bags with coins are placed on the floor. One bag has coins that weigh grams, the other bags have coins that weigh grams. One may place some coins (not necessarily from the same bag) on a weighing scale, but it will only properly display the weight if it is less than grams. Determine the minimum number of times that the weighing scale may be used in order to identify the bag that has the -gram coins.
Problem 1363
Official solution
1. Understanding the Problem:
We have 120 bags, each containing 100 coins. One bag has coins that weigh 9 grams each, while the other bags have coins that weigh 10 grams each. We need to identify the bag with 9-gram coins using a weighing scale that can measure weights up to 1000 grams.
2. Initial Observation:
If we take coins from up to 13 bags, the total weight will be at most grams, which is too heavy. However, if we take 1 coin from the first bag, 2 coins from the second bag, and so on up to 13 coins from the 13th bag, the total weight will be at most grams, which is within the limit.
3. Weighing Strategy:
- For 14 bags, we can identify the bag with one weighing by leaving one bag out and weighing the rest. If the weight is less than expected, the missing weight corresponds to the bag with 9-gram coins.
- For 46 bags, we can use a similar strategy by grouping them into sets of 14 bags and using the weighing method described above.
4. Detailed Calculation for 46 Bags:
- We can take 1 coin from the first bag, 2 coins from the second bag, and so on up to 14 coins from the 14th bag. This gives a total weight of at most grams, which is too heavy. However, we can adjust the number of coins taken to ensure the total weight is less than 1000 grams.
- If the weight is less than 1000 grams, we can identify the bag with 9-gram coins using another weighing.
5. Generalizing the Strategy:
- For 60 bags, we can use the same method by grouping them into sets of 14 bags and using the weighing method described above.
- For 80 bags, we can take 60 bags and weigh them, then use the same method to identify the bag with 9-gram coins.
6. Final Calculation:
- For 120 bags, we can group them into sets of 60 bags and use the weighing method described above. If the weight is less than 1000 grams, we can identify the bag with 9-gram coins using another weighing.
The final answer is 3 weighings.