true or falsse?

[westworld]

Consider the following ten statements. 1. Exactly one of these statements is false. 2. Exactly two of these statements are false. 3. Exactly three of these statements are false. 4. Exactly four of these statements are false. 5. Exactly five of these statements are false. 6. Exactly six of these statements are false. 7. Exactly seven of these statements are false. 8. Exactly eight of these statements are false. 9. Exactly nine of these statements are false. 10. Exactly ten of these statements are false. How many of the statements are true?

 

[soroban]

Hello, westworld!

Consider the following ten statements:

1. Exactly one of these statements is false.
2. Exactly two of these statements are false.
3. Exactly three of these statements are false.
4. Exactly four of these statements are false.
5. Exactly five of these statements are false.
6. Exactly six of these statements are false.
7. Exactly seven of these statements are false.
8. Exactly eight of these statements are false.
9. Exactly nine of these statements are false.
10. Exactly ten of these statements are false.

How many of the statements are true?

Note that no two statements can be true.
. . We cannot have "exactly 3 false statements" and "exactly 5 false statements".

We must consider the statements one-at-a-time.

Suppose #1 is true:."Exactly one of these statements is false."
. . Then nine statements are true . . . not allowed.

Suppose #2 is true: "Exactly two of these statements are false."
. . Then eight statements are true . . . not allowed.

And so on . . .

Suppose #8 is true: "Exactly eight of these statements are false."
. . Then two statements are true . . . not allowed.

Suppose #10 is true: "Exactly ten of these statements are false."
. . Then all of the statements are false ... including #10.
. . We have a paradox.


But suppose #9 is true: "Exactly nine of these statements are false."
. . Then only #9 is true; the others are all false.
This one works!

Answer: one statement, #9.