# Of 80 coins of the same denomination, one is conterfeit and lighter than the others. ?

Of 80 coins of the same denomination, one is counterfeit and lighter than the others. Identify the counterfeit coin by using a pan balance in no more than four weighings?

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Putting 26 coins aside, balance 27 coins with another 27 coins.

Case1: Not balanced

If not balanced, then you know the lighter one is in which 27 coins.

a) Putting 9 aside, balance 9 with another 9. So, you can tell in which 9 coins.

b) Putting 3 aside, balance 3 with another 3. So, you can tell in which 3 coins.

c) Putting 1 aside, balance 1 with another 1. So, you can find the lighter coin.

Case 2: Balanced

So, the lighter coin must be in the 26 coins.

a) Putting 8 aside, balance 9 with another 9. So, you can tell in which 9 or which 8 coins.

If it is in the 9 coins, following case 1, you can find the lighter coin in 4 steps.

If it is in the 8 coins, then

b) Putting 2 aside, balance 3 with another 3. So, in one more step you can find the lighter coin.

Put 27 coins in each side of the balance. If one side is lighter, the counterfeit is in that bunch. If not, it is in the 26 that is left over.

Take the group that has the counterfeit coin and put 9 coins in each side of the balance. Repeat the process and determine if one of your groups on the balance contains the fake or if it’s in the group to the side.

Repeat again with three coins in each side of the balance.

Now you have either a group of 3 coins, or if it was in one of the leftovers, a group of 2 coins. Regardless, put 1 coin on each side of the balance. If the balance is equal and you had 3 coins, it’s your leftover coin. Otherwise, the lighter coin on the balance is your counterfeit.