I was asked the following in an interview yesterday:
Let V(n) denote the volume of the unit n-sphere, and similarly, let A(n) denote the surface area of the unit n-sphere.
Let i and j be natural numbers greater than or equal to zero such that for all k not equal to i and all p not equal to j, V(i) > V(k) and A(j) > A(p), what is the value of i - j? My answer was 1, and I will provide my reasoning, let me know what you guys think.
First I noted that if i - j = x, I could deduce that i = j + x. Thus, I thought to look for a relationship between the volume and surface area of n-spheres. I realized that you could obtain a 3-sphere by fixing a 2-sphere (a circle) and rotating it 360 degrees, filling in the space the 2-sphere passes through.
Similarly, if you consider a 1-sphere to be a line of length 2, and rotate it 360 degrees about the midpoint using a similar thought process, you obtain a 2-sphere. So I conjectured that there is some constant c such that for a given n, V(n) = c * A(n-1), where c is related to the 360 degree rotation. Feel free to elaborate on this c, I didn't dig into it and am interested in what it is (if it does indeed exist).
So now, let i be such that for all k not equal to i, V(i) > V(k).
Then we know that c * A(i - 1) > c * A(k - 1) => A(i - 1) > A(k - 1).
Thus, let j = i - 1 and let p be some natural number not j. We obtain V(j + 1) = V(i) > V(p + 1) which gives us A(j) = A(i - 1) > A(p), thus, j = i - 1 is correct.
And thus, i - j = i - (i - 1) = 1.
What are your thoughts on my solution?