Completeness theorem on order-8 Nasik magic cubes (2/3)

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Completeness theorem on order-8 Nasik magic cubes (2/3)

The following lemma is the core of the proof of the completeness theorem.

Lemma 2
Let (a_ijk) be an order-8 (normal or non-normal) Nasik magic cube (namely, pan-2,3-agonal magic cube), where i, j, and k are integers from 0 to 7,
then the pair of the cells, a₀₀₀ and a₄₄₄, is a complement pair.

a₀₀₀	a₀₀₁	a₀₀₂	a₀₀₃	a₀₀₄	a₀₀₅	a₀₀₆	a₀₀₇
a₀₁₀	a₀₁₁	a₀₁₂	a₀₁₃	a₀₁₄	a₀₁₅	a₀₁₆	a₀₁₇
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a₀₄₀	a₀₄₁	a₀₄₂	a₀₄₃	a₀₄₄	a₀₄₅	a₀₄₆	a₀₄₇
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Proof
   It is enough to prove that the sum of the cells, a₀₀₀ and a₄₄₄, equals to a quarter of the magic constant.
Let s be the magic constant.
By the application of the Lemma 1 to slices of the cube, we get the following equations (every slice of a pan-2,3-agonal magic cube is a pandiagonal magic square):

(a₀₀₀ + a₀₄₀ + a₄₀₀ + a₄₄₀) + (a₂₂₀ + a₂₆₀ + a₆₂₀ + a₆₆₀) = s,    (2.1)
(a₀₀₄ + a₀₄₄ + a₄₀₄ + a₄₄₄) + (a₂₂₄ + a₂₆₄ + a₆₂₄ + a₆₆₄) = s,    (2.2)
(a₀₀₀ + a₀₀₄ + a₄₀₀ + a₄₀₄) + (a₂₀₂ + a₂₀₆ + a₆₀₂ + a₆₀₆) = s,    (2.3)
(a₀₄₀ + a₀₄₄ + a₄₄₀ + a₄₄₄) + (a₂₄₂ + a₂₄₆ + a₆₄₂ + a₆₄₆) = s,    (2.4)
(a₂₀₂ + a₂₀₆ + a₂₄₂ + a₂₄₆) + (a₂₂₀ + a₂₂₄ + a₂₆₀ + a₂₆₄) = s,    (2.5)
and
(a₆₀₂ + a₆₀₆ + a₆₄₂ + a₆₄₆) + (a₆₂₀ + a₆₂₄ + a₆₆₀ + a₆₆₄) = s.    (2.6)

By the calculation of ((2.1)+(2.2)+(2.3)+(2.4)-(2.5)-(2.6)) / 2, we get the equation
(a₀₀₀ + a₀₀₄ + a₀₄₀ + a₀₄₄) + (a₄₀₀ + a₄₀₄ + a₄₄₀ + a₄₄₄) = s.    (2.7)

On the other hand, we obtain the following equations by the application of the Lemma 1 to oblique slices of the cube:

(a₀₀₀ + a₀₀₄ + a₄₄₀ + a₄₄₄) + (a₂₂₂ + a₂₂₆ + a₆₆₂ + a₆₆₆) = s,    (2.8)
(a₀₄₀ + a₀₄₄ + a₄₀₀ + a₄₀₄) + (a₂₂₂ + a₂₂₆ + a₆₆₂ + a₆₆₆) = s,    (2.9)
(a₀₀₀ + a₀₄₀ + a₄₀₄ + a₄₄₄) + (a₂₂₂ + a₂₆₂ + a₆₂₆ + a₆₆₆) = s,    (2.10)
(a₀₀₄ + a₀₄₄ + a₄₀₀ + a₄₄₀) + (a₂₂₂ + a₂₆₂ + a₆₂₆ + a₆₆₆) = s,    (2.11)
(a₀₀₀ + a₄₀₀ + a₀₄₄ + a₄₄₄) + (a₂₂₂ + a₆₂₂ + a₂₆₆ + a₆₆₆) = s,    (2.12)
and
(a₀₄₀ + a₄₄₀ + a₀₀₄ + a₄₀₄) + (a₂₂₂ + a₆₂₂ + a₂₆₆ + a₆₆₆) = s.    (2.13)

By the calculation of ((2.7)+(2.8)-(2.9)+(2.10)-(2.11)+(2.12)-(2.13)) / 4, we get the equation
a₀₀₀ + a₄₄₄ = s/4.    Q.E.D

Here is the proof illustrated. (You might have to wait a moment until the page is displayed completely.)

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This page was last updated on October 1, 2007.
"Magic Cubes and Tesseracts" http://homepage2.nifty.com/googol/magcube/en/
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