Monday, October 27, 2008

Area visible outside in a hyper cube

In K-dimensional space, if a hyper cube of size N^k is formed with N^k unit hypercubes, how many hyper cubes are visible from outside?

There is a three dimensional cube of size NxNxN formed with NxNxN cubes. How many cubes are visible from outside?

For example,
In one dimensional space (i.e., in a line), there are two visible points.

In two dimensional space, for 3x3 square, there are 8 squares visible from outside.

In three dimensional space, for 3x3x3 cube, there are 26 cubes visible from outside.

If it is in k-dimensional space, how many hyper cubes are visible?


The hypercube is in K-Dimensional space. There are two visible sides in each dimension. The length of one side is N. If we remove the two visible cubes that are visible from outside, the remaining length is N-2. If we do the same in all the dimensions, then in all the dimensions, the length would be reduced to N-2. The volume of the remaining is (N-2)^k, and this is not visible. The original volume is N^k. So, the no.of hyper cubes that are visible is N^k - (N-2)^k.