Unfortunately, all you have is a scale of the classical type. It is the kind where there are two balanced dishes. By putting objects in, the scale will indicate that one side is heavier than the other (the dish will hang lower) or that each side weighs the same.

How can you find the counterfeit coin by only weighing

**three times**? And is the coin lighter or heavier than the others?

Solution

Allright! The first step is *not*to put 6 coins on each side. Sure, then we'll know that the counterfeit coin is in one of those groups of 6, but err... we already knew that.

So what we are going to do here is number the coins 1 to 12, and create a ‘tree’ of the different weighings. See, the outcome of every weighing determines your next move. The full tree encompasses every scenario. Don't be put off by its length, just start at the top and reason through!

Weigh 1,2,3,4 against 5,6,7,8.

__They weigh the same.__

Good! This is the easy scenario. The coin must be in 9,10,11,12. What you want to do now is weigh 9,10,11 against 1,2,3.

__They weigh the same.__

Then 12 must be the counterfeit. You have one weighing left to determine whether it is lighter or heavier: weigh it against any other coin to find out.__9,10,11 is heavier.__

So any of those must be the counterfeit, and it will always be heavier than the others. Weigh 9 against 10.

__They are the same.__

11 must be the counterfeit.__9 is heavier.__

Then 9 is the counterfeit.__10 is heavier.__

Then 10 is the counterfeit.

__9,10,11 is lighter.__

So any of those must be the counterfeit, and it will always be lighter than the others. The procedure is the same as above! Weigh 9 against 10.

__They are the same.__

11 must be the counterfeit.__9 is lighter.__

Then 9 is the counterfeit.__10 is lighter.__

Then 10 is the counterfeit.

__5,6,7,8 is heavier.__

The coin can be on either side. You must weigh 1,5,6 against 2,7,8.

__They are the same.__

Then coin 3 and 4 are the only possibilities left. Weigh one of those, like 3, against a known valid coin, such as 9.

__They are the same.__

Then the unweighed coin, number 4, is the counterfeit. From the first weighing you can tell whether it is heavier or lighter.__9 is heavier.__

Coin 3 is the lighter counterfeit.__9 is lighter.__

Coin 3 is the heavier counterfeit.

__2,7,8 is heavier.__

Argh, now it gets really complex. What this means is that either 7 or 8 is the heavier counterfeit, or 1 is the lighter counterfeit. Knowing that, just weigh 7 against 8.

__They are the same.__

Then 1 is the counterfeit.__7 is heavier.__

Then 7 is the counterfeit.__7 is lighter.__

Then 8 is the counterfeit.

__1,5,6 is heavier.__

Either 5 or 6 is the heavier counterfeit, or 2 is the lighter counterfeit. In the same way as above, weigh 5 against 6.

__They are the same.__

Then 2 is the counterfeit.__5 is heavier.__

Then 5 is the counterfeit.__5 is lighter.__

Then 6 is the counterfeit.

__1,2,3,4 is heavier.__

Weigh 1,2,5 against 3,4,6. In other words go through it similarly as above!

^{[source of answer]}

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