Office Applications and Entertainment, Magic Cubes

Vorige Pagina Volgende Pagina Index About the Author

4.9   Pantriagonal Magic Cubes, Moriyama (5 x 5 x 5)

4.9.1 Introduction

The cube shown below is a Pantriagonal Magic Cube of the 5th order, published by Yoshio Moriyama in 1967, with the additional property that the diagonals of the centerplanes sum to the Magic Sum.

Magic Cube, Pantriagonal (Moriyama)

Plane 11 (Top)

21 40 54 93 107
112 1 45 59 98
78 117 6 50 64
69 83 122 11 30
35 74 88 102 16

Plane 12

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

Plane 13

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

Plane 14

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

Plane 15

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


Due to the center planes property the cube is only member of a Class of 48 elements which can be obtained by rotation / reflection.

The Class of 48 elements which can be obtained by rotation / reflection of this Pantriagonal Cube is shown in Attachment 4.9.1.

4.9.2 Analytic Solution (1)

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 Moriyama 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);

  • All diagonals of the center planes (6) sum to the Magic Sum (315).

After deduction of the defining equations (181), the following set of linear equations - describing the Moriyama 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(97)=(-a(98)+a(99)-2*a(100)+3*a(117)+a(118)-a(119)+2*a(120))/3
a(96)=(945-2*a(98)-4*a(99)-a(100)-3*a(117)-a(118)+a(119)-2*a(120))/3
a(94)=(-1323+a(95)+a(99)+2*a(100)+3*a(117)+a(118)+4*a(120)+4*a(122)+a(123)+3*a(124)+3*a(125))/2
a(93)=(-189-2*a(98)+2*a(99)-a(100)+3*a(117)-a(118)+a(119)+a(120)+3*a(122)-3*a(123)+3*a(124))/3
a(92)=(5103-3*a(95)-2*a(98)-7*a(99)-4*a(100)-15*a(117)-7*a(118)-2*a(119)-14*a(120)-6*a(122)-3*a(123)-3*a(124)-9*a(125))/6
a(91)=189-a(95)+a(98)+a(118)-2*a(122)+a(123)-2*a(124)
a(90)=(-945-a(95)+2*a(98)+a(99)+3*a(117)+3*a(118)+4*a(120)+3*a(123)-a(124)+3*a(125))/2
a(89)=(-189+3*a(95)-2*a(98)-a(99)-a(100)+3*a(117)-a(118)+a(119)+a(120)+3*a(122)-3*a(123)+3*a(124))/3
a(88)=(567-4*a(98)-2*a(99)+a(100)-3*a(117)-2*a(118)-a(119)-a(120)+3*a(122)+3*a(124))/3
a(87)=(378-3*a(95)+2*a(98)+a(99)+a(100)+a(118)+2*a(119)-a(120)-3*a(122)-3*a(124))/3
a(86)=(3213+3*a(95)+2*a(98)+a(99)-2*a(100)-9*a(117)-5*a(118)-4*a(119)-10*a(120)-6*a(122)-3*a(123)-3*a(124)-9*a(125))/6
a(85)=(2835+3*a(95)-2*a(98)-a(99)-4*a(100)-3*a(117)-7*a(118)-2*a(119)-8*a(120)-9*a(123)+3*a(124)-9*a(125))/6
a(84)=(1134-3*a(95)-a(98)-2*a(99)-2*a(100)-3*a(117)+a(118)-a(119)-a(120)-3*a(122)+3*a(123)-3*a(124))/3
a(83)=(378+2*a(98)-2*a(99)+a(100)+a(118)+2*a(119)-a(120)-3*a(122)-3*a(124))/3
a(82)=(-378+3*a(95)-2*a(98)+2*a(99)+2*a(100)-a(118)-2*a(119)+a(120)+3*a(122)+3*a(124))/3
a(81)=(-3213-3*a(95)+4*a(98)+5*a(99)+2*a(100)+9*a(117)+5*a(118)+4*a(119)+10*a(120)+6*a(122)+3*a(123)+3*a(124)+9*a(125))/6
a(80)=(945-3*a(95)-2*a(98)-a(99)-a(100)-3*a(117)-a(118)+a(119)-2*a(120))/3
a(79)=(1323-a(95)+2*a(98)-a(99)-3*a(117)-a(118)-4*a(120)-4*a(122)-a(123)-3*a(124)-3*a(125))/2
a(78)=(189+a(98)+2*a(99)-a(100)+2*a(118)-2*a(119)+a(120)-3*a(122)+3*a(123)-3*a(124))/3
a(77)=(-3213+3*a(95)+4*a(98)-a(99)+2*a(100)+9*a(117)+5*a(118)+4*a(119)+10*a(120)+6*a(122)+3*a(123)+3*a(124)+9*a(125))/6
a(76)=(-567+3*a(95)-4*a(98)+a(99)+a(100)+3*a(117)-2*a(118)-a(119)+2*a(120)+6*a(122)-3*a(123)+6*a(124))/3
a(75)=(945-2*a(98)-a(99)-a(100)-3*a(117)-a(118)+a(119)-2*a(120)-3*a(125))/3
a(74)=+a(98)+a(100)-a(124)
a(73)=(-a(98)+4*a(99)-2*a(100)+3*a(117)+a(118)-a(119)+2*a(120)-3*a(123))/3
a(72)=(945+a(98)-4*a(99)-a(100)-3*a(117)-a(118)+a(119)-2*a(120)-3*a(122))/3
a(71)=(-945-a(98)+a(99)+a(100)+3*a(117)+a(118)-a(119)+2*a(120)+3*a(122)+3*a(123)+3*a(124)+3*a(125))/3
a(70)=(-945-a(95)+2*a(98)+a(99)+2*a(100)+3*a(117)+3*a(118)+2*a(120)+3*a(123)-a(124)+3*a(125))/2
a(69)=(-189+3*a(95)-2*a(98)+2*a(99)-a(100)+3*a(117)-a(118)-2*a(119)+a(120)+3*a(122)-3*a(123)+3*a(124))/3
a(68)=(567-a(98)-2*a(99)+a(100)-3*a(117)-5*a(118)-a(119)-a(120)+3*a(122)+3*a(124))/3
a(67)=(378-3*a(95)+a(98)+2*a(99)-a(100)+2*a(118)+a(119)+a(120)-3*a(122)-3*a(124))/3
a(66)=(3213+3*a(95)-2*a(98)-7*a(99)-4*a(100)-9*a(117)-a(118)+4*a(119)-8*a(120)-6*a(122)-3*a(123)-3*a(124)-9*a(125))/6
a(65)=(945+9*a(95)-2*a(98)-a(99)-4*a(100)+3*a(117)-a(118)-2*a(119)-2*a(120)-9*a(123)+3*a(124)-3*a(125))/6
a(64)=(-1701-3*a(95)-2*a(98)-a(99)+2*a(100)+3*a(117)-a(118)-2*a(119)+4*a(120)+6*a(122)+9*a(123)+9*a(124)+9*a(125))/6
a(63)=63
a(62)=(819+a(95)-2*a(98)-a(99)-3*a(117)-3*a(118)-2*a(120)+2*a(122)-a(123)+a(124)-3*a(125))/2
a(61)=(-189-9*a(95)+10*a(98)+5*a(99)+2*a(100)+3*a(117)+11*a(118)+4*a(119)+4*a(120)-12*a(122)+3*a(123)-15*a(124)+3*a(125))/6
a(60)=(945-9*a(95)+2*a(98)+a(99)-2*a(100)-3*a(117)+a(118)+2*a(119)+2*a(120)-6*a(122)+3*a(123)-3*a(124)+3*a(125))/6
a(59)=(1701+3*a(95)+2*a(98)-5*a(99)-2*a(100)-3*a(117)+a(118)+2*a(119)-4*a(120)-6*a(122)-3*a(123)-3*a(124)-3*a(125))/6
a(58)=-63-a(98)+a(122)+a(123)+a(124)
a(57)=(-567-3*a(95)+8*a(98)+a(99)+4*a(100)+3*a(117)+7*a(118)+2*a(119)+2*a(120)-6*a(122)+3*a(123)-9*a(124)+3*a(125))/6
a(56)=(63+3*a(95)-2*a(98)+a(99)+a(117)-3*a(118)-2*a(119)+4*a(122)-3*a(123)+3*a(124)-a(125))/2
a(55)=(945+3*a(95)-2*a(98)-a(99)+2*a(100)-3*a(117)-7*a(118)-2*a(119)-2*a(120)+6*a(122)-3*a(123)+3*a(124)-3*a(125))/6
a(54)=(1134-3*a(95)-a(98)+a(99)-2*a(100)-3*a(117)+a(118)+2*a(119)-a(120)-3*a(122)-3*a(124)-3*a(125))/3
a(53)=(378+5*a(98)-2*a(99)+a(100)+4*a(118)+2*a(119)-a(120)-6*a(122)-6*a(124))/3
a(52)=-441+a(95)-a(98)+a(99)+2*a(117)-a(119)+a(120)+2*a(122)+2*a(124)+a(125)
a(51)=(189-a(95)-a(99)-a(117)-a(118)+a(123)+a(124)+a(125))/2
a(50)=(-945+2*a(98)+a(99)+a(100)+3*a(117)+a(118)-a(119)+2*a(120)+3*a(122)+3*a(123)+3*a(125))/3
a(49)=315-a(98)-a(100)-a(123)-a(125)
a(48)=(945+a(98)-4*a(99)+2*a(100)-3*a(117)-a(118)+a(119)-2*a(120)-3*a(122)-3*a(124))/3
a(47)=(-945-a(98)+4*a(99)+a(100)+3*a(117)+a(118)-a(119)+2*a(120)+3*a(122)+3*a(124)+3*a(125))/3
a(46)=(945+a(98)-a(99)-a(100)-3*a(117)-a(118)+a(119)-2*a(120)-3*a(122)-3*a(125))/3
a(45)=(945+a(95)-2*a(98)-a(99)-2*a(100)-a(117)-a(118)-2*a(120)-3*a(123)+a(124)-3*a(125))/2
a(44)=(1134-3*a(95)+2*a(98)-2*a(99)+a(100)-3*a(117)-2*a(118)-a(119)-4*a(120)-3*a(122)+3*a(123)-3*a(124))/3
a(43)=(378+a(98)+2*a(99)-a(100)+2*a(118)-2*a(119)+a(120)-3*a(122)-3*a(124))/3
a(42)=(-378+3*a(95)-a(98)-2*a(99)+a(100)-2*a(118)+2*a(119)+2*a(120)+3*a(122)+3*a(124))/3
a(41)=(-3213-3*a(95)+2*a(98)+7*a(99)+4*a(100)+9*a(117)+7*a(118)+2*a(119)+8*a(120)+6*a(122)+3*a(123)+3*a(124)+9*a(125))/6
a(40)=(945-9*a(95)+2*a(98)+a(99)+4*a(100)-3*a(117)+a(118)+2*a(119)-4*a(120)-6*a(122)+3*a(123)-3*a(124)+3*a(125))/6
a(39)=(1701+3*a(95)+2*a(98)+a(99)-2*a(100)-3*a(117)+a(118)-4*a(119)-4*a(120)-6*a(122)-3*a(123)-3*a(124)-3*a(125))/6
a(38)=-63-a(118)+a(122)+a(123)+a(124)
a(37)=(-189-a(95)+2*a(98)+a(99)+a(117)+3*a(118)+2*a(120)-2*a(122)+a(123)-3*a(124)+a(125))/2
a(36)=(189+9*a(95)-10*a(98)-5*a(99)-2*a(100)+3*a(117)-5*a(118)+2*a(119)+2*a(120)+12*a(122)-9*a(123)+9*a(124)-3*a(125))/6
a(35)=(-945+9*a(95)-2*a(98)-a(99)+2*a(100)+3*a(117)-a(118)-2*a(119)+4*a(120)+6*a(122)-3*a(123)+3*a(124)+3*a(125))/6
a(34)=(-1701-3*a(95)-2*a(98)+5*a(99)+2*a(100)+3*a(117)-a(118)+4*a(119)+4*a(120)+6*a(122)+3*a(123)+9*a(124)+3*a(125))/6
a(33)=63+a(98)+a(118)-a(122)-a(124)
a(32)=(567+3*a(95)-8*a(98)-a(99)-4*a(100)+3*a(117)-7*a(118)-2*a(119)-2*a(120)+12*a(122)-3*a(123)+9*a(124)-3*a(125))/6
a(31)=(1197-3*a(95)+2*a(98)-a(99)-3*a(117)+a(118)-2*a(120)-6*a(122)+a(123)-5*a(124)-a(125))/2
a(30)=(945-3*a(95)+2*a(98)+a(99)-2*a(100)-3*a(117)+a(118)+2*a(119)+2*a(120)-6*a(122)+3*a(123)-3*a(124)-3*a(125))/6
a(29)=(-1134+3*a(95)+a(98)-a(99)+2*a(100)+3*a(117)+2*a(118)+a(119)+4*a(120)+3*a(122)+3*a(125))/3
a(28)=(-378-5*a(98)+2*a(99)-a(100)+3*a(117)-a(118)+a(119)+a(120)+6*a(122)-3*a(123)+6*a(124))/3
a(27)=756-a(95)+a(98)-a(99)-2*a(117)-2*a(120)-3*a(122)-2*a(124)-a(125)
a(26)=(-189+a(95)+a(99)+a(117)-a(118)-2*a(119)+2*a(122)+a(123)+a(124)+a(125))/2
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)=(945+a(98)-a(99)+2*a(100)-3*a(117)-a(118)+a(119)-2*a(120)-3*a(122)-3*a(124)-3*a(125))/3
a(21)=(-945+2*a(98)+4*a(99)+a(100)+3*a(117)+a(118)-a(119)+2*a(120)+3*a(122)+3*a(125))/3
a(20)=315-a(95)-a(117)-a(118)-a(120)
a(19)=(1323-a(95)-a(99)-2*a(100)-3*a(117)+a(118)-2*a(120)-4*a(122)-a(123)-3*a(124)-3*a(125))/2
a(18)=(189+2*a(98)-2*a(99)+a(100)+a(118)+2*a(119)-a(120)-3*a(122)+3*a(123)-3*a(124))/3
a(17)=(-3213+3*a(95)+2*a(98)+7*a(99)+4*a(100)+9*a(117)+7*a(118)-4*a(119)+8*a(120)+6*a(122)+3*a(123)+3*a(124)+9*a(125))/6
a(16)=-189+a(95)-a(98)+a(117)-a(118)+a(120)+2*a(122)-a(123)+2*a(124)
a(15)=(315+a(95)-2*a(98)-a(99)-a(117)-a(118)+2*a(122)-a(123)+a(124)-a(125))/2
a(14)=(1134-3*a(95)+2*a(98)+a(99)+a(100)-3*a(117)-2*a(118)+2*a(119)-4*a(120)-3*a(122)-3*a(124)-3*a(125))/3
a(13)=(378+4*a(98)+2*a(99)-a(100)+5*a(118)-2*a(119)+a(120)-6*a(122)-6*a(124))/3
a(12)=(-1323+3*a(95)-2*a(98)-a(99)-a(100)+6*a(117)-a(118)+a(119)+4*a(120)+6*a(122)+6*a(124)+3*a(125))/3
a(11)=(567-3*a(95)-2*a(98)-a(99)+2*a(100)-3*a(117)-a(118)-2*a(119)-2*a(120)+3*a(123)+3*a(124)+3*a(125))/6
a(10)=(945-3*a(95)+2*a(98)+a(99)+4*a(100)-3*a(117)+a(118)+2*a(119)-4*a(120)-6*a(122)+3*a(123)-3*a(124)-3*a(125))/6
a(9)=(-1134+3*a(95)+a(98)+2*a(99)+2*a(100)+3*a(117)+2*a(118)-2*a(119)+4*a(120)+3*a(122)+3*a(125))/3
a(8)=(-378-2*a(98)+2*a(99)-a(100)+3*a(117)-4*a(118)+a(119)+a(120)+6*a(122)-3*a(123)+6*a(124))/3
a(7)=(2268-3*a(95)+2*a(98)-2*a(99)-2*a(100)-6*a(117)+a(118)-a(119)-4*a(120)-9*a(122)-6*a(124)-3*a(125))/3
a(6)=(-567+3*a(95)-4*a(98)-5*a(99)-2*a(100)+3*a(117)+a(118)+2*a(119)+2*a(120)+6*a(122)+3*a(123)+3*a(124)+3*a(125))/6
a(5)=(-1890+3*a(95)+2*a(98)+a(99)+a(100)+6*a(117)+4*a(118)-a(119)+5*a(120)+3*a(122)+3*a(123)+6*a(125))/3
a(4)=(-693+a(95)-2*a(98)+a(99)+3*a(117)-a(118)+2*a(120)+4*a(122)-a(123)+5*a(124)+a(125))/2
a(3)=(756-a(98)-2*a(99)+a(100)-3*a(117)-2*a(118)-a(119)-a(120))/3
a(2)=(1323-3*a(95)-4*a(98)+a(99)-2*a(100)-3*a(117)-5*a(118)+2*a(119)-4*a(120)+6*a(122)-3*a(123)+3*a(124)-3*a(125))/6
a(1)=(2457-3*a(95)+4*a(98)-a(99)-a(100)-6*a(117)+2*a(118)+a(119)-5*a(120)-12*a(122)-9*a(124)-6*a(125))/3

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(98) ... a(100) and 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 ... 94, 96, 97, 101 ... 116, 121
Int(a(i)) = a(i)      for all broken equations
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 Moriyama Pantriagonal Magic Cubes, which will be discussed in Section 5.6a.

4.9.3 Analytic Solution (2)

Although not consisered in the analysis above, the Moryama Cube is an Associated Pantriagonal Magic Cube.

This results in following additional equations:

a(1) + a(125) = 126
a(2) + a(124) = 126
a(3) + a(123) = 126
a(4) + a(122) = 126
a(5) + a(121) = 126
a(6) + a(120) = 126
a(7) + a(119) = 126
a(8) + a(118) = 126
a(9) + a(117) = 126
a(10) + a(116) = 126
a(11) + a(115) = 126
a(12) + a(114) = 126
a(13) + a(113) = 126
a(14) + a(112) = 126
a(15) + a(111) = 126
a(16) + a(110) = 126

a(17) + a(109) = 126
a(18) + a(108) = 126
a(19) + a(107) = 126
a(20) + a(106) = 126
a(21) + a(105) = 126
a(22) + a(104) = 126
a(23) + a(103) = 126
a(24) + a(102) = 126
a(25) + a(101) = 126
a(26) + a(100) = 126
a(27) + a(99) = 126
a(28) + a(98) = 126
a(29) + a(97) = 126
a(30) + a(96) = 126
a(31) + a(95) = 126
a(32) + a(94) = 126

a(33) + a(93) = 126
a(34) + a(92) = 126
a(35) + a(91) = 126
a(36) + a(90) = 126
a(37) + a(89) = 126
a(38) + a(88) = 126
a(39) + a(87) = 126
a(40) + a(86) = 126
a(41) + a(85) = 126
a(42) + a(84) = 126
a(43) + a(83) = 126
a(44) + a(82) = 126
a(45) + a(81) = 126
a(46) + a(80) = 126
a(47) + a(79) = 126
a(48) + a(78) = 126

a(49) + a(77) = 126
a(50) + a(76) = 126
a(51) + a(75) = 126
a(52) + a(74) = 126
a(53) + a(73) = 126
a(54) + a(72) = 126
a(55) + a(71) = 126
a(56) + a(70) = 126
a(57) + a(69) = 126
a(58) + a(68) = 126
a(59) + a(67) = 126
a(60) + a(66) = 126
a(61) + a(65) = 126
a(62) + a(64) = 126


which can be added to the equations deducted in Section 4.9.2 above and results finally in following linear equations:

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(112) - a(113) - a(114) - a(115)
a(110) =       a(111) + a(118) - a(124)
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(116) - a(118) - a(121) - a(123)
a(106) = 315 - a(107) - a(108) - a(109) - a(110)
a(105) = 315 - a(110) - a(115) - a(120) - a(125)
a(104) = 315 - a(109) - a(114) - a(119) - a(124)
a(103) = 315 - a(108) - a(113) - a(118) - a(123)
a(102) = 315 - a(107) - a(112) - a(117) - a(122)
a(101) = 315 - a(106) - a(111) - a(116) - a(121)
a(100) = 189 - a(116) - a(123)
a( 99) = 189 - a(117) - a(122) + a(123) - a(124)
a( 98) =-126 - a(118) + 2 * a(122) + 2 * a(124)
a( 97) = 189 - a(119) - a(122) + a(123) - a(124)
a( 96) = 315 - a( 97) - a( 98) - a( 99) - a(100)
a( 95) = 504 - 2 * a(117) - a(118) - 2 * a(120) - a(122) - a(125)
a( 94) = 189 - a(116) + a(120) - a(121) - a(123)
a( 93) = 189 - a(122) - a(124)
a( 92) = 189 + a(116) - a(120) - a(123) - a(125)
a( 91) = 315 - a( 92) - a( 93) - a( 94) - a( 95)
a( 90) =-126 + 2 * a(117) + a(118) + 3 * a(120) - 2 * a(121) - a(124)
a( 89) = 189 - a(117) - a(118) - 2 * a(120) + a(121) + a(124)
a( 88) =-126 + a(118) + a(121) + a(125)
a( 87) =  63 - a( 95) + a(119)
a( 86) = 315 - a( 87) - a( 88)- a( 89) - a( 90)
a( 85) = 819 - 2 * a(117) - 2 * a(118) - a(119) - 3 * a(120) - a(122) - a(123) - 2 * a(125)
a( 84) = 189 - a(116) - 2 * a(119) + a(123) - a(124) + a(125)
a( 83) =  63 - a( 97) + a(117)
a( 82) = 504 - 2 * a(117) - a(120) - 2 * a(122) - a(124) - a(125)
a( 81) = 315 - a( 82) - a( 83) - a( 84) - a( 85)
a( 80) = 315 - a( 85) - a( 90) - a( 95) - a(100)
a( 79) = 315 - a( 84) - a( 89) - a( 94) - a( 99)
a( 78) = 315 - a( 83) - a( 88) - a( 93) - a( 98)
a( 77) = 315 - a( 82) - a( 87) - a( 92) - a( 97)
a( 76) = 315 - a( 81) - a( 86) - a( 91) - a( 96)
a( 75) = 378 - a(117) - a(120) - a(122) - a(124) - a(125)
a( 74) =-252 + a(117) + a(119) + a(120) + 2 * a(122) - a(123) + a(124)
a( 73) = 378 - a(117) - a(119) - 2 * a(122) + a(123) - 2 * a(124)
a( 72) =-126 + a( 79) + a(124) + a(125)
a( 71) = 315 - a( 72) - a( 73) - a( 74) - a(75)
a( 70) = 252 - a( 86) - a(121) - a(125)
a( 69) = 252 - a( 87) - a(122) - a(124)
a( 68) =  63 - a(118) + a(122) - a(123) + a(124)
a( 67) = 189 + a( 91) - a(117) - a(122) - a(124)
a( 66) = 315 - a( 67) - a( 68) - a( 69) - a( 70)
a( 65) = 189 + a( 86) - a(120) - a(122) - a(123)
a( 64) = 189 + a( 81) - a(119) - a(120) - a(122) 

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)

Also these equations can be applied in an efficient method to generate Associated Pantriagonal Magic Cubes, which will be discussed in Section 5.6b.


Vorige Pagina Volgende Pagina Index About the Author