Find fake coin

[Aus Lover]

You have 12 gold coins, except one is fake but the only difference is that it is a slightly different weight.
You do not know if it is heavier or lighter.
You have a set of balancing scales on which you can put the coins
You are allowed 3 weighs to be able to guarantee finding the fake one

How do you do it?

[Fibonacci]

divide 12 coins into 3 sets of 4.
weigh 1
Weigh 4 coins against another 4 coins.
- If they are equal: Label 4 unweighed coins C1C2C3C4 and the 8 equal coins all as X Go to Method A.
- If they are unequal: Label the heavy side H1H2H3H4 and the light side L1L2L3L4 and the remaining coins X; Go to Method B

Method A
Weigh 2
This is the easy method since you only have 4 candidates for the odd coin (C1-4).
Weight two coins from the candidate group C1C2 against two coins from the equal group XX. If they are unequal you know that either C1 or C2 is the odd coin; if they are equal you know it is C3 or C4. Either way you have only 2 candidates left.

Weigh 3
Weigh one of your candidates against one from the X group. If it is unequal it is your odd coin; if it is equal the other coin is you odd coin.

Method B
Ok now the slightly trickier one. We have 8 candidate odd coins; fortunately only 4 can be the coin if the odd coin is light and the other 4 can be the coin if the odd coin is heavy. So we have 4 light candidates (L1-4)and 4 heavy candidates (H1-4).

Weigh 2*
Weigh H1H2L1L2 against H3L3XX.(xx = two coins form unused set of 4)
- If the left side is heavier the odd coin is either H1 or H2 or L3; go to Weigh 3.1
- If the right side is heavier the odd coin is either L1 or L2 or H3; go to Weigh 3.2
- If the sides are equal: either L3 or L4 is the odd coin; go to Weigh 3.3

*There are quite a few permutations of this weighing. The key is to leave yourself a group which contains no more than 2 candidate coins from either the heavy or light group.

Weigh 3.1
Weigh H1 against H2 if one is heavier that one is the odd coin; if they are equal L3 is.

Weigh 3.2
Weigh L1 against L2 if one is lighter that one is the odd coin; if they are equal H3 is.

Weigh 3.3