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 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.

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:

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.