I am also failing the two "big" test cases. I have done a little memoization, avoiding to re-check all hops of previously checked rivers.
E.g. checking 3 as a possible start, if 3 -> 6 -> 12 -> 15 -> 21... doesnt meet my target number, then later I skip 6, 12, 15, 21 and all other hops of river 3. I thought this would solve it, but I am still timing out the big cases. Any ideas to help me along?