IBM Ponder This #244 · August 2018

There is a well-known riddle about two friends (Alice and Bob) who went walking in the desert.
Alice had two gallons of water and Bob had three gallons.
They met Charlie who had no water at all and they all (Alice, Bob, and Charlie) shared the five gallons of water evenly.
As a token of his gratitude, Charlie gave them five gold coins.
What is the fair way to split these five coins?
Hint: it is *not* the obvious two to Alice and three to Bob. (Why?)

Our challenge this month is to solve a similar problem:
In J.R.R. Tolkien's works, nine rings were given to humans. Our challenge is this: Each one of the nine ring holders brought W_i droplets of water (W_1, W_2, W_3, ..., W_9).
Sauron (who had the one ring to rule them all) came without any water at all. They all shared their water; and each one of the ring holders got G_i (G_1, G_2, G_3, ..., G_9) tiny nuggets of gold in return. Where W_1+W_2+...+W_9 = G_1+G_2+...+G_9.
However, the 18 W_i and G_i water droplet and gold nugget amounts, respectively, are all different and at least 17 of them are prime numbers.
Find the nine W_i's.

Solution

Best opened after a real attempt

To be added.