Boy, is this a stinker!

Ok...here goes...number coins 1 to 12.

1st weigh - 1 2 3 4 against 5 6 7 8

If balanced, fake must be from 9 10 11 12

2nd weigh - 9 10 11 against 5 6 7 (good coins)

If balanced, fake must be 12 and just need to determine whether heavier or lighter, so...

3rd weigh 1 against 12 will tell us

If, after second weigh, it is not balanced it tells us fake coin is 9 10 or 11 and whether heavier or lighter than the rest.

Then 3rd weigh would be 9 against 10

If balanced, 11 is the fake and we'll know from the second weigh if it was heavier or lighter

However, if 9 is heavier than 10 and 9 10 and 11 were heavier on the second weigh we know that 9 is the fake, which is heavier than the other coins. If 10 is heavier than 9 and 9 10 and 11 were also heavier on the second weigh then 10 is the fake, which is heavier than the other coins. The same principle applies if 9 10 and 11 were lighter.

Now the really tricky part

If after first weigh 1 2 3 4 were not balanced with 5 6 7 8 we know the fake coin is one of these eight coins and it will tell us which group is heavier or lighter

Then the 2nd weigh is 1 2 5 against 3 6

**12**(a good coin - the stinker

)

If balanced then we know 4, 7 or 8 is the fake,

So then 3rd weigh 7 against 8. If balanced 4 is the fake and from the 1st weigh we can tell if its heavier or lighter. Otherwise, if 7 is heavier than 8, the first weigh will have told us it if fake is heavier or lighter (for example, if 5 6 7 8 in first weigh was heavier and 7 is heavier than 8 in last then 7 is the fake, which is heavier than the good coins (you can see the other permutations on this one

)

If however, 1 2 5 is heavier than 3 6 12 we know the fake is 1 2 5 3 or 6

If on 1st weigh 1 2 3 4 was heavier than 5 6 7 8 then 3rd weigh 1 against 2 whichever is heavier is fake and is heavier than the good coins. If balanced then 6 is the fake and is lighter

If 1 2 5 is heavier than 3 6 12, but 5 6 7 8 was heavier than 1 2 3 4 (on 1st weigh) then 5 or 3 must be the fake so weigh 5 against 1 (a good coin), if unbalanced its the fake and we're told whether heavier or lighter. If balanced 3 is the fake and from 1st weigh tells us if heavier or lighter

I'm sure this is probably a convoluted way of solving the problem and

my explanations could be honed down, but is this right?