Sorry to be so slow getting here to judge the answer. We have had two entries with the same answer, both correct, with nippersean in first with a concise explanation which omitted a point (to be explained below) and with wercool coming in second with a more complete analysis which, I'm sorry, is a bit muddled from my point of view. I guess I will have to give it to nippersean, for a substantially complete and correct answer.
The answer found in Gardner's book covers any list of n such statements (provided n is greater than 1) follows:
At most one of the statements can be true because any two contradict each other. All the statements cannot be false, because this implies the list contains exactly zero false statements. Thus exactly one statement can be true. Thus exactly n - 1 are false and the (n - 1)st statement is true.
The bold statement is the point that nippersean missed, and if you do not get why that implication is a problem, ponder this: exactly zero false statements implies exactly
n true statements, and with
n greater than one, they will contradict each other.
Changing "exactly" to "at least" is pretty much the same as simply removing "exactly" (the "at least" would be understood). This variant was mentioned in Gardner's book and does in fact lead to 5 true and 5 false statements.
If you reduce the list to one statement, you get
- Exactly one statement on this list is false.
This is equivalent to the old liar's paradox: "This statement is false."
You can get around this by adding the zero statement:
- 0. Exactly none of the statements on this list is false.
Interestingly, this moves the one true statement from position (
n - 1) to position
n.
So, you are up, nippersean. Let's have another.
OK, a quick and simple one: {stolen from Raymond Smullyan}Hurry up then with with the next one!
