Perfect dissections
 

It is well know that is possible to partition a square into smaller squares (cf. below for an example to clarify the meaning). Prove that it is not possible to partition a cube into smaller cubes. (This is a well-known and famous problem but accessible to everyone here. I hope you enjoy it.)

http://i3.photobucket.com/albums/y73...are21_1100.gif

Re: Perfect dissections
 

I thought it was. For example you can partition a 9x9x9 cube into 27 3x3x3 cubes. Or do the cubes have to be different sizes?

Re: Perfect dissections
 

"Prove that it is not possible to partition a cube into smaller cubes."

Ummmm, you better stipulate your conditions better (i.e., must be different sizes).

Take 8 equal cubes and you have 1 big cube.

Re: Perfect dissections
 

If you look at the bottom of the cubed cube; you will see a squared square. The smallest square, S, in this squared square cannot be on the boundary. Therefore, the cube with S for a face is surrounded by four larger cubes, so its opposite face abuts another squared square. We can now look at the smallest square in this squared square. It is then the face of some cube surrounded by four larger cubes, which we can look at the opposite face of, and so on: since this process can be continued indefinitely, there can be no cubed cube.

The difference between 2D and 3D is in the starred statement. The smallest line in a segmented segment can be on the boundary, but the smallest square in a squared square cannot be.

Re: Perfect dissections
 

The cubes must be of different sizes. It's pretty clear that some sort of stipulation of this sort was implied from

1. It is [well-known] that is possible to partition a square into smaller squares. I wouldn't have said well-known if it was in fact trivial.
2. The figure indicates we're looking for something special
3. The problem is trivial and false without some sort of stipulation.

Thanks for helping me clarify it.

Re: Perfect dissections
 

[ QUOTE ]
"Prove that it is not possible to partition a cube into smaller cubes."

Ummmm, you better stipulate your conditions better (i.e., must be different sizes).

Take 8 equal cubes and you have 1 big cube.

[/ QUOTE ]

8, eh?

Re: Perfect dissections
 

I don't understand why it's necessary to force the shape of the cube or square to remain intact.

Re: Perfect dissections
 

[ QUOTE ]
[ QUOTE ]
"Prove that it is not possible to partition a cube into smaller cubes."

Ummmm, you better stipulate your conditions better (i.e., must be different sizes).

Take 8 equal cubes and you have 1 big cube.

[/ QUOTE ]

8, eh?



[/ QUOTE ]

Yes, you can create a cube with 8 equal cubes (2x2x2). Do you not agree?

Re: Perfect dissections
 

that would be 4 i think

Re: Perfect dissections
 

[ QUOTE ]
that would be 4 i think

[/ QUOTE ]

2 x 2 x 2 = 4? [img]/images/graemlins/confused.gif[/img]