Who stole the cookies from the cookie jar?

[sweedeljoseph]

Homework Statement

One of four girls has emptied a cookie jar. Their statements are as follow: ALICE: "Betsy did it." BETSY: "Martha did it." BARBARA: "I didn't do it." MARTHA: "Betsy lied when she said I did it." If only one statement is true, which of the girls took the cookies?

Homework Equations

Logic.

The Attempt at a Solution

Anyone know the answer?

 

[tiny-tim]

Hi sweedeljoseph!

Show us what you've tried, and where you're stuck, and then we'll know how to help!

 

[sweedeljoseph]

i think alice is telling the truth which makes barbara and betsy false
if betsy is telling the truth then martha did it
but that contradicts betsy
if barbara is telling the truth then martha is lying but since only 1 statement is true betsy is lying which means martha didnt do it and that contradicts

 

[Mark44]

Look at all four cases--one each for one statement being true and three being false.
For case 1 (assuming Alice's statement is the true one), you have
Alice: (T) Betsy did it.
Betsy: Martha didn't do it.
Barbara: I did it.
Martha: Betsy told the truth when she said she did it.

Notice that I left Alice's statement as-is and negated all the others. Do the same thing for each of the other three cases. There should be one case with no logical inconsistencies.

 

[sweedeljoseph]

If Betsy did it the other 3 would be true.
If Barbara did all would be false
If Martha did it two statements would be true two statements would be false.
So Alice is the only one left

but then who said the true statement? Barbara?

 

[HallsofIvy]

If Betsy did it the other 3 would be true.
If Barbara did all would be false
If Martha did it two statements would be true two statements would be false.
So Alice is the only one left

but then who said the true statement? Barbara?

Yes.

 

[sweedeljoseph]

so the answer is that Alice did it and Barbara said the truth?

 

[HallsofIvy]

Yes! I did this by choosing each in turn to be the "truth teller" rather than the "cookie stealer" and came to the same conclusion- that Barbara is telling the truth and Alice stole the cookies.

 

[JG89]

Yes! I did this by choosing each in turn to be the "truth teller" rather than the "cookie stealer" and came to the same conclusion- that Barbara is telling the truth and Alice stole the cookies.

But if Barbara is telling the truth and Alice stole the cookies, then the statement from Betsy: "Martha did it." would be false. But then Betsy lied, so the statement by Martha: "Betsy lied when she said I did it" is true.

 

[Kaimyn]

I get Martha...

Martha's statement, "Betsy lied when she said I did it" Is basically saying it wasn't me. The leaves us with:

Alice - Betsy did it
Betsy - Martha did it
Barbara - It wasn't me
Martha - It wasn't me

Now it becomes obvious that either Barbara or Martha are telling the truth, as otherwise we get two "It was me" statements. Should Barbara be telling the truth we get:

Alice - Betsy didn't do it
Betsy - Martha didn't do it
Barabara (T) - It wasn't me
Martha - It was me

As Betsy said that Martha didn't do it, we arive with a contradiction. Now, for Martha:

Alice - Betsy didn't do it
Betsy - Martha didn't do it
Barbara - It was me
Martha (T) - It wasn't me

Which has no contradictions. As it is the only one, it must be correct.

 

[tiny-tim]

Write it out properly …

oooooh!

is everyone doing maths or politics?

Write it out properly …

the four statements (abbreviating barbara to "R") are:

B M notR and notM

But one of M and notM must be true!

carry on from there … ​

 

[Borg]

Look at Betsy and Martha's statements. Can they both be wrong?