1. ## Slicing a parallelopipe

Hi, if there is a rectangular parallelopipe x*y*z ,made of 1*1*1 units then into how many parts can it be sliced by appling cuts at most n times parallel to it's parallel sides?
E.g if x=y=z=2 and n=3
then number of parts = 8.
If x=y=z then i think answer is x^n
but what for other cases?
Note: the cuts are made along integer co-ordinates.
Thanks.

2. Originally Posted by pranay
Hi, if there is a rectangular parallelopipe x*y*z ,made of 1*1*1 units then into how many parts can it be sliced by appling cuts at most n times parallel to it's parallel sides?
E.g if x=y=z=2 and n=3
then number of parts = 8.
If x=y=z then i think answer is x^n
but what for other cases?
Note: the cuts are made along integer co-ordinates.
Thanks.
I'm not sure that I understand your question completely and correctly. So I'm guessing:

1. Let $N_{xy}, N_{xz}, N_{yz}$ denote the number of cuts parallel to the xy-plane, xz-plane and yz-plane respectively. With each cut you'll get N +1 pieces.

Then the number A of pieces which you'll get by slicing the parallelepiped is calculated by:

$A = (N_{xy}+1) \cdot (N_{xz}+1) \cdot (N_{yz}+1)$

2. If $N_{xy} = N_{xz} = N_{yz}$ then $A = (N+1)^3$