All the weighings must divide the possibilities into three more-or-less equal groups, so that the result of the weighing divides the number of possibilities remaining by three.
So, each weighing must be one where “left scale heavier than right scale”, “right scale heavier than left scale”, and “scales equal”, are more or less equally likely.
Now you can work out what to do at the first weighing. There are only six options for the first weighing – one randomly-chosen ball against another one, two randomly-chosen against another two, 3 against 3, 4 against 4, 5 against 5, 6 against 6 – so if it comes to it you can just check out all six to find the one which narrows the possibilities from 24 to 8.
Work out for yourself, for each first-weighing outcome, a second-weighing operation to narrow down from 8 to 3, 3, or 2.
Then work out for each second-weighing outcome what third-weighing method to use to narrow down from 3 (or 2) to one. And you’re done! Now write down your answer, hand it in, and claim your prize!