From dik@cwi.nl Fri Nov 1 20:10:55 1991
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Date: Sat, 2 Nov 91 02:10:47 +0100
From: dik@cwi.nl
Message-Id: <9111020110.AA30522@paring.cwi.nl>
To: ccw@eql.caltech.edu, cube-lovers@ai.mit.edu
Subject: Re: What is the smallest cube which has all possible types of pieces?
> Some cube discussion has started in rec.games.misc. I am trying to
> re-direct it to rec.puzzles, or hopefully cube-lovers.
I follow up in cube-lovers.
>
> Actually the 7x7x7 has a type of piece that the 5x5x5 does not have.
Actually the 7x7x7 is also the first cube that can not be made.
>
> (Showing a pattern like: ABCD
> EFG
> HIJ
> K.
> Noting that B&C, E&I, G&J, H&F are similar and can be solved by the same
> set of procedures.)
But, all of E to J can be solved with the same set of procedures %. So
although there appear to be 7 essentially different cubelets, actually
there are only 5. And the 5x5x5 cube has them all. Solving these involves
rotating the three slices they are in originally. So if we rename cubelet
types we get:
1: A
2: B&C
3: D
4: E-J
5: K
and we find for the different cubes:
2x2x2: Type 1
3x3x3: Type 1, 3 and 5
4x4x4: Type 1, 2 and 4
5x5x5: All types.
--
% All of E to J occur 4 times on each face, so for each type there are 24
cubelets. B and C also occur 24 times, procedures working for these
two also work for E to J (although I use different procedures both on
4x4x4 and 5x5x5); but there are procedures for E to J that do not work
for B to C (these two have more constraints on position). This would be
different if the cube was patterned such that all cubelets of type B, C
and E to J had a unique position, in that case we would have only four
essentially different types.