$2+$

12 candidates for mayor were talking about themselves. After some time, one said: "They lied to me once." Another said: "Now it's twice." "Now it's three times" - said the third, and so on up to the 12th, who said: "Now they have lied 12 times." At this point, the host interrupted the discussion. It turned out that at least one candidate correctly counted how many times they had been lied to before them. So, how many times did the candidates lie in total?
## Solution

Assume that the first candidate lied. This means that before his statement, the number of false statements was not equal to 1. After his statement, this number increased by 1, so it became not equal to 2. Therefore, the second candidate also lied. Continuing this reasoning, we get that all candidates lied, which contradicts the condition of the problem. Therefore, the first candidate told the truth. Thus, the number of false statements was 1 both before and after his statement. Therefore, the second candidate lied. Continuing this reasoning, we get that all subsequent candidates also lied. Thus, all candidates except the first one lied, and one more lie was uttered before the described discussion. Therefore, the candidates lied a total of 12 times.