This is very amusing. Straight after you left I got the answer.
We know that d = kr;
We need to go about expressing each of the gems in terms of k and r.
2d = e + r
2kr = e + r
Therefore e = 2kr - r = r(2k-1)
s = d + e
s = kr + r(2k-1) = r(3k-1)
We have one more condition, 14r > s:
14r > s
14r > r(3k-1)
14r > 3kr - r
15r > 3kr
5r > kr
5 > k
That gives us somewhere to work with!
d + e > e + 3r, thus d > 3r:
This means that for d = kr, k > 3.
Obviously k is a positive integer such that 3 < k < 5, which means k = 4.
Therefore, Indiana Jones places four rubies on the other side.