You are a rare coins expert and have determined there are 7 fake coins out of 14 gold coins. Now you need to prove to the judge which ones are fake.

It is known that that real coins all weigh the same, fake coins all weigh the same, and fake coins weigh less than real ones (but are otherwise identical).

Using a traditional double-pan balance scale just 3 times, can you prove exactly which of the 14 coins are fake?

#### Solution

Yes:

- Put 1 fake coin on the left and 1 real coin on the right. The scale will be heavier on the right, so the left coin must be fake and the right coin must be real (if they were both real or both fake, the balance would show both sides were equal).
- You have proven 1 fake and 1 real coin. Keep these coins on the scale.

- Add 2 real coins to the left and add 2 fake coins to the right. The scale will now be heavier on the left (2 real + 1 fake > 1 real + 2 fakes), and this is only possible if you added 2 real to the left and 2 fake to the right – any other combination of additional coins would have resulted in the scale being balanced or heavier on the right.
- Now you have proven 3 fake and 3 real coins.

- Put the 3 proven fakes on the left and 3 proven real coins on the right. Add 4 real coins to the left and add 4 fake coins to the right. The scale will now be heavier on the left (4 real + 3 fakes > 3 real + 4 fakes), and this is only possible if you added 4 real to the left and 4 fake to the right – any other combination of additional coins would have resulted in the scale being balanced or heavier on the right.
- Thus you have proven exactly 7 coins to be real and 7 coins to be fake.