Office Applications and Entertainment, Magic Cubes

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4.10   Pan Diagonal/Triagonal Magic Cubes (5 x 5 x 5)

4.10.1 Introduction

The cube shown below, published by B. Violle in 1838, is a Pan Diagonal/Triagonal Magic Cube of the 5th order, meaning a Magic Cube for which all (Pan) Diagonals and (Pan) Triagonals sum to the Magic Constant.

Magic Cube, Pan Diagonal/Triagonal

Plane 11 (Top)

15 24 8 17 1
45 29 38 47 31
75 59 68 52 61
80 89 98 82 91
110 119 103 112 121

Plane 12

88 97 81 95 79
118 102 111 125 109
23 7 16 5 14
28 37 46 35 44
58 67 51 65 74

Plane 13

36 50 34 43 27
66 55 64 73 57
96 85 94 78 87
101 115 124 108 117
6 20 4 13 22

Plane 14

114 123 107 116 105
19 3 12 21 10
49 33 42 26 40
54 63 72 56 70
84 93 77 86 100

Plane 15

62 71 60 69 53
92 76 90 99 83
122 106 120 104 113
2 11 25 9 18
32 41 30 39 48


A Pan Diagonal/Triagonal Magic Cube can be transformed into another Pan Diagonal/Triagonal 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 Pan Diagonal/Triagonal Cube is shown in Attachment 4.10.1.

The Class of 125 elements which can be obtained by planar shifts of this Pan Diagonal/Triagonal Cube is shown in Attachment 4.10.2. Each cube of Attachment 4.10.1 can be used as a Base for Attachment 4.10.2.

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).

4.10.2 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 Pan Diagonal/Triagonal Magic Cube of the fifth order can be summarised as follows:

  • All (Pan) Diagonals (150) sum to the Magic Sum (315).

  • All (Pan) Triagonals (100) sum to the Magic Sum (315).

After deduction of the defining equations (250), the following set of linear equations - describing the Pan Diagonal/Triagonal Magic Cubes of the 5th order - can be obtained: 

a1  =-315+a85+a90 +a95 +a97 +a98 +a99-2*a100+a111+a114-a115-a116+a117+a118-a122+a125
a2  =-315+a85+a90 +a95 +a96 +a98 +a99-2*a100+a111+a114-a115+a118-a121-a123+2*a125
a3  =-315+a85+a90 +a95 +a96 +a97 +a99-2*a100+a111+a114-a115+a117-a122-a124+2*a125
a4  =-315+a85+a90 +a95 +a96 +a97 +a98-2*a100+a111+a114-a115+a117+a118-a119-a123+a125
a5  =-315+a85+a90 +a95 +a96 +a97 +a98 +a99-3*a100+a111+a114-a115+a117+a118-a120-a121-a124+2*a125
a6  =    -a85-a96 +a100-a111+a113+a119+a121
a7  =    -a85-a97 +a100-a111+a115+a116-a117+a119+a122
a8  =    -a85-a98 +a100-a114+a115+a116-a118+a119+a123
a9  =    -a85-a99 +a100+a112-a114+a116+a124
a10 =    -a85-a111+a112+a113-a114+a116+a119-a120+a125
a11 = 315-a90-a96 +a100-a110-a111-a114+a115-a117+a119-a120
a12 = 315-a90-a97 +a100-a110-a111-a114+a115-a117-a118+a119
a13 = 315-a90-a98 +a100-a110-a111-a114+a115+a116-a117-a118
a14 = 315-a90-a99 +a100-a110-a111-a114+a115+a116-a118-a120
a15 = 315-a90-a110-a111-a114+a115-a120
a16 = 315-a95-a96 +a100-a111-a117-a125
a17 = 315-a95-a97 +a100-a112-a116-a118+a120-a125
a18 = 315-a95-a98 +a100-a113-a117-a119+a120-a125
a19 = 315-a95-a99 +a100-a114-a118-a125
a20 = 315-a95-a115-a116-a119+a120-a125
a21 =    -a96+a110+a114-a116+a118+a121-a125
a22 =    -a97+a110+a111-a112+a114-a116+a120+a122-a125
a23 =    -a98+a110+a111-a113+a114-a119+a120+a123-a125
a24 =    -a99+a110+a111+a117-a119+a124-a125
a25 =    -a100+a110+a111+a114-a115-a116+a117+a118-a119
a26 = 315-a90 -a95 -a98 -a99+2*a100-a113-a114+a115-a118-a119+a120+a123+a124-2*a125
a27 = 315-a90 -a95 -a99 +a100-a114-a119+a124-a125
a28 = 315-a90 -a95 -a96 +a100-a111-a116+a121-a125
a29 = 315-a90 -a95 -a96 -a97+2*a100-a111-a112+a115-a116-a117+a120+a121+a122-2*a125
a30 = 315-a90 -a95 -a97 -a98+2*a100-a112-a113+a115-a117-a118+a120+a122+a123-2*a125
a31 = 315-a95 -a98 -a99 +a100-a118-a119+a120-a125
a32 = 315-a95 -a99 -a119-a125
a33 = 315-a95 -a96 -a116-a125
a34 = 315-a95 -a96 -a97+a100-a116-a117+a120-a125
a35 = 315-a95 -a97-a98+a100-a117-a118+a120-a125
a36 =-315+a85 +a90+a95+a96+a97-2*a100+a110+2*a111-a113+a114-a115+a117+a118-a119-a121+a125
a37 =-315+a85 +a90+a95+a96+a97+a98-3*a100+a110+2*a111+a114-2*a115-a116+2*a117+a118-a119-a122+a125
a38 =-315+a85 +a90+a95+a97+a98+a99-3*a100+a110+a111+2*a114-2*a115-a116+a117+2*a118-a119-a123+a125
a39 =-315+a85 +a90+a95+a98+a99-2*a100+a110+a111-a112+2*a114-a115-a116+a117+a118-a124+a125
a40 =-315+a85 +a90+a95+a96+a99-2*a100+a110+2*a111-a112-a113+2*a114-a115-a116+a117+a118-a119+a120
a41 =-315+a90 +a95+a96+a97-a100+a112+2*a116-a120-a121+2*a125
a42 =-315+a90 +a95+a96+a97+a98-2*a100+a111+a113-a115+2*a117-a120-a122+2*a125
a43 =-315+a90 +a95+a97+a98+a99-2*a100+a112+a114-a115+2*a118-a120-a123+2*a125
a44 =-315+a90 +a95+a98+a99-a100+a113+2*a119-a120-a124+2*a125
a45 =-315+a90 +a95+a96+a99-a100+a111+a114-a115+a120+a125
a46 = 315-a85 -a90-a98-a99+2*a100-a110-2*a111+a112-2*a114+2*a115+2*a116-a117-a118+a119-a120
a47 = 315-a85 -a90-a99+a100-a110-a111+a113-2*a114+a115+a116-a118+a119-a120
a48 = 315-a85 -a90-a96+a100-a110-2*a111+a112-a114+a115+a116-a117+a119-a120
a49 = 315-a85 -a90-a96-a97+2*a100-a110-2*a111+a113-2*a114+2*a115+a116-a117-a118+2*a119-a120
a50 = 315-a85 -a90-a97-a98+2*a100-a110-a111-a114+a115+a116-a117-a118+a119
a51 =-315+a90 +a95+a98+a99-2*a100+a110+a111+a114-a115-a116+a117+a118+a125
a52 =-315+a90 +a95+a99-a100+a110+a111+a114-a115+a118+a125
a53 =-315+a90 +a95+a96-a100+a110+a111+a114-a115+a117+a125
a54 =-315+a90 +a95+a96+a97-2*a100+a110+a111+a114-a115+a117+a118-a119+a125
a55 =-315+a90 +a95+a97+a98-2*a100+a110+a111+a114-a115+a117+a118-a120+a125
a56 =     a95 +a98+a99-a100-a110-a111+a113+a119-a120-a123-a124+2*a125
a57 =     a95 +a99-a110-a111+a115+a116-a117+a119-a120-a124+a125
a58 =     a95 +a96-a110-a114+a115+a116-a118+a119-a120-a121+a125
a59 =     a95 +a96+a97 -a100-a110+a112-a114+a116-a120-a121-a122+2*a125
a60 =     a95 +a97+a98 -a100-a110-a111+a112+a113-a114+a116+a119-2*a120-a122-a123+2*a125
a61 = 315-a85 -a90-a95 -a96-a97+2*a100-a111-a114+a115-a117+a119+a122-a125
a62 = 315-a85 -a90-a95 -a96-a97-a98+3*a100-a111-a114+a115-a117-a118+a119+a120+a121+a123-2*a125
a63 = 315-a85 -a90-a95 -a97-a98-a99+3*a100-a111-a114+a115+a116-a117-a118+a120+a122+a124-2*a125
a64 = 315-a85 -a90-a95 -a98-a99+2*a100-a111-a114+a115+a116-a118+a123-a125
a65 = 315-a85 -a90-a95 -a96-a99+2*a100-a111-a114+a115+a121+a124-2*a125
a66 = 315-a90 -a95-a96 -a97+a100-a111+a115-a117+a122-a125
a67 = 315-a90 -a95-a96 -a97-a98+2*a100-a112+a115-a116-a118+a120+a121+a123-2*a125
a68 = 315-a90 -a95-a97 -a98-a99+2*a100-a113+a115-a117-a119+a120+a122+a124-2*a125
a69 = 315-a90 -a95-a98 -a99+a100-a114+a115-a118+a123-a125
a70 = 315-a90 -a95-a96 -a99+a100-a116-a119+a120+a121+a124-2*a125
a71 =     a85 +a90+a98 +a99-2*a100+a114-a115-a116+a118-a123-a124+a125
a72 =     a85 +a90+a99 -a100+a111-a112+a114-a115-a116+a120-a124
a73 =     a85 +a90+a96 -a100+a111-a113+a114-a115-a119+a120-a121
a74 =     a85 +a90+a96 +a97-2*a100+a111-a115+a117-a119-a121-a122+a125
a75 =     a85 +a90+a97 +a98-2*a100+a111+a114-2*a115-a116+a117+a118-a119-a122-a123+a125
a76 = 315-a85 -a90-a95 -a97-a98-a99+2*a100-a114+a115-a118+a120+a123+a124-2*a125
a77 = 315-a85 -a90-a95 -a96-a98-a99+2*a100-a111+a112-a114+a115+a116-a117+a124-a125
a78 = 315-a85 -a90-a95 -a96-a97-a99+2*a100-a111+a113-a114+a115-a118+a119+a121-a125
a79 = 315-a85 -a90-a95 -a96-a97-a98+2*a100-a111+a115-a117+a120+a121+a122-2*a125
a80 = 315-a85 -a90-a95 -a96-a97-a98-a99+3*a100-a111-a114+2*a115+a116-a117-a118+a119+a122+a123-2*a125
a81 =     a85 +a96-a100+a111-a112+a114-a115-a116+a117-a119+a120
a82 =     a85 +a97-a100-a113+a114+a118-a119
a83 =     a85 +a98-a100+a111-a112-a116+a117
a84 =     a85 +a99-a100+a111-a113+a114-a115-a116+a118-a119+a120
a86 =     a90 +a96-a100+a111-a115-a121+a125
a87 =     a90 +a97-a100+a112-a115-a122+a125
a88 =     a90 +a98-a100+a113-a115-a123+a125
a89 =     a90 +a99-a100+a114-a115-a124+a125
a91 =     a95 +a96-a100+a116-a120-a121+a125
a92 =     a95 +a97-a100+a117-a120-a122+a125
a93 =     a95 +a98-a100+a118-a120-a123+a125
a94 =     a95 +a99-a100+a119-a120-a124+a125
a101= 315-a110-a114-a118-a122
a102= 315-a110-a111+a112-a114+a116-a117-a120-a121-a123+a125
a103= 315-a110-a111+a113-a114-a118+a119-a120-a122-a124+a125
a104= 315-a110-a111-a117-a123
a105= 315-a110-a111-a114+a115+a116-a117-a118+a119-a120-a121-a124+a125
a106=     a110+a111-a112+a114-a115-a116+a117-a119+a120+a121-a125
a107=     a110-a113+a114+a118-a119+a122-a125
a108=     a110+a111-a112-a116+a117+a123-a125
a109=     a110+a111-a113+a114-a115-a116+a118-a119+a120+a124-a125

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

The solutions can be obtained by guessing a(110) ... a(125), a(95) ... a(100), a(90) and a(85) 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 ... 109, 91 ... 94, 86 ... 89, 81 ... 84
a(i) ≠ a(j)           for i ≠ j

which can be incorporated in a guessing routine, which might be used to find other 5th order Pan Diagonal/Triagonal Magic Cubes.

However, the equations deducted above can be applied in a more efficient method to generate Pan Diagonal/Triagonal Magic Cubes, which will be discussed in Section 5.7.

4.11 Spreadsheet Solutions

The linear equations deducted in sections 4.2 through 4.10 above, have been applied in following Excel Spread Sheets:

Only the red figures have to be “guessed” to construct one of the applicable Magic Cubes of the 5th order (wrong solutions are obvious).

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