Integrals I: Feynman's Trick · Superchallenge problem

Six people around a table each hold some coins, $15$ in total. A move lets a person with at least $2$ coins give one coin to each neighbor. Which of these totals of $\sum k\cdot c_k$ (position $k = 1, \ldots, 6$ times coins $c_k$) modulo $6$ is invariant? Give the value of $\sum kc_k \bmod 6$ if initially all $15$ coins are at position $1$.