On Friday I posted this puzzle…..

Here are 10 numbered statements. How many of them are true?

1) Exactly one of these statements is true.

2) Exactly two of these statements are true.

3) Exactly three of these statements are true.

4) Exactly four of these statements are true.

5) Exactly five of these statements are true.

6) Exactly six of these statements are true.

7) Exactly seven of these statements are true.

8) Exactly eight of these statements are true.

9) Exactly nine of these statements are true.

10) Exactly ten of these statements are true.

If you haven’t tried to solve it, have a go now. For everyone else the answer is after the break.

All of the statements contradict one another, and therefore at most one of them can be true, in which case the other nine statements will all be false, which is what statement 1 asserts. Therefore statement 1 is the only true statement.

Note: I originally posted the wrong answer, saying it was statement 9. This is correct if you substitute the word ‘false’ for ‘true’ in the puzzle. Apologies for mix up.

Did you solve it?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

1. Joy says:

But statement 9 is saying “Exactly nine of these statements are true.” rather than “Exactly nine of these statements are false”. Maybe you mean statement 1 is true?

1. SofARMaths says:

I agree. 1 is true…

2. -M- says:

?

No…

If they contradict they are not true. Only one statement can be true and that is statement 1. An alternative is that none of the statements is true.

1. -M- says:

I think the puzzle was wrong and should have been “exactly -n- of these statements are false”. Pity. Would have been a nice puzzle.

2. ivan says:

You are correct, all of them being false is also a consistent interpretation. Therefore the correct answer to the puzzle as (mis)printed is “we cannot say”, since there are two consistent interpretations.

3. Slightly Less Wise One says:

M
Any chance of you explaining your “none of the statements is true” answer Oh Wise One?

4. -M- says:

Answer 1: if 1 statement is true, it is statement 1 that is true.
Alternative: If none of the statements is true, they are all false. That is a correct possibility. They CAN all be false.

Try any other alternatives:
“3 statements are true”. Then statement 3 is true, but no others. That is incorrect.

Wise enough?

5. Slightly Less Wise One says:

Not really

3. Ian says:

Isn’t this tactic of posting the wrong answer to provoke debate getting a bit old?

1. Steve says:

It seems to work for FtB…

2. I came to the comment section to make the same point 🙂

Just to follow up on “Slightly Less Wise One”‘s comment:
The question is “How many of these statements are true”. You can answer 1, which is true, because then statement one is true. You can also answer 0, because if 0 is the answer, then none of the statements are true, so 0 is a correct answer.

3. Slightly Less Wise One says:

Sorry
Still don’t understand
You seem to be saying that your answers themselves determine the truth of the statements in some wierd kind of Heisenberg scenario.
You will have to spell it out in a more simple way for a dunce like me to understand.

4. 9 says “exactly nine of these statements are true”, not “exactly nine of these statements are false. 1 is the only one that is true, as far as I can tell.

5. Bazooka says:

Statement 9 asserts that exactly 9 statements are true. If 9 are false, then statement 1 is the only possible correct answer.

6. Anders says:

I wonder if it was the question that was wrong, and that all the statements should really have said …are false. As it was phrased it was trivial, the other way would I think have been more confusing.

1. extremepoker says:

I suspect this is the case, as you say trivial puzzle in current form. So where are all the ‘there are two correct answers’ brigade now? Those comments had me fairly puzzled over the weekend.

2. -M- says:

I said there where 2 correct answers: Statement 1 is true or none of them.

3. Photon Boy says:

I like it -M- .

4. Martha says:

How can there be 2 correct answers if they contradict each other? There are 2 possible answers, but they can’t both be correct!

5. Davis says:

@Martha — two correct answers is okay here because one of the correct answers is not one of “these statements.”

7. Surely it depends on whether the tree is on this side of the island, or whether the river flows faster than the fire can spread?

Yes, but only of you are being epistemic

2. Wigan FC says:

Gonna explain where trees and rivers come into it ?

3. -M- says:

Guess he found them between two flag poles

4. Anders says:

How could he? The dog ate all the marbles

8. Mervulon says:

Yes. Only one can be true, and 1) is the only one that so states. Richard, do you not read these prior to posting the solutions?

9. Niva says:

Exactly one of the Richard’s solution to this puzzler is not true.

10. Anataboga says:

Statement 1!

11. Michael Sternberg says:

I agree with all above and had pegged 1, for exactly the same reasons stated.

Perhaps, it’s “opposite day” today.

12. Roland says:

What I said last week. Statement 9 3/4 is true 😉

14. edgar1975 says:

I think the answer and puzzle are both wrong.

NB THIS IS A COMMENT ON RW’s ORIGINAL ANSWER

RW’s answer works if you substitute “true” with “false”.

I think he has done an incomplete cut and paste from a similar question on the Internet

16. This doesn’t make sense, surely?

Statement 1 in this case is true.

If it were “exactly ____ of these statements are FALSE…”, then 9 would be correct.

Am I right? I feel like this might be a trick in itself…

17. Furie says:

Oh thank god for that. I thought I was being really thick then not understanding that. I had 1 as the only possible true one too.

18. Klaas De Smedt says:

Did I solved it? 5 minutes ago I thought so, then I saw your answer and your logic that I can’t follow. The read some of the comments.

The real question is: Did Richard Wiseman solved it 😉

19. Roland says:

The answer to Friday’s question: it took Richard the full weekend and a number of our comments to solve it 🙂

20. Martha says:

That one was easy!!!

21. Lazy T says:

I still have problems…
exactly doesn’t mean exclusivley, so 1, 0, any of them, all of them, could be true.
The statements only contradict each other if you interpret ‘exactly’ to mean more than it does.

“at most one of them can be true, ….Therefore statement 1 is the only true statement.”
‘can be true’ doesn’t mean ‘is true’

I have other problems too but they’re for HungoverPedant.nit

1. Anders says:

What are you talking about? “Exactly one” can never mean 0 or two. It can only mean “one”. It is exactly the meaning of “exactly”

When someone says to you “be here at exactly 2pm” and you show up at 5am and wait, you won’t get far by claiming that “exactly” doesn’t mean exclusively so you decided to include some more hours.

If you want to be pedantic, wait for something that actually has multiple meanings

You are right that “can be” isn’t the same as “is”, but in this puzzle it’s kinda moot

I do wish there had been one more statement “exactly 0 of these statements are true” to remove the second solution that they’re all false, as M pointed out

2. Photon Boy says:

Exactly DOES imply exclusively, Lazy T. Thats kind of what it means !

3. Lazy T says:

I’m not looking for mutiple meanings for ‘exactly’ I just think it was superfluous, it’s use actually adds to my confusion as to which solution was being asked for.

‘Only seven of these…’, ‘Just four of……’, or even ‘Three of..’ would have been clearer.

and
they didn’t say I couldn’t turn up at 5am, but it was a mistake to sleep in their herbacious border.

22. jumbleguitar says:

1 or 0, surely?

23. Duncan says:

Shame the wording was wrong, it made the puzzle too easy.

24. Anonymous says:

What is true about, “This statement is true.”? It implies neither truth nor false.

25. Eddie says:

Can I recommend listening to the ‘Logic’ episode of ‘In Our Time’ with Melvyn Bragg. It’s right up this street.

26. Christian says:

“here are 10 numbered statements” is also a true statement, so there are exactly 2 true statements.

1. Anonymous says:

…except it asks how many of the numbered statements are true, and “here are 10 numbered statements” is not numbered.

I’m tempted to say “Read the question” but if Richard Wiseman doesn’t RTQ, why should we expect Christian to?

27. Too many book tours, Richard. How many puzzles have required a correction after posting this year? Half?

28. Christian says:

I did read the question, and my statement still holds. I am not disputing the answer, but had the question been phrased just slightly different it could contain the twist I mentioned, thus making it a more interesting puzzle.

29. I said that there are two solutions.

One of them is that only one is true. The other solution is that none of them are true.

1. Just cause everyone disagrees, doesn’t mean one of them has to be right.

I am irresistibly prompted to mention religion at this point.

30. And “none are true” is a better solution (or, at least it’s more fun). A string of words does not have to be either true or false, consider: “are of true these Exactly five statements”, “are exactly five of these true?” or “flibble”.

One solution to the liar paradox uses this; “this sentence is false” as it refers to itself is taken to be just “flibble”, meaningless nonesense. All of the above numbered “statements” refer to themselves, are thus flibble and not statments at all. It is the assertion in the question, “here are 10 numbered statements”, that is false!

1. Photon Boy says:

Statements can be false (ask any politician). Therefore there is nothing false about “here are 10 numbered statements”.

31. Bob says:

Actually, according to widely accepted rules of logic (in mathematics) a statement cannot determine whether it’s true, nor can it determine the validity of any other stamement referring to the validity of the first statement. Therefore, none of these statements are either true or false. This rule was established in order to avoid paradoxes such as: 1. Statement number 2 is true. 2. statement numer 1 is false.

32. There is a subtlety here that Richard overlooked, and a quite interesting one: Statement number 1 is the only one that *can* be true, but it doesn’t therefore follow that it *is* true. In essence, the statement says “this sentence is true”, and there is no way to resolve whether or not it in fact is. This is kind of a converse to the Liar’s Paradox; the sentence “this sentence is false” has indeterminate truth value, because assuming it’s true implies it is false and vice versa. The given sentence, “this sentence is true”, confirms any assumption we make of it, but we can’t get anything else out of it, so it too has indeterminate truth value. So the correct answer is either zero or “at most one”, depending on your preferences.

33. joshthelegend says:

I’m not really sure what there is to argue about on this one. It’s clearly either just No. 1 or none at all. I’m 14, and I got that in about ten seconds.