n different spaces can divide the space into a)at most how many sub spaces?
b)at least how mant sub spaces?
You did not provide enough information for an answer. The phrase "n different spaces can divide the space ..." implies that there is some space being divided. What space are you trying to divide? How are the n different spaces supposed to divide it?
One plane will divide 3 dimensional space into two disjoint regions but neither is a "subspace".
n parallel planes will divide 3 dimensional space into n+ 1 regions. n parallel planes and m planes perpendicular to those will divide 3 dimensional space into (n+ 1)(m+ 1) regions. If we a have total of n+ m= N planes that will b maximized by taking m and n as close to N/2 as possible. If N is even, m= n= N/2 gives . If N is odd taking one of m and n equal to (N-1)/2 and the other (N+1)/2 gives [tex]\frac{N^2+ 4N+ 3}{4}[tex].