I decided I wanted one of those dartboard puzzles for this website, the kind where you are given the scores for each ring of the dartboard, and you have to determine how many darts fell in each ring to get a given total.

I chose 4 different scores for the 4 rings, then I picked a total N that could only be reached one way. It required 10 darts. I noticed that each of the totals N+1, N+2, N+3, ..., N+10 also could be reached only one way, and in each case fewer than 10 darts were needed.

What is the smallest value of N for which this is possible?