First Puzzle:

You have been given 9 weights, one of which is slightly heavier than the rest, and a balance so that you can compare them.

Can you always work out which weight is heavier in just two weighings of the balance?

Could you always work out which is heavier in just two weighings if you had been given 10 weights?

Second Puzzle:

You have now been given 27 weights, one of which is slightly heavier than the rest. 

Can you always work out which weight is heavier in just three weighings of the balance?
 

Third Puzzle:

You now have 9 weights again, and know that one is a slightly different weight, but you don't know if it is heavier or lighter.
Can you explain how you would always be able to tell which weight is different, and whether it is heavier or ligther, in just three weighings ?

Could you always work out which is the odd one out in just three weighings if you had been given 10 weights?

A very challenging follow-up puzzle that you might like to try is The Great Weights Puzzle.