Aparent Paradox:
Which of the Sentences are True?

Following are ten statements. How many of them are true? If any, which ones?

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.

Solution

Self-reference and apparent self-reference

1. Does It Blink?
2. Apparent paradox
3. Set of all subsets
4. An Impossible Page
5. Russell's paradox
6. An Impossible Machine
7. A theorem with an obvious proof
8. The Diagonal Argument
9. A link to a very similar puzzle

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Copyright © 1996-2018 Alexander Bogomolny

Solution

If all 10 are false then this is exactly what the 10th asserts. But if it's true, we get a contradiction since then not all 10 are false. Hence, at least one is indeed false. On the other hand, no two of the statements may be true simultaneously; for they make contradictory assertions. Therefore, exactly one of them is true while the remaining nine are false. This is what is claimed by the ninth statement. This is true too.

Reference

1. D.Wells, The Penguin Book of Curious and Interesting Puzzles, Penguin Books, 1992

Self-reference and apparent self-reference

1. Does It Blink?
2. Apparent paradox
3. Set of all subsets
4. An Impossible Page
5. Russell's paradox
6. An Impossible Machine
7. A theorem with an obvious proof
8. The Diagonal Argument
9. A link to a very similar puzzle

|Front page| |Contents| |Algebra| |Up|

Copyright © 1996-2018 Alexander Bogomolny