Question 1181761: James, Alex, and Linda were playing inside of their classroom. One of them broke the window. When their teacher asked who broke the window, she got these replies:
Linda: James broke the window!
James: I didn't break it!
Alex: Neither did I!
If two of the children lied, then who broke the window?
Found 4 solutions by MathLover1, math_tutor2020, MathTherapy, ikleyn:
Answer by MathLover1(20849)   (Show Source):
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if Linda and James lied, then Alex is telling the truth=> then Linda or James broke the window
if Alex and James lied, then Linda is telling the truth=>then James broke the window
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Answer: Alex broke the window
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Explanation:
Assume Linda is telling the truth. If so, then James is one of the two liars and he's the one that broke the window. At the same time, Alex is the other liar, which must mean he broke the window. We have a contradiction because only one kid broke the window.
This ultimately means Linda is not telling the truth, and we can conclude that James did not break the window.
So either Alex or Linda broke the window.
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Now assume that Alex is telling the truth, which means we assume he didn't break the window. That makes James and Linda the two liars.
If James is lying, then he broke the window.
If Linda is lying, then the truth is actually "James did not break the window".
We have another contradiction. So we rule out the scenario that Alex is being honest.
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The only thing left is to assume that James is the one telling the truth. He claims he didn't break the window, which matches with the conclusion made at the end of the first scenario (when we concluded that Linda wasn't telling the truth).
If Alex is lying, then he broke the window because it flips his original statement "I didn't break the window" (paraphrased) to "I did break the window". At the same time, Linda's original statement "James broke the window" is flipped to "James did not break the window".
In this third scenario, we can see that no contradictions happen. Since it's the last scenario left, and we've crossed out all the other previous ones, this must mean this scenario is the definitive case of what's going on. We've shown that James is the only kid telling the truth and the other students are lying. Furthermore, we've concluded that Alex is the one that broke the window.
Answer by MathTherapy(10552)  (Show Source):
You can put this solution on YOUR website!
James, Alex, and Linda were playing inside of their classroom. One of them broke the window. When their teacher asked who broke the window, she got these replies:
Linda: James broke the window!
James: I didn't break it!
Alex: Neither did I!
If two of the children lied, then who broke the window?
If Linda was truthful, then James lied, and Alex was truthful. Thus, 2 were truthful. This DOESN'T work, since 2 lied.
If Linda lied, then James was truthful, and Alex was either truthful or lied. Since 2 lied, then Linda and Alex may've lied.
Since Linda and Alex may’ve lied, then either one could’ve broken the window.
If Linda broke the window, then James and Alex were truthful. This DOESN’T work, as again, 2 lied!
If Alex broke the window, then Linda lied in saying that James did it, and Alex lied in saying he didn’t. Therefore, Alex broke the window!!
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
James, Alex, and Linda were playing inside of their classroom. One of them broke the window.
When their teacher asked who broke the window, she got these replies:
Linda: James broke the window!
James: I didn't break it!
Alex: Neither did I!
If two of the children lied, then who broke the window?
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I will construct my solution by another way.
(1) Let's assume Linda broke the window.
Then James and Alex are truthful, which contradicts to the condition; hence, this assumption is wrong.
(2) Let's assume that James broke the window.
Then Linda and Alex are truthful, which contradicts to the condition, again; hence, this assumption is wrong.
(3) So, if the problem is built in logically correct way, then the only possible version is that Alex broke the window.
(4) Nevertheless, we should check, if the problem is REALLY built in logically correct way.
So, let's assume that Alex broke the window.
Then Linda and Alex are the two liars and James is truthful, which is consistent with the condition.
The problem is just solved, and the ANSWER is: Alex broke the window.
The post-solution notice.
I'd like to explain why I wrote this solution, just N 4 in the list.
It is an entertainment logic problem.
There is some definite style of writing solutions to such problems.
The solution should not be wordy.
It must be short, clear, straightforward and convincing.
It should inspire the reader; it should entertain; it should not be boring or tedious.
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