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**Answer To The Logically Impossible Logical List Riddle**

In the first problem, line *n* states that exactly *n* statements are true.

#1. Exactly 1 of the statements in this list is false.

#2. Exactly 2 of the statements in this list are false.

#3. Exactly 3 of the statements in this list are false.

#4. Exactly 4 of the statements in this list are false.

#5. Exactly 5 of the statements in this list are false.

#6. Exactly 6 of the statements in this list are false.

#7. Exactly 7 of the statements in this list are false.

#8. Exactly 8 of the statements in this list are false.

#9. Exactly 9 of the statements in this list are false.

#10. Exactly 10 of the statements in this list are false.

Suppose #10 is true. This implies all statements are false, including #10. This is a contradiction, so #10 must be false. At least 1 of the statements has to be false.

Can 2 of the statements be true simultaneously? The answer is no because the statements are mutually exclusive. It is not possible to have exactly *x* statements and exactly *y* statements be true for *x* ≠ *y*. Therefore, at most 1 of the statements is true.

If 1 statement is true, then 9 statements are false.

This is what #9 states. If #9 is true, and all the other statements are false, there will be exactly 9 false statements and 1 true statement.

Therefore statement #9 is true and statements 1-8 and 10 are false.

**Part 2: “at least”**

Now we ask: how many are false, if line *n* reads that “at least” *n* items are false?

#1. At least 1 of the statements in this list is false.

#2. At least 2 of the statements in this list are false.

#3. At least 3 of the statements in this list are false.

#4. At least 4 of the statements in this list are false.

#5. At least 5 of the statements in this list are false.

#6. At least 6 of the statements in this list are false.

#7. At least 7 of the statements in this list are false.

#8. At least 8 of the statements in this list are false.

#9. At least 9 of the statements in this list are false.

#10. At least 10 of the statements in this list are false.

Suppose #10 is true. This would imply all 10 statements are false, including #10, which is a contradiction. So at least 1 statement has to be false, meaning statement #1 is true.

Now suppose #9 is true. Since statement #1 is true, that means all other statements (#2 to #10) have to be false, which contradicts that #9 is true. Therefore #9 has to be false, so there are at least 2 false statements. Hence statements #1 and #2 are true.

We can similarly reason that statements #8, #7, and #6 cannot be true, which implies #3, #4, and #5 are true, respectively. There must be at least 5 false statements and statements #1 to #5 have to be true.

This leaves the candidate that there are 5 false statements (#6 to #10) and there are 5 true statements (#1 to #5).

This works without contradiction. There are 5 false statements (#6 to #10) and 5 true statements (#1 to #5).