# Thirteen politicians

Yesterday evening there was a public discussion on TV that was dedicated to the upcoming elections. Thirteen politicians from the thirteen most important political parties participated in it.

• After half an hour, one of the politicians summarized the situation and said: One lie has been told.
• Another one of them said: Now two lies have been told.
• Then a third one said: Now three lies.
• A fourth one said: And now four lies have been told.
• The fifth politician: Actually, now there are already five lies.
• The sixth one: Now six lies have been told.
• The seventh one: And now seven lies have been told.
• The eight politician: Now eight lies have been told.
• The ninth one: Nine lies have been told.
• The tenth one: Up to now ten lies have been told.
• The eleventh politician: Now even eleven lies have been told.
• The twelfth politician: Twelve! Now twelve lies have been told.
• Then the thirteenth politician said: And now thirteen lies have been told.

At this moment the moderator got fed up and ended the discussion. The discussion was later carefully investigated by the political analysts, and it turned out that at least one of the politicians had correctly stated the total number of lies told up to the moment just before he made his claim.

Question: How many lies were altogether told by these thirteen politicians?

• I have a feeling this is a dupe, but can't find what it's a dupe of... – Rand al'Thor Mar 6 '15 at 10:33
• 13 political parties...wow...I am jealous... – kaine Mar 6 '15 at 13:51
• @randal'thor I find it similar to puzzling.stackexchange.com/questions/8718/how-many-knights but not quite a duplicate. – user3453281 Mar 6 '15 at 15:05
• 13 politicians = 13 lies. – Jodrell Mar 6 '15 at 16:18
• My intuition tells me there is some sort of trick in the question and the answer is not 13! :/ – The Dragonista Mar 6 '15 at 19:19

I have a much simpler argument (of course leading to the same answer):

Let $x$ be the number of the first politicians who says the truth. Then there are the $x$ lies mentioned in his statement plus the $13-x$ lies of the $13-x$ politicians speaking after him. This altogether gives

$x+(13-x)$ lies.

Exactly

one

lie must have been told before the quoted discussion, so the total number of lies altogether is

thirteen.

Explanation:

• if any other number of lies had already been told, then the first politician must have been wrong to say it was one, so by induction the $n$th politician was wrong to say it was $n$.

• given the total earlier number of lies, we know the first politician was right, so all the others are wrong (since by induction, each one could only be right if all the preceding ones were wrong).

• "Exactly lie must have been told before the quoted discussion" I guess this isn't necessary at all. – Xellos Mar 6 '15 at 10:34
• Although I find your logic to be correct, there's no way I'm willing to believe that in one half hour, thirteen politicians told that few lies. – Joel Rondeau Mar 6 '15 at 14:19
• Joel, most politicians don't actually lie. Generally, they say statements that are technically true but don't accurately convey the spirit of the truth. They would probably be really good at the riddles on this site!! – corsiKa Mar 6 '15 at 16:23
• @KSmarts "The new policy will allow us to save on foreign expenses by up to \$50M dollars or more!" – user3453281 Mar 6 '15 at 22:24
• @corsiKa Here in the UK some politicians lie like m***********s. I'm thinking of Aitken and Archer and their criminal convictions for perjury and fraud respectively. – A E Mar 22 '15 at 17:13

13

Because:

If there was no lie at all before that discussion, first one would be the first to lie, bringing lie counter to 1. Then second would bring it up to 2 and so on. But, in this case, none of them would state the total number of lies correctly before making his/her claim.
If there was 1 lie before that discussion, that would make first one to be correct about lie counting, making his/her statement true. Then second said Now two lies have been told., but actually there was only 1 lie, so now total lies is 2. Third would say that there were 3 lies, but his/her statement would bring total lies to 3, not before his statement. It sums up to 13 lies in total.
If there were 2 or more lies before that discussion, first one would lie and bring the counter to 3+, that would make second one lie too and so on, so none of them would state the total number of lies correctly before making his/her claim.

14 lies were told.

Since it's well known that politicians are incapable of telling the truth, it follows that every statement they made was a lie (13 lies). On top of that, the Analyst must also have been lying when he said that one of the politicians wasn't lying. That makes 14.

• There's no lateral-thinking tag! – Rand al'Thor Mar 7 '15 at 11:18
• @randal'thor - I demand that anyone who disagrees should tell me where my logic is faulty. – Richard Mar 7 '15 at 11:23
• The question is faulty, because it assumes the analysts are telling the truth. You're answering a more realistic scenario, rather than the question as stated. Also I see you haven't updated your avatar pic on this site! – Rand al'Thor Mar 7 '15 at 11:26
• But then I guess you must include the number of lies in the half hour before. – Paŭlo Ebermann Mar 8 '15 at 17:51
• @PaŭloEbermann - I like to think of it as a single meta-lie from each of them, like counting individual snowflakes in a drift. – Richard Mar 8 '15 at 18:12

13 lies

if there were no lies at the beginning, then 13 politicians lied. if there were one lie, then 12 polititians lied (all except the first one), total 13. if there were one lie, then 11 polititians lied (all except the first and the second one), total 13. etc...

• wrong. If there were 20 lies, then all of them lied, so total would be 33 lies. And how much is 1 + 11 again? – Novarg Mar 6 '15 at 10:42
• and how did the second not lie if there was 1 lie before first one? – Novarg Mar 6 '15 at 10:47
• ok, I guess there are many possible solutions because of your first comment. – Xellos Mar 6 '15 at 10:49
• ok, I understand, u r right, upvote ^^ – Xellos Mar 6 '15 at 10:55

Here's a short proof.

If the first politician was wrong, then they were all wrong, which isn't so. So he was right, which makes all the others wrong. So one lie was told before these statements, and 12 of the statements are wrong, so the answer is 13.

The discussion was later carefully investigated by the political analysts, and it turned out that at least one of the politicians had correctly stated the total number of lies told before he made his claim.

That's a lie!

Case 0: Not a single politician lied during the discussion.

In this case, the first politician interviewed after the discussion lied by saying there was one lie (there were none), the next to be interviewed lied by saying there were two lies (there was one), and so on. While none of the politicians exhibited their true colors during the discussion, every single one exhibited their true colors in the follow-up. In this case, not a single one of the politicians correctly stated the total number of lies told before he made his claim.

Case 1: Only one lie was told during the discussion.

In this case, the first politician interviewed after the discussion told the truth by saying there was one lie, but the subsequent politician to be interviewed lied, as did every politician who followed.

Case 2: Two or more lies were told during the discussion.

In this case, every politician interviewed after the discussion lied about the number of lies told before that politician was interviewed. As with case #0, not a single one of the politicians correctly stated the total number of lies told before he made his claim.

There is no way to ascertain the number of lies told by the politicians. Most likely the number of lies was in the hundreds during the discussion (how can you tell when politicians are lying? Answer: When you see their lips moving) plus thirteen more lies in the follow-up.

12. If we wish to answer the question as it is asked here, then the correct answer is not 13. We have been told three specific things: First, one of the politicians had correctly stated the total number of lies told before he made his claim. This has to be the first politician, since all others respond with an n + 1 statement (based on the preceding politician's stated number). So, the first lie was already told beforehand, and the first politician's statement was true. Secondly, by the logic I have just described, we have observed the subsequent twelve politicians tell lies. Thirdly, the question asked of us is to determine how many lies were altogether told by these politicians. We cannot infer or assume that the first lie was told by the politicians themselves. They could be discussing a lie that the moderator brought up as an anecdote. So, unless it is explicitly stated they told this lie, it cannot be counted. (The default value is false.)

I may be splitting hairs with this level of precision, but this is the logically correct answer to the question. If this is not the intended answer, then the question should be rephrased.

Edit: To extend the flaw a bit further, it doesn't explicitly state that these were the only possible lies. What about the preceding 30 minutes? They could have told hundreds. The riddle should be reworded.

All answers so far assume that a false statement is a lie. But this is mistaken, because a lie is a deliberate untruth. If someone makes a false statement that they believe to be true, they're not lying. If we go by this, then the answer is that the number of lies told was

between 1 and 13 inclusive

Proof:

If the first politician was right, then all the others were wrong, so the number of lies told was one plus however many of the other politicians were deliberately wrong, which could be any number between zero and 12. If the first politician was wrong, then at least one of the other politicians must have been right, giving us a number of lies between two and 13. So the number of lies altogether is between 1 and 13 inclusive.

this analysis is based on this note :

• one lie maximally was told before the conversation got started.

If the first guy was truthful , all other politicians wud have lied , n=13 lies , 12 political lies aside the first one.

Now , if the first guy told a lie , it means no lie was been told until first politican spoke, one lie until second politician lied and third , then forth , etc.

this does mean on the whole .

n=13 lies was been told

if we admit more than a lie from the first statement , n can go over 15 if first guy lied.

• But you know one of them told the truth, so if there were 2 or more lies to begin with, then none of them would have told the truth. – Trenin Mar 10 '16 at 13:11