You have nine brass rings, but three are actually gold. Can you pick these out using a balance scale three times at the most?

I think that the words 'one of' was left out of the problem statement:

You have nine brass rings, but three are actually gold. Can you pick **one of** these out using a balance scale three times at the most?

Now, finding one gold ring out of the group is possible in only three weighings. I will use the fact that gold is heavier than brass.

Make three groups of three: A, B and C. Weigh A vs B.

If A>B then A must have at least one gold ring.

If A<B then B must have at least one gold ring.

If A=B then C must have at least one gold ring.

Take which group is known to have a gold ring and label the three coins in the group D, E and F. Weigh D vs E.

If D>E then D is a gold ring.

If D<E then E is a gold ring.

If D=E then weigh D vs F. If not equal, the heavier is a gold coin; and if equal, all three are gold coins.