2. A conference hall has a round table with $n$ chairs. There are $n$ delegates to the conference. The first delegate chooses his or her seat arbitrarily. Thereafter the $(k+1)$th delegate sits $k$ places to the right of the $k$th delegate, for $1\le k\le n-1$. (In particular, the second delegate sits next to the first.) No chair can be occupied by more than one delegate.

Find the set of values $n$ for which this is possible.

For question two, my answer was that n must be a power of 2. This wasn't proved very well at all. I treated each delegate as a triangle number (0,3,6,10,15etc), and tried to show that the triangle numbers up to the nth triangle number were all different (mod n) only for n = 2^r . I'm not even sure this is right, but I can prove n can't be odd.

You seem to have the right idea for Q2; the way I did it was to consider factorising n as 2^a (2b+1) for integers a and b, and showing that if b > 0 there always exist j and k with j(j+1)/2 congruent to k(k+1)/2 mod n. (In fact I think you can always make j(j+1)/2 - k(k+1)/2 = n; take two cases according to whether or not b > 2^a ).

David