# Thread: splitting a cube into 4 parts

1. ## splitting a cube into 4 parts

hi, if we have a cube of size n*n*n then how can we say if it can be splitted into 4 pieces (not necessarily cubes) by two cuts (one horizontal and other vertical)? .e.g if we have a cube of size 3*3*3 and the parts are 12,3,6,6 then we can say that the cube can be split into these parts by 2 cuts..however if the cube is 2*2*2 and the parts are 5,1,1,1 then not.. Is it to do with lcm ? Thanks.

2. ## Re: splitting a cube into 4 parts

There are a lot of conditions you are not stating. You are obviously requiring that the slices be by integer steps.
If you have a cube n by n by n and split it horizontally at height h, and vertically at width k, then the four parts you divide the cube into are h by k by n, n-h by k by n, h by n-k by n, and n-h by n-k by n. those have volume nkh, (n-h)kn, h(n-k)n, and (n-h)(n-k)n. h and k can be any integers greater than 0 and less than n.

What you are noting is that if n= 3, then we can take you can take h= 1 and k= 1 so that nkh= 3(1)(1), (n-h)kn= 2(1)(3)= 6, h(n-k)n= (1)(2)(3)= 6, and (n-h)(n-k)(n)= 2(2)(3)= 12. But if n= 2 the only possible value for h and k is 1. Then nkh= 2(1)(1)= 2, h(n-k)n= 1(1)(2)= 2, h(n-k)(n)= 1(1)(2)= 2, and (n-h)(n-k)n= 1(1)(2)= 2. That is, the only possible way to divide a 2 by 2 by 2 cube into four pieces by integers is to divide two sides in the middle.

3. ## Re: splitting a cube into 4 parts

thanks Hallsoflvy..so basically i need to check for every combination of h and k such that 1<h,k<n and calculate the volume of the pieces thus comparing if the formed volumes are the same as given?...Isn't there any relationship possible between h,k so that i can reduce the number of checks..