I'm very sick so this is a very bad solve. This solution converts the input from binary to base 6. The solve starts by grabbing X quicksilver where X is the value of the input. It then sets one metal as X mod 6, and the other metal as the remainder. To do modulo, my system divides by 6 repeatedly, making a chain of length X mod 6. That chain then rotates Rivari's wheel that many times. For the remainder, I just take the leftover stick and use it to purify a lead. To save sanity, I encode the 8th place and the 16th place with salt instead of 6 quicksilver. It's not strictly necessary but I didn't feel like programming the division arm 5 times, nor dealing with the geometry that would imply. I hope this X mod 6 and remainder method will be better used by someone else because this is tragic. 

Thanks for an amazing tournament Haxton, although having no real cost puzzle is definitely the reason I have Pneumonia right now :P