a(100) + a(122) + a(123) + a(124) + a(125)
a( 70) =  315 - a(92) - a(93) - a(95) - a(120)
a( 69) =        a(93) + a(95) - a(119)
a( 68) =        a(92) + a(94) - a(118)
a( 67) =  315 - a(92) - a(94) - a(95) - a(117)
a( 66) = -315 + a(92) + a(95) + a(117) + a(118) + a(119) + a(120)
a( 65) = -630 + a(92) + a(93) + 2*a(95) + a(97) + a(98) + a(100) + a(117) + a(118) + a(120) + a(125)
a( 64) =  315 - a(93) + a(94) - a(95) - a(98) - a(100) - a(118) - a(120) + a(124)
a( 63) =  315 - a(92) + a(93) - a(94) - a(97) - a(99) - a(117) - a(119) + a(123)
a( 62) = -630 + 2*a(92) + a(94) + a(95) + a(97) + a(99) + a(100) + a(117) + a(119) + a(120) + a(122)
a( 61) =  945 - 2*a(92) - a(93) - a(94) - 2*a(95) - a(97) - a(100) - a(117) - a(120) - a(122) - a(123) - a(124) - a(125)
a( 60) =  945 - a(92) - a(93) - 2*a(95) - a(97) - a(98) - 2*a(100) - a(117) - a(118) - a(120) - a(122) - a(123) - a(125)
a( 59) = -315 + a(93) - a(94) + a(95) + a(98) - a(99) + a(100) + a(118) + a(120) + a(123) + a(125)
a( 58) = -315 + a(92) - a(93) + a(94) + a(97) - a(98) + a(99) + a(117) + a(119) + a(122) + a(124)
a( 57) =  945 - 2*a(92) - a(94) - a(95) - 2*a(97) - a(99) - a(100) - a(117) - a(119) - a(120) - a(122) - a(124) - a(125)
a( 56) = -945 + 2*a(92) + a(93) + a(94) + 2*a(95) + 2*a(97) + a(98) + a(99) + 2*a(100) + a(117) + a(120) + a(122) + a(125)
a( 55) = -630 + a(92) + a(93) + a(95) + a(97) + a(98) + 2*a(100) + a(120) + a(122) + a(123) + a(125)
a( 54) =  315 - a(93) - a(95) - a(98) + a(99) - a(100) + a(119) - a(123) - a(125)
a( 53) =  315 - a(92) - a(94) - a(97) + a(98) - a(99) + a(118) - a(122) - a(124)
a( 52) = -630 + a(92) + a(94) + a(95) + 2*a(97) + a(99) + a(100) + a(117) + a(122) + a(124) + a(125)
a( 51) =  945 - a(92) - a(95) - 2*a(97) - a(98) - a(99) - 2*a(100) - a(117) - a(118) - a(119) - a(120) - a(122) - a(125)
a( 50) = -315 + a(97) + a(98) + a(100) + a(122) + a(123) + a(125)
a( 49) =  315 - a(98) - a(100) - a(123) - a(125)
a( 48) =  315 - a(97) - a(99) - a(122) - a(124)
a( 47) = -315 + a(97) + a(99) + a(100) + a(122) + a(124) + a(125)
a( 46) =  315 - a(97) - a(100) - a(122) - a(125)
a( 45) = -315 + a(92) + a(93) + a(95) + a(117) + a(118) + a(120)
a( 44) =  315 - a(93) - a(95) - a(118) - a(120)
a( 43) =  315 - a(92) - a(94) - a(117) - a(119)
a( 42) = -315 + a(92) + a(94) + a(95) + a(117) + a(119) + a(120)
a( 41) =  315 - a(92) - a(95) - a(117) - a(120)
a( 40) =  945 - a(92) - a(93) - 2*a(95) - a(97) - a(98) - a(100) - a(117) - a(118) - 2*a(120) - a(122) - a(123) - a(125)
a( 39) = -315 + a(93) - a(94) + a(95) + a(98) + a(100) + a(118) - a(119) + a(120) + a(123) + a(125)
a( 38) = -315 + a(92) - a(93) + a(94) + a(97) + a(99) + a(117) - a(118) + a(119) + a(122) + a(124)
a( 37) =  945 - 2*a(92) - a(94) - a(95) - a(97) - a(99) - a(100) - 2*a(117) - a(119) - a(120) - a(122) - a(124) - a(125)
a( 36) = -945 + 2*a(92) + a(93) + a(94) + 2*a(95) + a(97) + a(100) + 2*a(117) + a(118) + a(119) + 2*a(120) + a(122) + a(125)
a( 35) = -945 + a(92) + a(93) + 2*a(95) + a(97) + a(98) + 2*a(100) + a(117) + a(118) + 2*a(120) + a(122) + a(123) + 2*a(125)
a( 34) =  315 - a(93) + a(94) - a(95) - a(98) + a(99) - a(100) - a(118) + a(119) - a(120) - a(123) + a(124) - a(125)
a( 33) =  315 - a(92) + a(93) - a(94) - a(97) + a(98) - a(99) - a(117) + a(118) - a(119) - a(122) + a(123) - a(124)
a( 32) = -945 + 2*a(92) + a(94) + a(95) + 2*a(97) + a(99) + a(100) + 2*a(117) + a(119) + a(120) + 2*a(122) + a(124) + a(125)
a( 31) = 1575 - 2*a(92) - a(93) - a(94) - 2*a(95) - 2*a(97) - a(98) - a(99) - 2*a(100) - 2*a(117) - a(118) - a(119) +
- 2*a(120) - 2*a(122) - a(123) - a(124) - 2*a(125)
a( 30) =  945 - a(92) - a(93) - a(95) - a(97) - a(98) - 2*a(100) - a(117) - a(118) - a(120) - a(122) - a(123) - 2*a(125)
a( 29) = -315 + a(93) + a(95) + a(98) - a(99) + a(100) + a(118) + a(120) + a(123) - a(124) + a(125)
a( 28) = -315 + a(92) + a(94) + a(97) - a(98) + a(99) + a(117) + a(119) + a(122) - a(123) + a(124)
a( 27) =  945 - a(92) - a(94) - a(95) - 2*a(97) - a(99) - a(100) - a(117) - a(119) - a(120) - 2*a(122) - a(124) - a(125)
a( 26) = -945 + a(92) + a(95) + 2*a(97) + a(98) + a(99) + 2*a(100) + a(117) + a(120) + 2*a(122) + a(123) + a(124) + 2*a(125)
a( 25) =  315 - a(100) - a(122) - a(123) - a(125)
a( 24) =      - a(99) + a(123) + a(125)
a( 23) =      - a(98) + a(122) + a(124)
a( 22) =  315 - a(97) - a(122) - a(124) - a(125)
a( 21) = -315 + a(97) + a(98) + a(99) + a(100) + a(122) + a(125)
a( 20) =  315 - a(95) - a(117) - a(118) - a(120)
a( 19) =      - a(94) + a(118) + a(120)
a( 18) =      - a(93) + a(117) + a(119)
a( 17) =  315 - a(92) - a(117) - a(119) - a(120)
a( 16) = -315 + a(92) + a(93) + a(94) + a(95) + a(117) + a(120)
a( 15) = -630 + a(92) + a(93) + a(95) + a(100) + a(117) + a(118) + 2*a(120) + a(122) + a(123) + a(125)
a( 14) =  315 - a(93) - a(95) + a(99) - a(118) + a(119) - a(120) - a(123) - a(125)
a( 13) =  315 - a(92) - a(94) + a(98) - a(117) + a(118) - a(119) - a(122) - a(124)
a( 12) = -630 + a(92) + a(94) + a(95) + a(97) + 2*a(117) + a(119) + a(120) + a(122) + a(124) + a(125)
a( 11) =  945 - a(92) - a(95) - a(97) - a(98) - a(99) - a(100) - 2*a(117) - a(118) - a(119) - 2*a(120) - a(122) - a(125)
a( 10) =  945 - a(92) - a(93) - a(95) - a(97) - a(98) - a(100) - a(117) - a(118) - 2*a(120) - a(122) - a(123) - 2*a(125)
a(  9) = -315 + a(93) + a(95) + a(98) + a(100) + a(118) - a(119) + a(120) + a(123) - a(124) + a(125)
a(  8) = -315 + a(92) + a(94) + a(97) + a(99) + a(117) - a(118) + a(119) + a(122) - a(123) + a(124)
a(  7) =  945 - a(92) - a(94) - a(95) - a(97) - a(99) - a(100) - 2*a(117) - a(119) - a(120) - 2*a(122) - a(124) - a(125)
a(  6) = -945 + a(92) + a(95) + a(97) + a(100) + 2*a(117) + a(118) + a(119) + 2*a(120) + 2*a(122) + a(123) + a(124) + 2*a(125)
a(  5) = -630 + a(95) + a(97) + a(98) + a(100) + a(117) + a(118) + a(120) + a(122) + a(123) + 2*a(125)
a(  4) =  315 + a(94) - a(98) - a(100) - a(118) - a(120) - a(123) + a(124) - a(125)
a(  3) =  315 + a(93) - a(97) - a(99) - a(117) - a(119) - a(122) + a(123) - a(124)
a(  2) = -630 + a(92) + a(97) + a(99) + a(100) + a(117) + a(119) + a(120) + 2*a(122) + a(124) + a(125)
a(  1) =  945 - a(92) - a(93) - a(94) - a(95) - a(97) - a(100) - a(117) - a(120) - 2*a(122) - a(123) - a(124) - 2*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(122) ... a(125), a(117) ... a(120), a(97) ... a(100) and a(92) ... a(95) 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 ... 91, 96, 101 ... 116, 121 a(i) ≠ a(j)           for i ≠ j which can be incorporated in a guessing routine, which might be used to find other 5th order Pantriagonal Magic Cubes. However, the equations deducted above can be applied in a more efficient method to generate Pantriagonal Magic Cubes, which will be discussed in Section 5.5.