Three Treasure Chests
by Nick Yee

Most logic problems lose their mystique and initial charm once the solution is given, but one particular logic problem fascinates me because it retains its charm even when the solution is given, and this is precisely because the solution is completely counter-intuitive to most people. Every time I have mentioned this particular logic problem, the solution inevitably leads to a heated discussion of whether the solution is in fact correct.

The king shows you three treasure chests. He tells you that one of the three chests is filled with priceless jewels, while the other two are empty. He then allows you to pick any one of the three chests. After you have picked a chest, the king stops you before you open it. He says, "Wait". The king then opens one of the other two chests and shows you that it is empty. The king then tells you that you can switch your choice if you want to. Is it to your advantage to switch at this point? Or does it not make any difference?

It feels counter-intuitive that switching would be to your advantage, but that is the case here. One explanation goes like this. When you first choose, you only have a 1/3 chance of choosing the right chest and a 2/3 chance of choosing an empty chest. Because you are more likely to have chosen an empty chest in the beginning, you are also therefore more likely to switch to the chest with jewels if you choose to switch. And because there is a 2/3 probability that you chose the wrong chest in the beginning, there is also a 2/3 probability that you will switch to the right chest. Thus, it is to your advantage to switch, because there is only a 1/3 chance that you chose the chest with jewels in the beginning.

Another explanation focuses on the fact that the king can always open an empty chest. Since there are two empty chests, he is always able to show you at least one empty chest. Because of this, his opening of an empty chest does not change the fact that when you made the choice, there was only a 1/3 chance of you choosing the right chest. After he opens an empty chest, there is still only a 1/3 chance of you having chosen the right chest. And thus there is a 2/3 chance that the other chest holds the jewels. Again, this is because the king can always show you an empty chest, thus his doing so does not increase the odds that you have the right chest.

Most people have difficulty accepting that it can be to your advantage to switch because they argue that there are only two boxes left, and that the probability should be 50/50 at that point, and thus it makes no difference whether you switch or not. Or they might argue that if someone else had to choose after the king had opened an empty chest, then it would be 50/50 and that the original 1/3 does not hold any longer.

The problem with this argument is that you initially made your choice when there were three chests, not when there were two. Let's say that there are 100 chests, and 100 princes. Each prince is assigned to one chest. Only one prince will have correctly chosen the chest with jewels in the beginning. The king can always open 98 empty chests for each prince. Thus out of 100 princes, 99 will benefit from switching to the other chest, while only one prince will switch inappropriately. It is clearly not the case that 50 princes benefit from switching and the other 50 do not benefit.

Because you know that the king can always open an empty box, it actually doesn't matter whether he does it or not, because in either case you haven't learned anything new. Moreover, you have already made a choice at this point, so your odds do not suddenly get better. The odds are exactly as if he didn't show you an empty box, and this is precisely because no new information is presented.

I first came across this logic problem in Raymond Smullyan's The Riddles of Scheherezade.