It is tempting to reason that since each box has an equal probability of having the fattest stack, your odds will always be one in four. Knowing what the first box contained doesn't tell you a thing about whether the second will be larger or smaller.
You'd be right about the second part. The second box has a fifty-fifty chance of being larger or smaller than the first.
And once you've seen the first two boxes, there is only a fifty-fifty chance of the fattest stack being in one of those boxes.
But let's say you knew it was- you could say with certainty which box it was. The larger of the two, of course. But this still doesn't help you, because if the larger of the two boxes at this point turned out to be the first, it's too late to go back.
Wait a minute, though: although it's too late to go back, it isn't too late to press ahead. If the second box is smaller than the first, you know with 100% certainty that the second box is a loser. And while the winner might have passed by, pushing on until you see a box larger than the first can only help.
If you employ the strategy of opening the first box and passing on it, then settling on the next box you see that isn't a guaranteed loser (in other words, the first box that contains a stack larger than the first), you can improve your odds over one in four.
In fact, your odds of winning rise to almost 46%!