Hint 3

Given that you only have one weigh, you need to find some way of "remembering" which coins come from which bag. One way to do this, is to try and attach a label. Numbers make for a good label. Let's label the ten bags with the numbers 1 through 10. Now you need to think of a way of "attaching" the number 1 to the coins which come from bag 1, the number 2 to the coins which come from bag 2, and so on.

Attaching some kind of algebra, like numbers, to objects is one of the most powerful tools of modern mathematics. That's what algebraic geometry and algebraic topology are all about.