Suppose you have 8 marbles and a two-pan balance used to compare the weight of 2 things. All of the marbles weigh the same except for one, which is heavier than all of the others. How would you find the heaviest marble if you are only allowed to weigh the marbles 2 times?
Let’s start solving this puzzle by keeping it simple. What if we just divide the 8 marbles into 2 groups of 4 each, and we put 4 marbles on one pan and 4 marbles on the other. What do we know after the first weighing? Well, we know which group of 4 marbles is heavier, which means that we also know that the heaviest marble is in that group.
Now, we’ve narrowed it down to 4 marbles, and we know one of those marbles is the heaviest – and this is after one weighing. Can we find out the heaviest marble in one more weighing with just 4 remaining marbles? What if we just compared 2 marbles – one marble on each pan? Well, if one of those happened to be the heavier marble then we would know which marble is the heaviest in 2 weighings. But, one of those 2 marbles is not necessarily the heavier one – and we don’t know which of the other 2 marbles that were not weighed is heavier. So that is an assumption that we can not safely make, and our solution is not valid.
What if we just put 2 marbles on each pan and do another weighing? Well, one side would be heavier, and we would be able to narrow it down to 2 marbles – but we still don’t know which of those 2 marbles is heavier. This would require one more weighing, for a total of 3 – when the question specifically asks us to find the heaviest marble in just 2 weighings. This means that our solution so far has been invalid, and we must come up with a new and better solution.
Challenge our assumptions
We started with the assumption that we should put all the marbles on the scale. What if we left some off of the scale? Could that possibly tell us something? Well, yes, it actually does tell us something by the process of elimination. Because, if we know that the marbles on the scale weigh the same, then we also know that the heaviest marble is one of the marbles not on the scale (so we can eliminate the marbles on the scale). And if the marbles on the scale do not weigh the same, then we know that one of the marbles on the scale is the heaviest, and we can eliminate the marbles that are not on the scale.
So, let’s do this: we put 3 marbles on each pan – for a total of 6 marbles on the pan, and we leave 2 marbles off the pan. Then, we compare the 6 marbles on the pan – if one side is heavier than the other then we only have 3 marbles left. We can compare 2 of those 3 marbles to each other, and if they are the same weight then the 3rd is the heaviest, and if one is heavier than the other then we have the heaviest in just 2 weighings. If, when comparing the 6 marbles we find that both sides are equal, then we know that the heaviest marble has to be in the 2 marbles that are not on the pan. This then means that we only have to compare those 2 remaining marbles and we have the heaviest marble. So, we have found our answer!