We have 10 bags with 10 marbles each. All marbles weigh 10 grams except those from one of the bags that weigh 9 grams. We have a scale,
How many weights should we do at least to detect the bag containing the marbles that weigh 9 grams?
A heavy one is enough for us. We put in the balance a marble of the first bag, two of the second bag, three of the third bag and so on until completing the 10 bags.
In this way we will have 1 + 2 + 3 +… + 9 + 10 = 55 marbles. If all weighed 10 grams we would have 550 grams in total. But there will be X balls that weigh only 9 grams so the weight that will mark the scale will be 550 - X where X is the bag that contains the marbles that weigh 9 grams.