Office Applications and Entertainment, Magic Cubes

V   Associated Magic Cubes, Magic Center Planes

V-1 Introduction

In this Exhibit V, the variables of a Magic Cube of order 6 are assigned as shown below:

All rows, columns, pillars and the main diagonals of the six center planes sum to the Magic Sum s1.
The associated pairs sum to s1/3.

V-2 Horizontal Magic Center Planes

The equations describing the Horizontal Magic Center Plane 13 can be written as:

c(103) = s1 - c(104) - c(105) - c(106) - c(107) - c(108)
c( 73) = s1 - c( 80) - c( 87) - c( 94) - c(101) - c(108)
c( 77) = s1 - c( 83) - c( 89) - c (95) - c(101) - c(107)
c( 78) = s1 - c( 83) - c( 88) - c( 93) - c( 98) - c(103)
c( 74) = s1 - c( 80) - c( 86) - c( 92) - c( 98) - c(104)
c( 76) = s1 - c( 73) - c( 74) - c( 75) - c( 77) - c( 78)
c( 82) = s1 - c( 76) - c( 88) - c( 94) - c(100) - c(106)
c( 81) = s1 - c( 75) - c( 87) - c( 93) - c( 99) - c(105)
c( 91) = s1 - c( 92) - c( 93) - c( 94) - c( 95) - c( 96)
c( 85) = s1 - c( 86) - c( 87) - c( 88) - c( 89) - c( 90)
c( 79) = s1 - c( 80) - c( 81) - c( 82) - c( 83) - c( 84)
c( 97) = s1 - c( 73) - c( 79) - c( 85) - c( 91) - c(103)
c(102) = s1 - c( 84) - c( 90) - c( 96) - c( 78) - c(108)

with the independent variables (23 ea) highlighted in red.

Solutions for these equations can be generated quite fast by calculating sequentially the bottom row, the main diagonals and the remaining rows and columns.

The variables of the Horizontal Center Plane 14 are complementary to Plane 13 (Associated).

V-3 Vertical Magic Center Planes (L/R)

The equations describing the Vertical Magic Center Plane 23 (third plane from the left) can be written as:

c( 33) = s1 - c( 63) - c( 93) - c(123) - c(153) - c(183)
c(  3) = s1 - c( 45) - c( 87) - c(129) - c(171) - c(213)
c(207) = s1 - c(171) - c(135) - c( 99) - c( 63) - c( 27)
c(189) = s1 - c(153) - c(117) - c( 81) - c( 45) - c(  9)
c( 15) = s1 - c(  3) - c(  9) - c( 21) - c( 27) - c( 33)
c(195) = s1 - c(183) - c(189) - c(201) - c(207) - c(213)
c(165) = s1 - c(201) - c(129) - c( 93) - c( 57) - c( 21)
c(159) = s1 - c(195) - c(123) - c( 87) - c( 51) - c( 15)
c( 69) = s1 - c( 39) - c( 45) - c( 51) - c( 57) - c( 63)
c(177) = s1 - c(213) - c(141) - c(105) - c( 69) - c( 33)
c(147) = s1 - c(153) - c(159) - c(165) - c(171) - c(177)

With the independent variables (13 ea) highlighted in red and the previously defined variables hatched (grey).

Solutions for these equations can be generated quite fast by calculating sequentially the main diagonals and the remaining rows and columns.

The variables of the Vertical Center Plane 24 (forth plane from the left) are complementary to Plane 23 (Associated).

V-4 Vertical Magic Center Planes (B/F)

The equations describing the Vertical Magic Center Plane 34 (forth plane from the back) can, be deducted to:

c(199) = s1 -  c(164) - c(129) - c( 94) - c( 59) - c( 24)

c(167) = s1 -  c( 56) - c(164) - c( 59) +
            - (c(165) + c(166) - c(127) + c(129) + c(130) - c(132) - c(91) + 
                               + c( 93) + c( 94) - c( 96) + c( 57) + c(58)) / 2

c(204) = s1 -  c(167) - c(130) - c( 93) - c( 56) - c( 19)
c( 20) = s1 -  c( 23) - c( 19) - c( 24) - c( 21) - c( 22)
c( 55) = s1 -  c( 60) - c( 56) - c( 59) - c( 57) - c( 58)
c(168) = s1 -  c( 60) - c(204) - c( 24) - c(132) - c( 96)
c(163) = s1 -  c( 55) - c( 19) - c(199) - c(127) - c( 91)
c(203) = s1 -  c( 23) - c(167) - c( 59) - c(131) - c( 95)
c(200) = s1 -  c( 20) - c( 56) - c(164) - c(128) - c( 92)

With the independent variables (7 ea) highlighted in red and the previously defined variables hatched (grey).

Solutions for these equations can be generated quite fast by calculating sequentially the main diagonals and the remaining rows and columns.

The variables of the Vertical Center Plane 24 (forth plane from the left) are complementary to Plane 23 (Associated).

V-5 Horizontal Top and Bottom Planes

Finally the remainder of the cube, consisting out of eight Corner Cubelets, has to be determined.

The corresponding equations can be deducted to:

c( 31) = s1 - c(32) - c(35) - c(36) - c(33) - c(34)
c( 25) = s1 - c(26) - c(29) - c(30) - c(27) - c(28)
c(  6) = s1 - c(12) - c(30) - c(36) - c(18) - c(24)
c(  5) = s1 - c(11) - c(29) - c(35) - c(17) - c(23)
c(  7) = s1 - c( 8) - c(11) - c(12) - c( 9) - c(10)
c(  2) = s1 - c( 8) - c(26) - c(32) - c(14) - c(20)
c(  1) = s1 - c( 7) - c(25) - c(31) - c(13) - c(19)

c(181) = s1/3 - c(36)   c(182) = s1/3 - c(35) 
c(185) = s1/3 - c(32)   c(186) = s1/3 - c(31)
c(187) = s1/3 - c(30)   c(188) = s1/3 - c(29) 
c(191) = s1/3 - c(26)   c(192) = s1/3 - c(25)
                    
c(205) = s1/3 - c(12)   c(206) = s1/3 - c(11) 
c(209) = s1/3 - c(8)    c(210) = s1/3 - c(7)
c(211) = s1/3 - c(6)    c(212) = s1/3 - c(5)  
c(215) = s1/3 - c(2)    c(216) = s1/3 - c(1)

c(37) = s1/3 - c(180)   c(38) = s1/3 - c(179)
c(43) = s1/3 - c(174)   c(44) = s1/3 - c(173)
c(65) = s1/3 - c(152)   c(66) = s1/3 - c(151)
c(71) = s1/3 - c(146)   c(72) = s1/3 - c(145)

c(61) = s1 - c(205) - c(169) - c(133) - c( 97) - c(25)
c(62) = s1 - c(206) - c(170) - c(134) - c( 98) - c(26)
c(67) = s1 - c(211) - c(175) - c(139) - c(103) - c(31)
c(68) = s1 - c(212) - c(176) - c(140) - c(104) - c(32)

c(41) = s1 - c(185) - c(149) - c(113) - c(77) - c(5)
c(42) = s1 - c(186) - c(150) - c(114) - c(78) - c(6)
c(47) = s1 - c(191) - c(155) - c(119) - c(83) - c(11)
c(48) = s1 - c(192) - c(156) - c(120) - c(84) - c(12)

c(175) =   s1   - c(176) - c(177) - c(178) - c(179) - c(180)
c(173) = 2*s1/3 - c(174) - c(179) - c(180)

c(145) =   s1   - c(181) - c(109) - c( 73) - c(37) - c(1)
c(146) =   s1   - c(182) - c(110) - c( 74) - c(38) - c(2)
c(151) =   s1   - c(187) - c(115) - c( 79) - c(43) - c(7)
c(152) =   s1   - c(188) - c(116) - c( 80) - c(44) - c(8)

c(170) =   s1   - c(146) - c(152) - c(158) - c(164) - c(176)
c(169) =   s1   - c(145) - c(151) - c(157) - c(163) - c(175)

c(149) =   s1/3 - c(68)    c(150) = s1/3 - c(67)
c(155) =   s1/3 - c(62)    c(156) = s1/3 - c(61)

With the independent variables (13 ea) highlighted in red and the previously defined variables hatched (grey).

V-6 Conclusion

The linear equations deducted in previous sections can be incorporated in a guessing routine for the Magic Sum s1 = 21, and the integers 0, 1 ... 7.

The solutions can be obtained by guessing the 56 independent variables, while ensuring that the 27 Cubelets contain each these 8 different integers.