# Bag of Coins Puzzle

## Bag of Coins Puzzle

You are given 10 bags, with each bag containing 1000 coins. However one of the bags contains forgeries. All the coins are identical in every aspect except that the forged coins weigh 1.1 gram while the real coins weigh 1 gram. What is the minimum number of times you need to weigh the bags to figure out the bag with the forged coins?

## Bag of Coins Puzzle

You are given 10 bags, where each bag containing 1000 coins. However, one of the bags contains forgeries. All the coins are identical in every aspect except that the forged coins weigh 1.1 gram while the real coins weigh 1 gram. What is the minimum number of times you need to weigh the bags to figure out the bag with the forged coins?

## Solution

• We are allowed to weigh the bags only once to figure out the forged bags.
• Let us number the bags, from bag 1 to bag 10.
• Now from bag 1, we take 1 coin, and from bag 2 we take 2 coins, from bag 3 we take 3 coins, and so on.
• Now we take all these coins and weigh them together.
• Between all the 10 bags we will have 55 coins.

Now we know that the real coins all weigh 1 gram while the forged coins weigh 1.1 gram or have an extra 0.1 gram of weight.

Therefore by observing the extra weight in our picked coins, we can figure out the forged bag.

Say the weight of the collective coins comes as 55.5 grams. Then we can say that there is an additional 0.5-gram weight, which is $0.1\times5$, therefore bag 5 from where we took 5 coins is the forged bag.

Similarly, if the weight was 55.7, then it would be bag 7. To sum it up, if the weight is 55, then the bag numbered n is the bag with the forgeries.

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