Hint 1 for the 12 billiard balls maths prize
Often in maths it is good to start with a simplified version of the problem, or to start at the end and work backwards.
And it’s often good first to think about the shape of the answer before plunging into trying to work out details.
Those bits of advice all point the same way here. Think first about the third (last) weighing.
You’ve started with 12×2=24 possibilities: the odd ball may be any one of 1 to 12, and it may be heavy or light.
Your two first weighings must have narrowed the range. To how few possibilities?