Solution to Tuesdays Riddle

Solution to Riddle of the Week: The Spies With Stamps on Their Heads

Difficulty level: Hard

By Jay Bennett

The best way to solve this riddle is to get it down on paper. Sketch out all the possibilities, including the stamps in the guard's pocket, and then start eliminating them based on the captives' answers. You can see the original question here.

SOLUTION

From the information given, you cannot say what color stamps spies A and C have. But we can tell that Spy B has one red stamp and one green stamp. Here's why:

Each time one of the spies answers, you learn a little more about what combinations of stamps are possible. Let's start at the beginning, when spy A says for the first time that he doesn't know what stamps are on his head. This means that he does not see all four red stamps on the other men, because if he did, then he would know that he must have two green. He also does not see all four green stamps for the same reason. When spy B answers "No" for his first response, you know that he also does not see four red or four green. The same is true for spy C when he answers "No."

Here is a chart of all the possible combinations at the beginning, and the response that eliminates them as possibilities is provided on the right.



As you can see, spy A's first negative response eliminates possibilities #1 and #7. Spy B's first response eliminates possibilities #2 and #8. Spy C's response eliminates possibilities #3 and #9.

The next important thing to realize is that spy C's response also eliminates possibilities #4 and #10. The reason for this is that spy C, perfectly logical being that he is, considers possibility #4 and realizes that if he sees two green stamps on spy B and two red stamps on spy A, and neither of them answer correctly on their first responses, then he (spy C) cannot have two red or two green stamps, so he must have one red stamp and one green stamp. Spy C does not figure this out, though, so we can eliminate possibility #4 (and possibility #10, which is the same with the colors inverted).

Now that everyone has answered once, we cycle back around to spy A, who again says he does not know. This eliminates possibilities #13 through #16. We can eliminate #13 and #14 for the same reason as #4 and #10. By the time spy A answers for a second time, he would know that he cannot have two of the same color if he sees both spy B and spy C with two of the same color, leaving only one red and one green as a possibility. He does not know that he has one red and one green, so #13 and #14 must be incorrect.

Possibilities #15 and #16 are a little trickier to eliminate. Imagine that you are spy A, and you look and see that spy B has two green stamps, and spy C has one red and one green stamp (possibility #16). You know that there are four possible stamps that you can have on your head, and three of them are red while one of them is green, so you can only have two green or one red and one green. But you know that you cannot have two green, because if you had two green, that would be possibility #4, and spy C would have figured it out on his last response. Therefore, if possibility #16 were the true arrangement of stamps, then spy A would figure it out on his second response. He does not, so possibility #16 must be incorrect (along with possibility #15, which is the same with the colors inverted).

Now all of the possibilities are eliminated except for #5, #6, #11, #12, #17, #18, and #19. All of these possibilities have spy B with one red and one green stamp, and so regardless of which one it is, he figures out the stamps on his head and gets to go free. The other two spies, unfortunately, were doomed the moment the stamps were placed on their heads.