إجابة مرجعية
This question would be really difficult to figure out on the spot. Fortunately, it's a puzzle with answers all over the place online.
The identifying factor for each of these bags of marbles is weight; fortunately, we have only one different bag. Unfortunately, we only have one chance to weigh, so we couldn't just weigh each bag individually.
Instead, we can solve the problem if we put a different number of marbles from each bag into a new bag to weigh it and reverse engineer the identity of the heavier bag.
Let's take 1 marble from the first bag, 2 from the second bag, 3 from the third bag, and so on. This way each bag we've drawn from is uniquely identifiable by the number of marbles missing. I've used my kindergarten-level illustration skills to draw this process.
The total number of marbles in the bag can be calculated now using the series sum formula alluded to in question 5: n(n+1)/2. If we plug the numbers in, we should get 55. Now we have to multiply it by the weight of each marble, which is 10g. That means the total weight of the marbles should be 550g, in a perfect world.
But we're not in a perfect world. One of these bags is different. Let's say, for argument's sake, the third bag is the one that has the heavier 11g marbles. The weights would look like this: 10, 20, 33, 40, 50, 60, 70, 80, 90, 100. If you weighed this, in total, it would add up to 553. Clearly, one of these bags has botched things up. To find out which one, we can subtract 550 from 553, getting 3. In other words, the third bag is the odd one out. The formula, then, would look like this: W – w(n(n+1)/2), where W = total weight and w = weight of each marble (except the odd ones).
Note that we've labeled the bags 1-10 based on the number of marbles taken from it. The difference won't necessarily be this number, however. If the bag were more than 1g heavier or lighter, we'd have to do more math. Say, for example, the odd marbles weighed 12g instead; the difference would have been 6. This still points to the third bag because we know that the odd marbles are 2g heavier than the other marbles. If we divide 6 by 2, we get 3.