Return to my Puzzle pages
Go to my home page

Who Are Bill & Susan? - Answer

© Copyright 1999, Jim Loy

The puzzle:

We are taking one statement from each of eight people. The last two are Bill and Susan, or maybe Susan and Bill. Anyway, one of those two is Susan, the other is Bill. See if you can deduce which is Bill and which is Susan:

1. Exactly seven of us are lying.
2. Exactly six of us are lying.
3. Exactly five of us are lying.
4. Yes, exactly five of us are lying.
5. Exactly four of us are lying.
6. Exactly three of us are lying.
7. My name is Susan.
8. My name is Bill.

The answer:

#7 is Bill, and #8 is Susan.

The reasoning:

We know that the last two are Bill and Susan, in either order. So, either they are both telling the truth, or they are both lying. For the moment, let us ignore person #4, as he/she is just parroting person #3. Then people 1, 2, 3, 5, and 6 are making statements that are incompatible with each other. So, at least four of them are lying. So, at least four of the eight people are lying. Let us look at each possible case:

1. Four people are lying: That means that four people are telling the truth. Of the first six people, Person #5 is the only one telling the truth. That is too many lies. This case cannot be.
2. Five people are lying: Three people are telling the truth. People 3 and 4 are telling the truth. Since Bill and Susan are either both lying or both telling the truth, we have either two or four people telling the truth, instead of three. So, this case cannot be.
3. Six people are lying: Two people are telling the truth. Person #2 is telling the truth. Since Bill and Susan are either both lying or both telling the truth, we have either one or three people telling the truth, instead of two. So, this case cannot be.
4. Seven people are lying: One person is telling the truth. That person is person #1. So, Bill and Susan are both lying. This case may be true.
5. Eight people are lying: No one is telling the truth. So, Bill and Susan are lying. This case may be true.

So, we find that both Bill and Susan are lying. #7 is Bill, and #8 is Susan.

The first six people, above, are making self-referential statements. These are statements which refer to themselves, such as, "This statement is true," or, "This statement is fairly short." Such statements can be true or false, or they can sometimes have no meaning, whatsoever. Well, in the context of the above puzzle, those statements do have meaning. When we assume something, like, "Exactly four people are lying," we encounter a contradiction. It is impossible that exactly four people are lying. Similarly, assuming that 5 people or 6 people are lying also lead to contradictions.

In the end, it doesn't matter whether person #1 is lying or telling the truth, we get the same answer, regardless. Person #1's statement, "Exactly seven of us are lying," can be either true or false. We cannot pin it down. But, we encounter no problems, the puzzle has the same solution, no matter whether we call it true or false.

Return to the original puzzle.

Return to my Puzzle pages
Go to my home page