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S-magic cubes of order 4 are impossible Walter Trump, 2003-06-17 Definitions


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s-magic cubes of order 4 are impossible Walter Trump, 2003-06-17

Definitions

A s-magic cube is a magic cube where all squares in the six surface planes are (non-normal) magic squares.

In this paper a "perfect" magic cube is a magic cube in which all squares in the orthogonal planes are magic.

Abstract


In Scientific American, January, 1976 Martin Gardner shows a proof that Richard Schroeppel published in a memorandum in 1972.
Schroeppel proved the non-existence of "perfect" order-4 cubes.

In this paper Schroeppel's proof is generalized in order to prove that magic surface cubes of order 4 are impossible.


Lemma 1


In his first lemma Schroeppel shows that the sum of the integers in the four corners of a magic square equals the magic constant.

We generalize this lemma and show that the sum of the corners equals the magic constant S in a 4-by-4 number square where the two inner columns, the two diagonals, the first and the last row are magic.



a1

b1

b2

a2




c1

c2







c3

c4




a3

b3

b4

a4

Illustration 1

Let a = a1 + a2+ a3+ a4 , b = b1 + b2+ b3+ b4 and c = c1 + c2+ c3 + c4 .

We obtain three equations:
(1) a + b = 2S because two rows are magic
(2) a + c = 2S because the diagonals are magic
(3) b + c = 2S because two columns are magic

Now subtract (3) from (1) to obtain


a – c = 0  a = c

and replace C in equation (2):


a + a = 2S  a = S (Lemma 1)

With this new lemma the further proof of Schroeppel can be applied to magic surface cubes:



Assumption

Assume that all surfaces of a magic order-4 cube are magic squares.

Then all oblique squares have the properties of the above shown number square, because two of their edges are the diagonals of magic squares on the cube surface and the diagonals of an oblique square are the triagonals (space digonals) of the cube. Therefore the corners of any oblique square must sum up to the magic constant. (In illustration 2 for exsample the corners E, A, C, G belong to an oblique square.)

Of course also the corners of all squares in the surfaces have the sum S.



Lemma 2

Now Schroeppel proves that any two corners connected by an edge must have the sum S/2.




Illustration 2

Consider the front square, the right square and the above mentioned oblique square. Again we build three equations:

(1) A + B + E + F = S  (A + E) + (B + F) = S
(2) B + C + F + G = S  (C + G) + (B + F) = S
(3) A + E + C + G = S  (A + E) + (C + G) = S

Subtract (2) from (1) to obtain:

(A + E) – (C + G) = 0  (A + E) = (C + G)
Replace (C + G) in equation (3):

(A + E) + (A + E) = S  (A + E) = S/2 (Lemma 2)



Conclusion

Now consider the corner A and conclude from Lemma 2:

A + E = A + B  E = B

But a normal magic cube cannot have the same integer in two different corners.



Thus the proof is complete. A s-magic cube of order 4 is impossible.


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