Puzzle | 100 wrong statements

Puzzle: There are 100 men in a village. One day they gather at a place and started making statements.

1st man said: "At least 1 is wrong."
2nd man said: "At least 2 are wrong."
3rd man said: "At least 3 are wrong."
4th man said: "At least 4 are wrong."
5th man said: "At least 5 are wrong."
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Similarly,
98th man said: "At least 98 are wrong."
99th man said: "At least 99 are wrong."
100th man said: "At least 100 are wrong."

So the question remains, how many of them are correct and how many are wrong?

Solution: Let’s try to analyse each of the men’s statement starting backwards from the 100th man.

• 100th man said: “At least 100 are wrong.”
  Now the man’s statement is contradicting itself. If what he said is assumed to be correct, that would make the 100th statement itself as false. According to 100th statement, if at least 100 are wrong, then that must include the 100th man himself, but assuming him to be correct would raise discrepancy. Therefore, it can be concluded that:
  
  
  • 100th statement is wrong
  • 1st statement is correct i.e., “at least 1 is wrong”. Here it is the 100th man.
• 99th man said: “At least 99 are wrong.”
  Assuming this to be a correct statement, would make 2 men correct i.e, the 1st one and the 99th himself. Thus the 99th statement would be proven false which claimed at least 99 of them to be wrong and one correct. Therefore, it can be concluded that:
  • 99th statement is wrong.
  • 2nd statement is correct i.e., “at least 2 are wrong”. Here they are 99th and the 100th men.
• 98th man said: “At least 98 are wrong.”
  Assuming this statement to be correct would make 3 men correct, the 1st, 2nd and the 98th himself. Now he says that at least 98 are wrong and 2 are correct which is again false. Thus, it can be concluded that:
  • 98th statement is wrong.
  • 3rd statement is correct i.e., “at least 3 are wrong”. Here they are 98th, 99th and the 100th men.
• .
• .
• .

Moving on in a similar way, it can be found that the first 50 men make correct statements and the last 50 men make the wrong statements.

• .
• .
• .
• 50th man said: “At least 50 are wrong.”
  Assuming this statement to be correct would make 50 men correct, the 1st, 2nd, .. and the 50th himself. Now he says that at least 50 are wrong and 50 are correct which is again true. Thus, it can be concluded that:
  • 50th statement is wrong.
  • 50th statement is correct i.e., “at least 50 are wrong”. Here they are 1st, 2nd, .. and the 50th men.