Results of Who's Saying What?
The problem challenge:
Eight patrons at Zeno's one night last week made the following statement each. The last two were made by our friends Maggie and Charles, but we're not told which staement was made by whom. See if you can deduce which is Charles and which is Maggie! Show your reasoning:)
A nice response from David Buxton:
comments: This is pretty easy, when approached the right way. The "x of us are lying" statements are examinable individually, by assuming that each in turn is true.
The easiest place to start is with statement 6. This cannot be true, because there are 5 statements that say something different (and so if it is true, there would be at least 5 liars which is an obvious contradiction).
Statement 5 presents us with a similar dilemma if we assume that it is true, so we know that it is false.
Statements 3 and 4 can be treated as a pair. If we assume that they are true, we can say that 5 of the other 6 statements are false. However, as in this case statements 1, 2, 5 and 6 and one of 7 and 8 would have to be false, this situation cannot be the correct one, as this would mean that both the names given in 7 and 8 would be the same (ie both "Maggie" or both "Charles". Therefore these statements can be discounted.
Statement two leaves us in exactly the same position, as both names given would be the same if it were true.
And so we come to the first statement. This is the only one remaining that could be true. However, there are some other possibilities that need to be discarded before we can make our final decision. One is that fewer than 3 people were lying. This is fairly obviously impossible. Another is that all 8 people are lying. This is possible, but, as we will see, makes no difference to the end result. And so, if we assume that statment 1 is true, and there is no reason to discard it, we end up that both Maggie and Charles were lying. Therefore, statement 7 was made by Charles and statement 8 by Maggie. This is the same if all 8 people were lying, as this simply means that statement 1 was not true either.
Statement 7: Charles
Statement 8: Maggie
Note: Daniel B. Cristofani wrote Zeno's to protest the above finding. I agreed to include his response. Here it is, and fell free to offer your comments:
Most people probably said that Maggie and Charles lied, and explained
why. Actually it is impossible to solve the puzzle on the basis of
the information given. Notice that if the puzzle were soluble, then
after the first six statements were made as described, if Maggie and
Charles then decided, on a whim, to simply tell the truth about their
names, then it would be logically impossible for them to do so. Which
is absurd. So where did the reasoning go astray?
The problem is the implicit assumption that each of the eight
statements must be either true or false. Actually, once statements
get self-referential, it's easy to make statements, or sets of
statements, that are neither true nor false. One classic example,
"1. Exactly one of me is lying."
Once we get into these waters, we can no longer conclude that a
statement is false merely because its truth leads to a contradiction.
Its falsity might lead to a contradiction also--or just to an untruth.
This last possibility is what makes the trap in this puzzle so
subtle. Even after recognizing the unpleasant fact that not every set
of statements can be consistently assigned truth-values, there is
still the tendency to assume that if there IS a way to assign them
consistently, then anything every consistent assignment agrees on
must be right. But consider the following distilled version of the
"1. Exactly one of us is lying.
2. Most fish live in water."
Although statement 2 is obviously true, the pair cannot consistently
be assigned truth-values unless 2 is assigned the value "false".
(Check it out!) If statement 2 were a less obvious truth, it might
well be believed false simply because that's the only way to avoid
open paradox. The "criminal" statement would then have managed to
implicate an innocent statement, while having a good chance of
escaping conviction itself.
(Note that in the consistent assignments for both the "fish" puzzle
and the original one, statement 1 can consistently be considered
either true OR false. Given all the other truth-values, it ends up
asserting its own truth and nothing else. This should be a warning
flag. The statement "I am telling the truth" is just as problematic
as "I am lying", the problems are just more subtle. The only
advantage of "I am telling the truth" is that assigning it an
arbitrary truth-value doesn't engender contradiction. But what are we
doing assigning truth-values arbitrarily anyway?)
Anyway, congratulations on an excellent puzzle.
-Daniel B. Cristofani.