Three is the correct answer. Here's the math. Twelve coins translates to 24 possible outcomes - Coin 1 is heavier, Coin 1 is lighter, Coin 2 is heavier, Coin 2 is lighter, . . . Using the balance, there are three possible outcomes - Tilts to the left, balances, tilts to the right. The lowest power of 3 that exceeds 24 is 3
3 = 27. So three weightings should cover it.
The next thing is to ensure that no two ball have the same weigh pattern.
Here are the weigh patterns for each ball. The first denotes right scale for weighing 1, left for 2, right for 3.
- R | L | R
- R | R | L
- L | R | R
- L | L | -
- - | L | L
- R | - | R
- L | R | -
- - | R | L
- R | - | L
- - | - | R
- - | L | -
- L | - | -
Results:
Left down, right down, left down = Coin 1 is heavier
Right down, left down, right down = Coin 1 is lighter
Left down, left down, right down = Coin 2 is lighter
Right down, right down, left down = Coin 2 is heavier
. . . .
Left heavier, even, left heavier = Coin 6 is lighter
Right heavier, even right heavier = Coin 6 is heavier
. . . .
Even, even, right is heavier = Coin 10 is heavier.
. . . .
And so on.