# Propositional Logic “Riddle/Puzzle”

I have this kind of 'riddle' as a question that i need to complete, however I'm not sure what to do of it.

This is the question:

Determine who out of the following is guilty of doping. The suspects are: Sam, Michael, Bill, Richard, Matt.

1) Sam said: Michael or Bill took drugs, but not both.

2) Michael said: Richard or Sam took drugs, but not both.

3) Bill said: Matt or Michael took drugs, but not both.

4) Richard said: Bill or Matt took drugs, but not both.

5) Matt said: Bill or Richard took drugs, but not both.

^ Of these 5 statements, 4 are true, one is false.

6) Tom said: If Richard took drugs, then Bill took drugs.

^ This statement is guaranteed to be true.

From this information, I deduced:

p : Michael took drugs

q : Bill took drugs

r : Richard took drugs

s : Sam took drugs

t : Matt took drugs

So given this I came up with this:

1) (p ^ ~q) v (~p ^ q)

2) (r ^ ~s) v (~r ^ s)

3) (t ^ ~p) v (~t ^ p)

4) (q ^ ~t) v (~q ^ t)

5) (q ^ ~r) v (~q ^ r)

6) (~r v q)

However, I'm not sure where to go from here. I suppose, I could connect each statement with an ^ as that seems like the next step to do. Then that entire equation would essentially tell me who was guilty? The next step, would obviously be to simplify, and come up with a name. However, I'm not sure how to do this.

Could anyone please shed some light and give me some tips on how to do this?

Thanks.

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is one of them lying? statements 1,3 and 4 can't all be true? –  Tim May 19 '13 at 11:07
@Tim, sorry, I missed some vital information in my post, I edited it, to show that one person is lying, and one person is guaranteed to be telling the truth. (Tom is telling the truth), of the other 5 statements, one is a lie. –  Anteara May 19 '13 at 11:09
From Tim's observation, you know that the false statement is 1, 3 or 4. This means that the second and fifth must be true as well as the sixth. That looks like progress to me. –  Mark Bennet May 19 '13 at 11:20

There are two things you need to notice.

First, statements one three and four are mutually contradictory. So the false statement must be one of those three.

So both statements 5 and 6 are true. If Richard took drugs then Bill both did and didn't take drugs. Therefore Richard did not and Bill did.

So by statement 2 (which we know to be true) Sam also took drugs.

There's no way of solving it from here, we can declare any one of statements 1,3 and 4 to be false and work out the rest.

The only thing you can say for sure is that at least one of Matt or Michael took drugs.

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How do you know that 1, 3 and 4 contradict eachother? –  Anteara May 19 '13 at 12:25
Statements 1,3 and 4 together claim that every pair of statements $p$, $q$ and $t$ has one true and one false, that can't hold. formally you can show $p\Leftrightarrow p^\complement$ quite easily. –  Tim May 19 '13 at 13:11