perplexus.info :: Numbers : Cubic Diagonals and Edges

Cubic Diagonals and Edges (Posted on 2009-04-30)

This is the net map of the outer surfaces of the 20 edge cubes that represents
a 3 x 3 x 3 'parent' cube (like in a Rubik cube).

A	B	C
D		E
F	G	H

A	D	F
I		J
M	N	O

F	G	H
J		K
O	P	Q

H	E	C
K		L
Q	R	S

C	B	A
L		I
S	T	M

O	P	Q
N		R
M	T	S

Assign unique values from 1 to 20 to the letters A-T such that the sum of each pair of diametrically opposite cubes is to be the same as all others while the sum of each set of edge cubes may not differ from that of any other set by more than one.

Eg, diagonals:
                    A + Q = N + E = C + O ...(etc)
and edge cubes:     A + B + C = C + E + H = F + G + H ...etc,
                or (A + B + C) ±1 = C + E + H
                                  = F + G + H ....(etc)

Note: The problem's development and my solution used a spreadsheet; as such a well-constructed sheet could enable a solution. Although this problem may lend itself to a programmed solution I would appreciate seeing attempts of a more manual basis within the first 24-48 hrs.

See The Solution Submitted by brianjn

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re(2): computer solutions

| Comment 5 of 6 |

(In reply to re: computer solutions by brianjn)

They are unique as placing the lowest numbered corner at A and rotating/reflecting so that B<D<I assures that a given configurations appears only once.

For example, solution 1 is the only one where there is a 1 in a corner spot and it's surrounded by 11, 13 and 17.

To take another example, while solutions 29, 30 and 31 all have the least corner as a 3, which is surrounded by 11, 16 and 20, when oriented in this way there are other differences, and getting those other differences into some semblance of a same place as another solution, the 3, 11,16,20 relationship would be destroyed.

Edited on May 3, 2009, 12:24 pm

Posted by Charlie on 2009-05-03 12:13:31

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