Office Applications and Entertainment, Magic Cubes

3.9    Associated Magic Cubes

3.9.1  Introduction

An Associated Magic Cube is a Simple Central Symmetric Magic Cube, for which the defining properties can be summarised as follows:

• All Rows, Columns and Pillars sum to the Magic Sum s1 and
• all Central Symmetric Pairs sum to s1/2.

3.9.2  Analytic Solution

The defining properties result, after deduction of the corresponding equations, in following set of linear equations describing the Associated Magic Cubes.

a(61) =   s1 - a(62) - a(63) - a(64)
a(57) =   s1 - a(58) - a(59) - a(60)
a(53) =   s1 - a(54) - a(55) - a(56)
a(52) =   s1 - a(56) - a(60) - a(64)
a(51) =   s1 - a(55) - a(59) - a(63)
a(50) =   s1 - a(54) - a(58) - a(62)
a(49) =-2*s1 + a(54) + a(55) + a(56) + a(58) + a(59) + a(60) + a(62) + a(63) + a(64)
a(45) =   s1 - a(46) - a(47) - a(48)
a(42) = 2*s1 - a(43) - a(46) - a(47) - a(58) - a(59) - a(62) - a(63)
a(41) = - s1 - a(44) + a(46) + a(47) + a(58) + a(59) + a(62) + a(63)
a(40) =      - a(44) + a(46) + a(47) - a(56) - a(60) + a(62) + a(63)
a(39) = 2*s1 - a(43) - a(46) - a(47) - a(55) - a(59) - a(62) - a(63)
a(38) =        a(43) - a(54) + a(59)
a(37) = - s1 + a(44) + a(54) + a(55) + a(56) + a(60)
a(36) =   s1 - a(46) - a(47) - a(48) + a(56) + a(60) - a(62) - a(63)
a(35) = - s1 + a(46) + a(55) + a(59) + a(62) + a(63)
a(34) = - s1 + a(47) + a(54) + a(58) + a(62) + a(63)
a(33) = 2*s1 + a(48) - a(54) - a(55) - a(56) - a(58) - a(59) - a(60) - a(62) - a(63)

with a(43), a(44), a(46) ... a(48), a(54) ... a(56), a(58) ... a(60), a(62) ... a(64) the independent variables.

The equations deducted above can be applied in an efficient method to generate Associated Magic Cubes, which will be discussed in Section 3.12.6.

3.10   Pantriagonal/Associated Magic Cubes

3.10.1 Introduction

A Pantriagonal/Associated Magic Cube is a Central Symmetric Magic Cube for which also the Pantriagonals sum to the Magic Sum.

3.10.2 Analytic Solution

The defining properties result, after deduction of the corresponding equations, in following set of linear equations describing the Pantriagonal/Associated Magic Cubes.

a(61) = s1 - a(62) - a(63) - a(64)
a(57) = s1 - a(58) - a(59) - a(60)
a(55) =      a(56) + a(58) + a(60) - a(62) - a(63)
a(54) = s1 - a(56) - a(58) - a(60)
a(53) =    - a(56) + a(62) + a(63)
a(52) = s1 - a(56) - a(60) - a(64)
a(51) = s1 - a(56) - a(58) - a(59) - a(60) + a(62)
a(50) =      a(56) + a(60) - a(62)
a(49) =-s1 + a(56) + a(58) + a(59) + a(60) + a(64)
a(47) =    - a(48) + a(59) + a(60)
a(46) = s1 + a(48) - a(58) - 2 * a(59) - a(60)
a(45) =    - a(48) + a(58) + a(59)
a(44) =    - a(48) + a(59) + a(63)
a(43) =      a(48) - a(59) + a(64)
a(42) = s1 - a(48) + a(59) - a(62) - a(63) - a(64)
a(41) =      a(48) - a(59) + a(62)
a(40) = s1 + a(48) - a(56) - a(58) - 2 * a(59) - a(60) + a(62)
a(39) = s1 - a(48) - a(56) + a(59) - a(60) - a(64)
a(38) =-s1 + a(48) + a(56) + a(58) + a(60) + a(64)
a(37) =    - a(48) + a(56) + a(59) + a(60) - a(62)
a(36) =    - a(48) + a(56) + a(58) + a(59) + a(60) - a(62) - a(63)
a(35) =      a(48) + a(56) - a(59)
a(34) =    - a(48) - a(56) + a(59) + a(62) + a(63)
a(33) = s1 + a(48) - a(56) - a(58) - a(59) - a(60)

with a(48), a(56), a(58) ... a(60) and a(62) ... a(64) the independent variables.

Based on the above listed equations, a routine can be written to generate Pantriagonal/Associated Magic Cubes of order 4 (ref. PrimeCubes4e option 'Natural Numbers').

Subject guessing routine produced, with a(64) constant, 28368 cubes within 11,25 hours, of which 120 are shown in Attachment 3.10.2. The total number of Pantriagonal/Associated Magic Cubes is 64 * 28368 = 1815552 (= 48 * 37824).

An alternative efficient method to generate Pantriagonal/Associated Magic Cubes, based on the equations deducted above, will be discussed in Section 3.12.7.

3.11a  Spreadsheet Solutions (1)

The linear equations deducted in sections 3.2.1, 3.2.2, 3.7.2, 3.8.2, 3.9.2 and 3.10.2 above, have been applied in following Excel Spread Sheets:

• CnstrSngl4a, Almost Perfect Magic Cubes
  
  
• CnstrSngl4b, Almost Perfect Magic Cubes with Symmetrical Horizontal Planes
  
  
• CnstrSngl4c, Pantriagonal Magic Cubes, 2D Compact and Complete
  
  
• CnstrSngl4d, Pandiagonal/Triagonal Magic Cubes
  
  
• CnstrSngl4e, Simple Associated Magic Cubes
  
  
• CnstrSngl4f, Pantriagonal/Associated Magic Cubes

Only the red figures have to be “guessed” to construct one of the applicable Magic Cubes of the 4^th order (wrong solutions are obvious).

3.11b  Summary

The obtained results regarding the miscellaneous types of order 4 Magic Cubes as deducted and discussed in previous sections are summarized in following table:

The Quaternary Solutions will be discussed in Section 3.12.
More 4^th order Magic Cubes will be discussed in Section 3.13 and Section 3.14.