Problem #4 MEDIUM
Heavy Gold Gold coins
Amazon Math Logic
Problem Statement
You are given 10 bags of gold coins. Nine of the bags contain genuine gold coins, each weighing exactly 10 grams. One bag, however, contains counterfeit coins, each weighing exactly 9 grams.
You have a digital weighing scale that shows the exact weight of whatever is placed on it. You want to identify the counterfeit bag.
What is the minimum number of weighings required on the digital scale to find the counterfeit bag?
Answer & Quick Explanation
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You only need exactly 1 weighing by taking a progressive number of coins (1 from bag 1, 2 from bag 2, etc.) and checking the total weight deficit.
Detailed Editorial Solution
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The solution uses a progressive sampling technique, allowing us to pinpoint the counterfeit bag in a single weighing.
Label the bags from 1 to 10. Take coins from each bag as follows:
- 1 coin from Bag 1
- 2 coins from Bag 2
- 3 coins from Bag 3
- ...
- 10 coins from Bag 10
In total, you will have 1 + 2 + 3 + ... + 10 = 55 coins. Place all 55 coins on the scale at once.
If all coins were genuine (10g each), the scale would read exactly 550 grams. However, since the counterfeit coins weigh 9 grams (a deficit of 1 gram per coin), the total weight will be less than 550 grams.
The deficit will tell us exactly which bag is counterfeit:
- A deficit of 1 gram (549g total) means Bag 1 is counterfeit (since we took 1 coin from it).
- A deficit of 2 grams (548g total) means Bag 2 is counterfeit.
- A deficit of N grams means Bag N is counterfeit.