Propositional Logic "Riddle/Puzzle"

Question

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.

Answer 1

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.

Answer 2

Based on what @Tim said I have framed the following answer:

Out of statements from $1$ to $5$, exactly one is false. While statement 6 is true. On perusal of statements $1$ to $5$, we see that statements, $1, 3$ and $4$ are mutually contradictory i.e. one of these three statements is false. This implies statements $2, 5$ and $6$ are true.

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

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

6: If Richard took drugs then Bill also took drugs.

We deduce that Sam and Bill took drugs, because if Richard is to take drugs then as per statement $6$, Bill also took drugs however that will violate statement $5$, hence Richard didn’t take drugs. Thus going by statement $2$, Sam has taken drugs for sure.

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

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

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

As Bill took drugs, then going by statements $1$ and $4$ Matt and Michael didn’t take drugs.

Thus, statement $3$ is false, in the sense that neither of the two took drugs (this is where my logic is contradicting that of Tim's). Therefore, Sam and Bill are the ones who took drugs.