1. Aug 26, 2009

### sweedeljoseph

1. The problem statement, all variables and given/known data
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?

2. Relevant equations
Logic.

3. The attempt at a solution

2. Aug 26, 2009

### tiny-tim

Hi sweedeljoseph!

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

3. Aug 26, 2009

### sweedeljoseph

i think alice is telling the truth which makes barbara and betsy false
if betsy is telling the truth then martha did it
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

4. Aug 26, 2009

### Staff: Mentor

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.

5. Aug 26, 2009

### 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?

6. Aug 26, 2009

### HallsofIvy

Staff Emeritus
Yes.

7. Aug 26, 2009

### sweedeljoseph

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

8. Aug 26, 2009

### HallsofIvy

Staff Emeritus
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.

9. Aug 26, 2009

### JG89

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.

10. Aug 27, 2009

### 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.

11. Aug 27, 2009

### 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 …

12. Aug 27, 2009

### Borg

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