There are 3 people placed in a room. They all have perfect logic. The 3 people are told by a host that a number has been written on each of their foreheads. Each of the 3 numbers are unique, they are all positive, and they relate to each other such that A + B = C (i.e. one is the sum of the other two). In the room, each person can only see the other two people’s numbers, as they cannot see their own foreheads.

Suppose you are one of the 3 and you see one person with “20” on their forehead and the other person with “30.” The host asks you, then the person with “20,” and then the person with “30” what number is on their heads and all 3 say that they don’t know. The host then asks, again, you, then the person with “20,” and then the person with “30” what number is on their head, and all 3 answer correctly. How does this happen?

Answer:

The key to this brainteaser is to calculate the logic of each person’s point of view, i.e. put yourself in each of their shoes. The annoying part of solving this brainteaser, then, is having to keep track of 3 different points of view.

“First” person: If you see “20” and “30,” that means you are either 10 or 50. So you don’t know what’s on your forehead among these two numbers.

The “20” person: You see either 1.) “30” and “10” or 2.) “30” and “50.” In case 1.) you are either 20 or 40. In case 2.) you are either 20 or 80. So you don’t know.

The “30” person: You see either 1.) “20” and “10” or 2.) “20” and “50.” In case 1.) you *must* be 30 because you cannot be 10 as well the “First” person. So the key here is that if you see one person has number “x” and another has number “2x,” you know you cannot also have “x” on your forehead. You must be “3x.” So in this case, the “30” person would know the answer that he or she has 30 on his head. In case 2.) the “30” person has either 30 or 70, and so he or she wouldn’t know.

Since after the first round, everyone answered that he or she did not know, that means that we cannot have the “30” person’s case 1.), which is that he sees “20” and “10.” In other words, our “First” person cannot have 10. He has 50 on his forehead. So when the host asks the “First” person the second time, he or she will answer 50.

The most illuminating and clean part of the problem is just up to here, but in an attempt for completeness, I kept going.

From the “20” person’s point of view, we assume that he or she is able to figure out the above sort of logic on his or her own. What the “20” person sees is “30” and “50,” which means that he or she is either 20 or 80. Somehow, the “50” person figured out on his or her own on the second round of questioning that they have 50 on their head. The logic is that in order to find out what your number is on the second round, you are using someone’s “I don’t know” answer in the first round of questioning. So if the “20” person indeed has 20 on his or her head, they can deduce that the “50” person is able to figure out all the above and that his or her number is 50 on the second round. If the “20” person has 80 instead, the “First” person sees “80” and “30” and is thus wondering if his or her number is 50 or 110 and the “30” person sees “80” and 50 and wondering if they’re 30 or 130. In none of these cases is a person announcing that they are not seeing an “x” and “2x” situation (which is what the “First” person experiences: seeing a “2x” and “3x” situation, and then seeing that the “3x” person doesn’t immediately say that he or she knows that his or her number is “3x.”). If the “20” person has 20, then, again, the “First” person sees that the “30” person is announcing that they aren’t seeing an “x” and “2x” situation, which means that the “First” person can’t have 10 and must have 50. This causes the “20” person to know that his or her number is 20.

Similarly, the “30” person sees “50” and “20” initially doesn’t know if he or she is 30 or 70. If it’s 70, the other people either see “70” and “50” or “70” and “20,” which doesn’t allow the situation described above of someone announcing that he or she doesn’t see an “x” and “2x” situation. If it’s 30, then everything that’s been discussed happens, and so it must be 30.

The key basically is that if someone sees “x” and “2x,” they should know immediately that they are 3x. If someone sees “2x” and “3x,” they are immediately on high alert to see if the “3x” person immediately knows that he or she is 3x. If the “3x” person doesn’t know, that is an announcement that the “3x” person did not see an “x” and “2x” situation, which means that the person we started with must be “5x.” So, in an “x” and “2x” situation, you know immediately that you are 3x. In a “2x” and “3x” situation, if everyone says that he or she doesn’t know in the first round, that announces that no one saw (and the “3x” person in particular did not see) an “x” and “2x” situation, which means that you must be 5x.