Solution

[Obviously we don't publish the answer till next time, but the table should make it clear what kind of interactivity I was thinking of. We don't need two different colours -- that's just for the explanation.]

Let's arrange the numbers in rows of 11. We know we can get any score in the right-hand column as it's a multiple of 11, so we'll shade that column. We can also certainly make any multiple of 7, so let's colour those numbers green.

Once we can make a particular score, we can also make any score below it in the table by collecting additional jewels. So we can make all the shaded scores in the table, and any number higher than 60. The highest "impossible" score is 59.