Office Applications and Entertainment, Magic Cubes

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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.

Magic Cube, Pantriagonal

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:

Magic Cube (5 x 5 x 5)

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.

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