False Statements
None of these statements is correct.
At most 1 of these statements is correct.
At most 2 of these statements are correct.
…
At most 98 of these statements are correct.
At most 99 of these statements are correct.
How many of these statements are correct?
SOLUTION
If the number of true statements is X, then statements 1, 2, … , X are wrong, and the rest are correct. Therefore X = 100 – X and X = 50. Thus, there are 50 correct statements.
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This appears to compound the mathematical gibberish of the shrinking cucumber problem given earlier. OK, so say the number of true statements is substituted by “X” in a formula. You then render the term “X” valueless by saying that it equally represents the number of false statements! This would be true ONLY if there were 50 correct statements, which you assert to be the case without any evidence whatever to support it. One might as well say 13 are correct, or 13 are wrong, or any number at all between 1 and 99 are correct or wrong, for there is no evidence as to the correctness or falsity of any single proposition. The number of true and false statements can correctly be stated to lie in the range of 1 99, with no particular number be ascertainable either way. Unless previous comments have been deleted, I am surprised no one has pointed this out before!
Hi Element. The claim that the first X statements are false follows from the fact that they contradict our assumption. For example, if there are 2 correct statements, then the first statement is wrong, since it says that no statements are correct. Similarly, the second statement, “there is at most 1 correct statement”, is also wrong.