Put answers/explanations in a tab, like below, to avoid spoiling it for others.
[tab]Use the “tab” button in your reply[/tab]
Exactly 1 of these statements is false.
Exactly 2 of these statements are false.
Exactly 3 of these statements are false.
Exactly 4 of these statements are false.
Exactly 5 of these statements are false.
Exactly 6 of these statements are false.
Exactly 7 of these statements are false.
Exactly 8 of these statements are false.
Exactly 9 of these statements are false.
Exactly 10 of these statements are false.
Exactly 11 of these statements are false.
Exactly 12 of these statements are false.
Which statements, if any, are false?
Part 2: Replace ‘Exactly’ with ‘At least’. Now which statements, if any, are false? Part 3: Try ‘At most’. Same question.
[tab]I’m thinking 11 has the only chance of being true, so that would make 1 to 10 and 12 false (the possibility of 0 statements being true is still open).
The general formula is: for any statement “Exactly n of these statements are false,” if it is true, then exactly n of these statements must be false. Which means that any statement that says “Exactly m of these statements are false,” if m != n, then it must be false. So if statement 11 is saying that exactly 11 of these statements are false, and 11 of these statements say otherwise, then they are false given that statement 11 is true. This logic does not apply to any other statement (try it!).
I’ll try part 2 and 3 later. Right now, my brain will explode if I try them.[/tab]
[tab]Since the general rule is: each statement contradicts each other statement, there can only be at most 1 statement that’s true, that being statement #11. So at least 11 statements must be false. There can be 12 statements that are false, but not at least 12, so that rules out statement #12. Statement #11 remains the only viably true statement.[/tab]
Part 3:
[tab]If there are more than 11 statements that are false, then there are at most 12 statements that are false (there are only 12 statements in the list). But then statement #12 would be true, reducing the number of false statements to 11. So really, there can’t be 12 false statements. Statement #11 remains the only viably true statement. So at most there can only be 11 false statements.[/tab]
[tab]with “at least x of these statements are false”… if this statement is true, then everything less than whatever number x is must also be true. if “at least 2 of these statements are false” is a true statement, then “at least 1 of these statements is false” must also be a true statement. So i think the highest true statement is 6… “at least 6 of these statements are false” is a true statement. statements 7,8,9,10,11,12 are all false statements. statements 1 2 3 4 5 6 are true statements[/tab]
stab at part 3
[tab]gets tricky here semantically i think. “at most 12 of these statements are false” will ALWAYS be a true statement, assuming each statement can be assigned a true/false value. if 0 of the statements are false, the statement “at most 12 of these statements are false” is still a true statement. if all 12 of the statements are false “at most 12 of these statements are false” is still at true statement. same with any number inbetween 1 and 12.
we know that “at most 11 of these statements is false” is a true statement, because statement 12 is true
this cascades down
they’re all true. each statement would handle the case of 0 false statements “at most 1 of these statements is false”[/tab]
As we know, women are airheads.
I will wrap my brain around the “at least and at most” bit in the morning when I have hard wood. Hard wood and mathematics go together.
I cannot logic when my mind is female.
I just want to vegetate.
UP1001,
[tab]nice, you got part 1.[/tab]
gib,
[tab]correct, every statement except number 11 must be false. interesting formula, i’ll have to think about that.
your answer for part 2, however, is not correct.
and I’ll get back to you on part 3.[/tab]
Ecmandu,
[tab]not quite. some or all of the statements must be false for part 1.[/tab]
incorrect,
[tab]awesome, you got part 2! and your explanation is pretty much is exactly what my thought process was for solving it, too.
I’ll get back to you on part 3.[/tab]