Hint 2 for the 12 billiard balls maths prize (follows hint 1)

A weighing can at best sort possibilities into *three* groups: when the right-hand scale is heavier, when the right-hand scale is lighter, when the scales are even.

So before the third weighing you must have reduced the possibilities to *three* (or fewer).

You start with 24 possibilities before the first weighing. You have reduced them to three possibilities after the second weighing. How few must you have narrowed them down to after the *first* weighing?

Click here for hint 3

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