# 24 Coins

You have a pile of 24 coins. Twenty-three of these coins have the same weight, and one is heavier. Your task is to determine which coin is heavier and do so in the minimum number of weighings. You are given a beam balance (scale), which will compare the weight of any two sets of coins out of the total set of 24 coins. How many weighings are required to identify the heavier coin?

### 8 guesses to 24 Coins

• As little as 5, no more than 6 weighings. 2 weighings of 12 coins each, the heavier group to be divided into 2 groups of 6, only one of which needs to be weighed. If the group of 6 weighs more than 1/4 of the total of the first two weighings, it contains the heavier coin, and should be separated into two groups of 3. If the group of 6 weighed has less mass than 1/4 of the total from the first two weighings, then the other 6 coins should be divided into two groups of three. (3 so far). If the first group of 3 weighs more than 1/8 of the total from the first two weighings, it holds the heavier coin. If less than 1/8, the other group. (4 weighings so far. 2 coins from the group containing the heavy coin should be weighed. If the weight is less than 1/12 of the total from the first weighing, the remaining coin is heaviest. (5 weighings) If more than 1/12, then one of the two coins being weighed is the heaviest. Weighing either one will tell if it is the heaviest, if it is greater than 1/24 of the total form the first weighing, it is the heaviest. If it is less than 1/24 of the total form the first weighing, the other coin is the heaviest (6 weighings)

• oops too hasty. I see now that it is a balance scale.

• 4 weighings. 12/12, 6/6, 3,3. weigh any two of the remaining 3 coins. if it is equal, the remaining coin is the heaviest.

• Dude

The RiddleDude can do it in less. Explain how.

• Roolstar

3 weighings max for up to 27 coins…

Working with 9 on each scale; if they weigh the same, the heavier coin is among the remaining 9 (not on the scale), otherwise the scale will tip towards the lot that contains the heavy coin.

Either way, we end up with 9 coins with the heavy coin among them.

3 coins on each scale will tell us either that the remaining 3 coins have the heavy one, or simply which of the 3 on the scale have it.

From these 3, put 1 coin on each side of the scale and you’re done. Balanced means it’s the one not on the scale, tipping will point to the heavier one.

Generalization form:
1 weighing for 3 coins (Start with 1 on each side of the scale)
2 weighings for 9 coins (Start with 3 on each side of the scale)
3 weighings for 27 coins (Start with 9 on each side of the scale)
4 weighings for 81 coins (Start with 27 on each side of the scale)
…….
n weighings for 3 to the power n coins (Start with 3 Pow (n-1) on each side of the scale)

Note:
It will take one extra weighing if we don’t know wether the coin is heavier or lighter simply by comparing one of the lots that has a discrepancy to another lot of regular coins already found in previous weighings…

• Dude

As usual Roolstar, you “Rool” the day!

It can indeed be done in three weighings.

Weighing 1: Break the coins into three piles of eight. Weigh one group of eight against another group of eight. If the scale balances, then the group that hasn’t been weighed has the heavier coin. If the scale tips, then that group contains the heavier coin.

Weighing 2: Break the group of eight that has the heavier coin into three groups (three coins, three coins, and two coins). Weigh one set of three against the other set of three. If it balances, the group of two has the heavier coin. If the scale tips, then that group has the heavier coin.

Weighing 3: If the heavier coin is in the group of two, then just weigh one coin against the other to determine the heaviest coin. If the heavier coin is in a group of three, then take two of those coins and weigh them against each other. If the scale balances, the coin that hasn’t been weighed is the heavier coin. If the scale tips, then that is the heavier coin.

You’re today’s winner.