mathematics - 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?

Answer

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.