• #### Re: 17 Piece puzzle cube problem

From Rizki Fadillah@21:1/5 to All on Sun Mar 5 23:21:01 2023
In article <steve-01090...@newtyr21.tyler.net>,
st...@puzzlecraft.com (Steve Strickland) wrote:
In article <8onooc\$sab\$1...@nnrp1.deja.com>, kans...@my-deja.com wrote:

As I said above there are 6 2x2x3, 6 2x1x4 and 5 1x1x1 pieces. There
are no irregular-shaped pieces at all. They fit (supposedly) into a
5x5x5 cube.
I'll run it this weekend and let you know.
Oh come on. Why use a computer program on such an easy puzzle?
You can pretty much deduce the location of every piece. I wrote
down the solution in pen and I don't even have the pieces in
front of me.
Fact 1: Each of 15 layers of the cube (5 layers in each of the
three perpendicular directions) must contain exactly one 1x1x1
piece.
Reason: The simple parity argument already mentioned a few times.
The most obvious way of achieving this is to place the 1x1x1s along
a main diagonal of the cube but there are other ways.
Fact 2: Each of the 1x2x4s must be placed so that its 2x4 rectangle
is in an outer face of the cube and furthermore, each of the outer
faces must contain exactly one such 2x4 rectangle.
Reason: There must be a 1x1x1 piece in each second layer. What piece
can fill the adjacent cubicle in the outer layer? A 2x2x3 piece
can't fit and we can't use a 1x1x1 without violating fact 1. Therefore
it must be a 1x2x4 oriented so that the 2x4 rectangle is in the
outer face. This is true for each of the 6 outer faces and there
are precisely 6 1x2x4s. Fact 2 follows.
Fact 3: From the reasoning in fact 2, we see that of the 8 cubicles
in a second layer adjacent to the outer 1x2x4, 1 of these cubicles
is filled by a 1x1x1 piece. 6 of the other cubicles must be filled
by a 2x3x3 and one must be filled by part of a 1x2x4.
Reason: left as an exercise (I'm too lazy to write it out)
Fact 4: Each 1x2x4 in an outer layer fills up all of two rows/columns
except two cubicles. These cubicles must be filled with a 1x1x1 piece
and part of a 1x2x4.
Reason: left as an exercise.

Now the bottom layer must have a 1x2x4 oriented with a 2x4 rectangle
in the bottom face. There are essentially two distinct ways of
doing this. For one of the placements, its not hard to see that the
above constraints cannot be satisfied which leaves only one
possibility. This placement and the above constraints force the
locations of half the pieces. The other half is symmetric and
voila! you are done. Now go solve your puzzle.
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