### Brain Teasers

# Liars

Twelve men are in a room, some of them always tell lies, while the others always tell the truth.

The first man says: "None of us is a truthteller."

The second man says: "At most one of us is a truthteller."

The third man says: "At most two of us are truthtellers."

...

The twelfth and last man says: "At most eleven of us are truthtellers."

Who are the liars?

The first man says: "None of us is a truthteller."

The second man says: "At most one of us is a truthteller."

The third man says: "At most two of us are truthtellers."

...

The twelfth and last man says: "At most eleven of us are truthtellers."

Who are the liars?

### Hint

It's OK to use trial and error, however, there might exist better methods.### Answer

If there are exactly N liars, then the last N men are truthtellers, leaving the first (12-N) men to be liars.Therefore, N=12-N, N=6. The first 6 men are liars, and the last 6 men are truthtellers.

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## Comments

Good one. 6 is the only answer that works for all statements because they are saying at MOST. Very good. 7-12 are truthtellers while 1-6 are the liars.

This leaves me SOOO confused .

ok at first i nvr got it but it would of helped 2 put sum1 who waz a truth teller (anthor 1)

No, it has to be wrong. N = 12 - N isn't true at all. If you subtracted the number of people lying from the total number of people, you would get the number of people telling the truth, not the number of people lying.

The logic on this is flawed. The first man HAS to be a liar, but as far as I can see, the other eleven could be telling the truth. At least one has to be telling the truth, but other than that, it can go any number of ways.

Also, you can't use the same variable for the people who are lying and the people who are telling the truth. The answer doesn't work.

I've given this some thought. The answer is right, but the reasoning is wrong. Here is my answer:

Let's rewrite each man's statement so that it is clearer:

1st man: All of us are liars.

2nd man: At least 11 of us are liars.

3rd man: At least 10 of us are liars.

4th man: At least 9 of us are liars.

5th man: At least 8 of us are liars.

6th man: At least 7 of us are liars.

7th man: At least 6 of us are liars.

8th man: At least 5 of us are liars.

9th man: At least 4 of us are liars.

10th man: At least 3 of us are liars.

11th man: At least 2 of us are liars.

12th man: At least 1 of us is a liar.

If the first man is telling the truth, then his own statement is false, therefore he is lying. This makes the 12th man's statement true. Now, if #2 is telling the truth, then there are 2 people who tell the truth and only 10 liars, so #2 is lying. This makes #11 true, so we have at least 2 truth tellers and 2 liars. Now, if #3 is telling the truth, then there are 3 people who tell the truth and only 9 liars, so #3 is lying. This makes #10 true, so we have at least 3 truth tellers and 3 liars. Now, if #4 is telling the truth, then there are 4 people who tell the truth and only 8 liars, so #4 is lying. This makes #9 true, so we have at least 4 truth tellers and 4 liars. Now, if #5 is telling the truth, then there are 5 people who tell the truth and only 7 liars, so #5 is lying. This makes #8 true, so we have at least 5 truth tellers and 5 liars. Now, if #6 is telling the truth, then there are 6 people who tell the truth and only 6 liars, so #6 is lying. This makes #7 true, so we have 6 truth tellers and 6 liars.

Sorry it's so long, but this is the only way I could explain it.

Let's rewrite each man's statement so that it is clearer:

1st man: All of us are liars.

2nd man: At least 11 of us are liars.

3rd man: At least 10 of us are liars.

4th man: At least 9 of us are liars.

5th man: At least 8 of us are liars.

6th man: At least 7 of us are liars.

7th man: At least 6 of us are liars.

8th man: At least 5 of us are liars.

9th man: At least 4 of us are liars.

10th man: At least 3 of us are liars.

11th man: At least 2 of us are liars.

12th man: At least 1 of us is a liar.

If the first man is telling the truth, then his own statement is false, therefore he is lying. This makes the 12th man's statement true. Now, if #2 is telling the truth, then there are 2 people who tell the truth and only 10 liars, so #2 is lying. This makes #11 true, so we have at least 2 truth tellers and 2 liars. Now, if #3 is telling the truth, then there are 3 people who tell the truth and only 9 liars, so #3 is lying. This makes #10 true, so we have at least 3 truth tellers and 3 liars. Now, if #4 is telling the truth, then there are 4 people who tell the truth and only 8 liars, so #4 is lying. This makes #9 true, so we have at least 4 truth tellers and 4 liars. Now, if #5 is telling the truth, then there are 5 people who tell the truth and only 7 liars, so #5 is lying. This makes #8 true, so we have at least 5 truth tellers and 5 liars. Now, if #6 is telling the truth, then there are 6 people who tell the truth and only 6 liars, so #6 is lying. This makes #7 true, so we have 6 truth tellers and 6 liars.

Sorry it's so long, but this is the only way I could explain it.

I like, no I love my reasoning which is logic. Logic is mathematics and the only real numbers given are l and 12 which makes the answer 13. Either all are liars or one is a liar, the rest of the information cannot matter because no real numbers exist, only 1+ 12 = 13. Either one or all, no one can say for 2 thur ll for those are not real numbers. I'm sorry, I didn' t hear that comment. Just having fun, the answer 6 is probably Wrong

kwelchan is right.. you cannot use the same variable to represent the liars and the truthtellers..

thanks kwelchans, that explanation works very nicely.

thanks for the quiz.

thanks for the quiz.

Kweich's way works, this teaser's does not. It's completely flawed.

Kweich's way works, this teaser's does not. It's completely flawed.

Feb 12, 2009

This answer to this teaser is completely wrong. I agree with the comment above that the same variable cannot be used for both the number of liars and truth tellers.

Furthermore, the statement begins with "at most" which implies that there can be NO MORE THAN the specified number. Ergo, the only individual who would be telling the truth would be the second man who states that "at most one of us is a truthteller". It is the only possible real solution.

Look at it another way, two or more men cannot possibly be telling the truth. Consider only the first two men (or any two men, really): one man says "at most one of us is a truthteller" whereas the next one says "at most two of us are truthtellers". If the second man is telling the truth, then the first man HAS to be lying because he states that AT MOST there is only one man telling the truth (assuming that somehow the second man could be telling the truth). But because every man makes the same statement plus one, then only the one statement "at most one of us is a truthteller" can possibly be true.

I hope this explaination helps!!!

Furthermore, the statement begins with "at most" which implies that there can be NO MORE THAN the specified number. Ergo, the only individual who would be telling the truth would be the second man who states that "at most one of us is a truthteller". It is the only possible real solution.

Look at it another way, two or more men cannot possibly be telling the truth. Consider only the first two men (or any two men, really): one man says "at most one of us is a truthteller" whereas the next one says "at most two of us are truthtellers". If the second man is telling the truth, then the first man HAS to be lying because he states that AT MOST there is only one man telling the truth (assuming that somehow the second man could be telling the truth). But because every man makes the same statement plus one, then only the one statement "at most one of us is a truthteller" can possibly be true.

I hope this explaination helps!!!

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