I recently came across a practice maths interview question, which asked: 'what is the maximum number of slices one can obtain from n straight cuts'. I soon realised the answer was 1 + (the sum of the first n integers). I was intrigued to see if this could be generalised, and so I thought of a straight line, and realised that the maximum number of 'slices' of the line from n 'cuts' (i.e. dots), was 1 + (the first n integers raised to the power 0). However i was very dissatisfied when I realised that for a sphere and plane cuts, the answer isn't 1 + (the sum of the first n integers squared).
My question is the following. Is there a general formula, for the maximum numbers of k-dimensional portions one can obtain from 'slicing' a k-dimensional sphere with n (k-1)-dimensional cuts?
Apologies for the formatting. I need to learn LaTeX soon. Also, if anyone has any good tags I can use on this question, please do let me know. Thanks.