I like to play with a wooden puzzle I have. It consists of the 12 pieces you can make when you'd glue 5 small squares together in every possible way. So you would have a piece like this:
xxxxx
and this:
xxxx x
and this
xxx xx
and so on.
Anyway, it is not very hard to put the pieces together in the box again. But you can also make all sorts of three dimensional figures out of it. I used to have a booklet with examples, but I lost it. Now I am trying to make this figure:
bottom: 6 x 6 first floor: 4x4 in the middle of the 6x6 second and third floor: 2x2 in the middle of the 4x4
Well, now the question: I don't succeed, how could you prove mathematically whether it is possible or not to make this figure?
Posted by Icarus (Member # 3162) on :
I'm confused: 12 pieces, or 56? Posted by ginette (Member # 852) on :
Oh... 12 Because you have mirrors, they don't count Posted by Icarus (Member # 3162) on :
o_O
I see.
Not.
Posted by FlyingCow (Member # 2150) on :
It's 12 pieces of 5 blocks each, Ic, for a total of 60 blocks.
Base has 36 blocks, first floor has 16 blocks, and the next two floors have 4 blocks each - for a total of 60 blocks.
Each piece has a different configuration of 5 blocks, sort of like Tetris (only with 5 squares instead of 4... Pentris, perhaps?).
He's trying to make a certain configuration of three dimensional pieces (made up of 5 blocks per pice) into a three dimensional tower shape, I think.
If there's a mathematical way to prove it can or can't be done, I don't know what it is. Without the exact configuration of each of the 12 pieces, I don't know if that's even possible.
Posted by James Tiberius Kirk (Member # 2832) on :
(She.)
--j_k
Posted by Sharpie (Member # 482) on :
quote:Originally posted by ginette: Oh... 12 Because you have mirrors, they don't count
I have to say that I enjoy this answer VERY much!
Posted by ginette (Member # 852) on :
Thank you so much, Flying Cow, to help explain. It's exactly right what you are saying. Except that I am a she indeed
The configuration of the pieces is:
xxxxx
xxxx x
xxx xx
x xx _xx
x xxx x
_x xxx _x
xx x xx
xxxx _x
xx _xxx
xx _x _xx
x x xxx
_x xxx x
(Edited for lost spaces )
Yeah... I thought so Sharpie
Hi James! Nice to see you again
Are you REALLY going to think about this Icarus ?
[ October 25, 2006, 02:10 PM: Message edited by: ginette ]
Posted by The Rabbit (Member # 671) on :
The pattern
x xxx x
occurs three times in that list.
The pattern
xxxx x
occurs twice.
Is this a mistake or do you have three replicates of these patterns.
Posted by Icarus (Member # 3162) on :
And there's no (significant) third dimension?
(I dunno, Ginette. It's hard for me to imagine this without something to physically manipulate. )
Posted by The Rabbit (Member # 671) on :
I can only come up with ten unique permutations. What are the last two
XXXXX
XXXX X
XXXX _X
XXX XX
XXX X_X
XXX X X
XXX _X _X
XX _X _XX
XX _XX _X
_X XXX _X
Posted by rivka (Member # 4859) on :
quote:Originally posted by Icarus: (I dunno, Ginette. It's hard for me to imagine this without something to physically manipulate. )
Yeah, I'm having the same trouble.
Posted by The Rabbit (Member # 671) on :
quote:Originally posted by Icarus: And there's no (significant) third dimension?
As I understand it Icarus, Each of the blocks is a cube.
I think to get the 12 permutations you have to add a third dimension
X(2X)X __X
XX(2X) X
XX(2X) ___X
where the 2X means that there is a two blocks stacked in the third dimension. But now I'm getting more than 12 permutations.
Posted by ginette (Member # 852) on :
Oops. Something gone wrong while typing those figures. Well, The Rabbit showed the right ones. Then we have this one:
X XX _X _X
and this one:
XX _XX __X
to make it twelf
Posted by ginette (Member # 852) on :
And the pieces are flat. Just 5 squares glued together in a flat figure.
Posted by aspectre (Member # 2222) on :
[ October 25, 2006, 02:25 PM: Message edited by: aspectre ]
Posted by ginette (Member # 852) on :
Yeah!!! Thank you so much for the links aspectre! I tried to find them myself ofcourse, but I couldn't find anything, that's why I posted my question here.
Posted by The Rabbit (Member # 671) on :
If you allow out of plane arrangements you start getting quite a few more permutations. Such as
XXX _(2X)
XXX (2X)
Additionally, I think you get stereoisomers this way which give you even more combinations.
ginette. I need help to figure out which pieces are actually in the puzzle.
Also note that
XXX XX
and
XX XX X
are identical.
Posted by The Rabbit (Member # 671) on :
Thanks for the edit ginette. That make it clear. No out of plane cubes and two I was missing
are
XX _XXX
and
X XX _XX
Posted by ginette (Member # 852) on :
If only I'd have known those things are called Pentominos! Here is another link Pentomino's They even have a computer program for solving pentomino problems. Hmm.
Posted by King of Men (Member # 6684) on :
(The following is my understanding of a field I don't work in; may contain inaccuracies.) The little blocks are called pentominoes, and are a fairly popular field for amateur mathematicians; however, it is my understanding that it is quite difficult to prove any general results with them, because the tools don't really exist to do so. (And probably won't until someone with some good formal maths training gets annoyed and really takes a full-on swing at it.) All you can do is play around with visualising it, which is fine for constructive solutions but doesn't do so well at proving there are none.
![[Smile]](smile.gif)
![[Confused]](confused.gif)
![[Big Grin]](biggrin.gif)