With the preliminaries out of the way, let's now describe a
strategy for solving the puzzle.

First weighing  weigh 4 coins against 4 coins. There are
two possible outcomes  the scale is in balance or the scale is out of
balance. It may be tempting to say that there are three possible
outcomes  the left side of the scale is heavier, the right side of the
scale is heavier, or the scale is in balance. But if we are careful,
we can simplify our writeup by treating either of the out of balance
outcomes as being equivalent.

Second weighing  the scale was in balance on
the first weighing. Therefore, the 8 coins
that were weighed are known
to be genuine and the 4 unweighed
coins remain unknown. Weigh 3 of the genuine coins
against 3 of the remaining 4 unknown coins. There are
two possible outcomes  the scale is in balance or the scale is out of
balance.

Third weighing  the scale was in balance on
the first two weighings. Therefore, all 11 coins
that have been weighed are
known to be genuine and the 12^{th} coin that has not yet
been weighed is known to be
counterfeit. It remains only to determine whether the counterfeit
coin is heavier than normal or lighter than normal. Weigh 1 of the
genuine coins against the counterfeit coin to determine if the
counterfeit coin is heavier or lighter than normal.

Third weighing  the scale was in balance on
the first weighing and out of balance on the second weighing.
Therefore, 9 coins are
known to be genuine  the 8 coins that were in balance on the first weighing
and the 1 unknown coin that was not weighed on the second weighing.
The 3 unknown coins that were weighed on
the second weighing are still
unknown. However, it is now known whether the counterfeit
coin is heavier or lighter than normal. Weigh 2 of the
3 remaining unknown
coins against each other. If the 2 are in balance, the counterfeit coin
is the 3^{rd} unknown coin that was not weighed on the third weighing,
and its heavier or lighter status is already
known. If the 2 coins are out of balance,
one of them is counterfeit and we know
which one it is because its heavier or lighter status is already known.

Second weighing  the scale was out of
balance on the first weighing. Therefore,
the 4 coins we didn't weigh are now known to be genuine, and the
8 coins we weighed are all still unknown. And we know a little
bit about the 8 unknown coins, because we know which 4 coins
were on the heavy side
of the scale and which 4 coins were on the light side of the scale. So if
later in the process we are able to determine whether the counterfeit
coin is heavier or lighter than normal, the number of unknown coins
immediately drops from 8 to 4.
This is the most
challenging point in the solution. It's tempting to do something
like weighing 3 of the coins from the heavy side of the first weighing
plus 1 of the coins from the light side of the first weighing
against the 4 coins that are now known to be genuine. That is close
to the correct solution, but is not quite right. It works if by luck
the second weighing is unequal, but it gets us into needing four weighings
if the second weighing is equal.
The correct solution is very similar to the incorrect solution that
we might have been tempted to try. Weigh 3 of the
coins from the heavy side of the first weighing
plus 1 of the coins from the light side of the first weighing
against 3 of the coins that are now known to be genuine plus the
4^{th} coin from the heavy side of the first weighing.
The counterintuitive aspect of the solution is that we need to split
the 4 coins from the heavy side of the first weighing and put some of them
on one side of the scale and the other one on the other side of the
scale.

Third weighing  the scale was out of
balance on the first weighing and was in balance on the second
weighing. All the coins involved in the second
weighing are known to be genuine.
Therefore, 9 coins are known to genuine 
the 4 coins that were not weighed in the first weighing, the 4 coins
that were on the heavy side of the scale in the first weighing, and the
1 coin from the light side of the first weighing
that was weighed again as a part
of the second weighing.
The 3 coins that remain unknown were all on the light side of the first
weighing. Weigh 2 of the remaining 3 unknown coins against each
other. If they are out of balance, the coin from the lighter side
of the third weighing
is the counterfeit coin and it is lighter than normal.
If they are in balance, the counterfeit
coin is the remaining unknown coin that was not weighed on the third
weighing, and it is lighter than normal.

Third weighing  the scale was out of
balance on the first weighing and was out of balance on the second
weighing. The heavy side of the second weighing was the side
containing the 3 coins that were on the heavy side of the first
weighing The counterfeit coin must be one of those 3 coins.
Weigh 2 of them against each other. If they are out of balance, the
heavier coin is the counterfeit coin. If they are in balance, the
counterfeit coin is the remaining unknown coin and it is heavier than normal.

Third weighing  the scale was out of
balance on the first weighing and was out of balance on the second
weighing. The heavy side of the second weighing was the side
containing 3 genuine coins plus 1 coin that was on the heavy side of the first
weighing. The light side of the second weighing was the side
containing 3 coins that were on the heavy side of the first
weighing and 1 coin that was on the light side of the
first weighing 10 coins are known to be genuine  the
4 coins that were not weighed on the first weighing, the 3 coins that
were on the heavy side of the first weighing and on the light side
of the second weighing, and the 3 coins that were on the light side of the
first weighing and that were not weighed on the second weighing.
The problem is therefore reduced to 2 coins  1 that was on the light side
of the first weighing and that was also on the light side of the second
weighing, and 1 that was on the heavy side of the first weighing
and that was also on the heavy side of the second weighing. It might
be tempting simply to weigh the last 2 coins against each other, but the
1 that was on the heavy side of both of the first two weighings will be heavier
than the 1 that was on the light side of both of the first two weighings.
So weighing the 2 remaining unknown coins against each other will not
tell us anything about which one of them is genuine. Therefore, weigh
either of the 2 remaining unknown coins against a genuine coin.
If the third weighing is balanced, the counterfeit coin is the unknown
coin that was not weighed on the third weighing,
and it is already known whether it's heavier or lighter than normal.
If the third weighing is not balanced, the counterfeit coin is the unknown
coin that was weighed on the third weighing
and it is already known whether it's heavier or
lighter than normal.

