Before reading the answer can I interest you in a clue?
Most people seem to think that the thing to
do is weigh six coins against six coins, but if you think about it, this would
yield you no information concerning the whereabouts of the only different
coin or the nature of it's difference as we already know that one side will be heavier than the other. It's worth mentioning that if this is used as an interview question
this observation, by itself, is likely to get you a good portion of the way there.
So that the following plan can be followed, let us number the coins from 1 to
12. For the first weighing let us put on the left pan coins 1,2,3,4 and on the
right pan coins 5,6,7,8.
There are two possibilities. Either they balance, or they don't. If they
balance, then the different coin is in the group 9,10,11,12. So for our second
one possibility is to weigh 9,10,11 against 1,2,3
(1) They balance, in which case you know 12 is the different coin, and you just weigh it against any other to determine whether it is heavy or light.
(2) 9,10,11 is heavy. In this case, you know that the different coin is 9, 10, or 11, and that that coin is heavy. Simply weigh 9 against 10; if they balance, 11 is the heavy coin. If not, the heavier one is the heavy coin.
(3) 9,10,11 is light. Proceed as in the step above, but the coin you're looking for is the light one.
That was the easy part.
What if the first weighing 1,2,3,4 vs 5,6,7,8 does not balance? Then any one
of these coins could be the different coin. Now, in order to proceed, we must
keep track of which side is heavy for each of the following weighings.
Suppose that 5,6,7,8 is the heavy side. We now weigh 1,5,6 against 2,7,8. If
they balance, then the different coin is either 3 or 4. Weigh 4 against 9, a
known good coin. If they balance then the different coin is 3, otherwise it is
4. The direction of the tilts can tell us whwther the offending coin is heavier or lighter.
Now, if 1,5,6 vs 2,7,8 does not balance, and 2,7,8 is the heavy side, then
either 7 or 8 is a different, heavy coin, or 1 is a different, light coin.
For the third weighing, weigh 7 against 8. Whichever side is heavy is the
different coin. If they balance, then 1 is the different coin. Should the
weighing of 1,5, 6 vs 2,7,8 show 1,5,6 to be the heavy side, then either 5 or
6 is a different heavy coin or 2 is a light different coin. Weigh 5 against 6.
The heavier one is the different coin. If they balance, then 2 is a different