A sheet of paper has statements numbered from 1 to 70. For all values of n…
2023
A sheet of paper has statements numbered from 1 to 70. For all values of n from 1 to 70, statement n says, 'At least n of the statements on this sheet are false.' Which statements are true and which are false?
- A.
The even numbered statements are true and the odd numbered are false.
- B.
The odd numbered statements are true and the even numbered are false.
- C.
The first 35 statements are true and the last 35 are false.
- D.
The first 35 statements are false and the last 35 are false.
Show answer & explanation
Correct answer: C
Concept
Each statement n asserts "at least n of the 70 statements are false." These claims are nested: if "at least n are false" holds, then "at least (n-1) are false" must also hold, since a bigger false-count implies every smaller false-count. So whenever statement n is true, every earlier statement 1 through n-1 is also true - the true statements always form an unbroken opening block, and the false statements form the remaining block after it.
Application
Let k be the number of true statements at the start of the block, so statements 1 to k are true and statements k+1 to 70 are false - meaning exactly 70 - k statements are false in total.
Statement k is true, so its claim "at least k are false" must hold: 70 - k >= k, which gives k <= 35.
Statement k+1 is false, so its claim "at least k+1 are false" must NOT hold: 70 - k < k + 1, which gives k >= 35 (since k must be a whole number, k > 34.5 means k >= 35).
Both bounds together force k = 35: the first 35 statements are true and the remaining 35 are false.
Cross-check
With k = 35, exactly 35 statements are false. Statement 35 claims "at least 35 are false" - true, since the count is exactly 35. Statement 36 claims "at least 36 are false" - false, since only 35 are false, not 36 or more. Every statement's own claim is satisfied, confirming the split is consistent.