# Who committed the crime?

A crime has been carried out by one person, there are 5 suspects. Each suspect is asked under polygraph who they think committed the crime.

• Terry : It wasn't Carl, It was Steve
• Steve : It wasn't Matt, It wasn't Carl
• Matt : It was Carl, It wasn't Terry
• Ben : It was Matt, It was Steve
• Carl : It was Ben, It wasn't Terry

The polygraph showed that each suspect told one lie and one truth. Who committed the crime?

• All of the answers here depend on the false assumption that what a polygraph shows has any connection to what is true ... or even to what the subject thinks is true. Aug 5 '16 at 8:27
• @HenningMakholm would it be better instead of polygraph, I used the word "lie detector" ? The intent was a device that could determine if a statement is true or false. Aug 5 '16 at 9:15
• Can we all just stop dissecting riddles and problems? I've seen that a lot on this website and I think it prevents some people from seeing the main purpose of a question. Everybody understood what the question was about. The fact that a polygraph is not 100% accurate is not important in this case. If it was, then it would have the lateral-thinking tag most probably. Aug 5 '16 at 11:14
– user20
Aug 5 '16 at 15:56
• @JacobiJohn Re-phrasing to "lie detector" wouldn't help: if you have used this device that determines if a statement is true or false, you already know which statements were truth and which were lies. It's clear how you want your puzzle to work, and it's probably okay to leave it as it is, but if you want to address the concerns anyway, then you could avoid mentioning the device: "Each suspect is asked who they think committed the crime." and "If each suspect told one lie and one truth, who committed the crime?"
– hvd
Aug 7 '16 at 17:25

Matt.

Reasoning:

"Ben : It was Matt, It was Steve". So it has to be Matt or Steve.
If it would be Steve then Terry would tell the truth twice (It wasn't Carl, It was Steve).
Hence, it was Matt.

And it fits the requirement of one truthful answer and one lie for each.

Terry : It wasn't Carl (T), It was Steve (F)
Steve : It wasn't Matt (F), It wasn't Carl (T)
Matt : It was Carl(F), It wasn't Terry(T)
Ben : It was Matt(T), It was Steve(F)
Carl : It was Ben(F), It wasn't Terry(T)

• Also, Terry's first statement can't be a lie (two culprits), so Steve's first has to be.
– Carl
Aug 5 '16 at 10:41
• @Carl. You're not allowed to comment anymore. You already took the polygraph test. it says so in the question :) Aug 5 '16 at 11:16

It was

Matt

The easiest way to this conclusion is

using only Ben's and Steve's words:
Because exactly one of Ben's sentences is true, it has to be either Matt or Steve.
But also one of Steve's sentences has to be a lie, so it was either Matt or Carl.
Matt is the only one in the intersection of these sets, and as @Marius already showed, the others' sentences support this too.

The criminal was

Matt

Why?

Start with Terry, Matt, and Carl's statements. Each states person A did it and person B didn't do it. If "B didn't do it" was a lie, then this would mean that B did it, however this would contradict the "A did it" part of the statement. Therefore in each of the three statements, the "B didn't do it" is the truth.

From here we can gather that

It was not Steve, Carl, or Ben. It must have been Matt or Terry. Let's move to Ben's statement. Ben said "Steve did it" however we've already proved that Steve could not have done it. So this is a lie. Therefore Matt did it is the true part of his statement

• nice observation about the "A did, B didn't" kind of answers! Aug 5 '16 at 8:30
• You can go even further: "A did not and B did" answers not only lead to "B did not do it", but also "A did not do it". So using just that, you already know not steve, not carl, not terry, not ben... Aug 5 '16 at 13:13

EDIT: this answer references an older version of the question that did not specify the crime was carried out by one person.

A new answer different from the others given so far. The question "who committed the crime" does not require that a singular person committed the crime so another valid answer is:

Terry, Steve, Ben and Carl. Good lord!

Due to the fact that:

• Terry : It wasn't Carl (lie), It was Steve (true)
• Steve : It wasn't Matt (true), It wasn't Carl (lie)
• Matt : It was Carl (true), It wasn't Terry (lie)
• Ben : It was Matt (lie), It was Steve (true)
• Carl : It was Ben (true), It wasn't Terry (lie)

It's interesting that the above is the negation of the popular solution. For fun for the code-heads out there, here's a snippet that exhaustively searches the space:

https://repl.it/CjuJ

• This might have been edited since you answered, but... "A crime has been carried out by one person" seems to disqualify your answer. Aug 5 '16 at 14:44
• @CharlesWatson dang it! That's what I get for not hitting refresh. Thanks for the heads-up. Aug 5 '16 at 14:49

You can resolve this by logical calculus only. Using standard notation, $\land$ is the "logical and", $\lor$ is the" logical or", and $\lnot$ is negation. Let's use the variables $t,s,m,b,c$ to represent different people being the culprit. The first line can be written : $(\lnot c \land \lnot s )\lor( c \land s)$ (one of the two statements is true and the other is false).

So if we apply this to all this people:

$\hphantom{\lor}((\lnot c \land \lnot s) \lor( c \land s))\\ \land((\lnot m \land c) \lor (m \land \lnot c))\\ \land((c\land t )\lor( \lnot c \land \lnot t))\\ \land((m \land \lnot s )\lor( \lnot m \land s))\\ \land((b \land t )\lor( \lnot b \land \lnot t))$

Since there's only one culprit, things like $c \land s$ are always false, so we can remove them: so we obtain :

$(\lnot c \land \lnot s \land \lnot b \land \lnot t) \land (\lnot m \land c \lor m \land \lnot c)\land((m \land \lnot s )\lor( \lnot m \land s))$

If you distribute $\land$ over $\lor$ you get:

$\hphantom{\lor}(\lnot c \land \lnot s \land \lnot b \land \lnot t \land \lnot m \land c \land m \land\lnot s)\\ \lor( \lnot c \land \lnot s \land \lnot b \land \lnot t \land\lnot m \land c \land \lnot m \land s)\\ \lor( \lnot c \land \lnot s \land \lnot b \land \lnot t \land m \land\lnot c \land m \land \lnot s)\\ \lor( \lnot c \land \lnot s \land \lnot b \land \lnot t \land m \land \lnot c \land \lnot m \land s)$

Things like $c \land \lnot c$ are false (nothing is itself and his contrary) so we obtain :

$\lnot c \land \lnot s \land \lnot b \land \lnot t \land m$

so :

$m$

So the culprit is:

Matt

I didn't found any notation better than this, sorry if it's not very readable. Also, sorry if my English is not very good.

It's easier to do it yourself on paper.

Add this answer beacause there is no calculus only answer to this and I find it so powerfull so you don't have to read 5 times the thing to find the answer, just 3 lines of "Maths".

• Welcome to Puzzling! I've changed your notation to mathematical rather than programming notation (using $\land$, $\lor$, and $\lnot$ instead of &&, ||, and !), and cleaned up some of your equations for you.
– Deusovi
Aug 5 '16 at 18:46
• Thanks a lot. I did this during a pause at work so I didn't take the time to look for mathematical notations. Aug 6 '16 at 10:39

Late answer, but I like to play.

It was Matt.

Evaluating the statements:

Ben: It was Matt, it was Steve

One of these have to true, so any "Not x" statement about a person other than these two is true, meaning the other statement from the speaker is false.

Steve: It wasn't Matt, it wasn't Carl

Since we know it wasn't Carl from the reasoning in the paragraph above, "It wasn't Matt" is a false statement. Therefore, the culprit was Matt.

With this, we only need two of the accounts to draw our conclusions. (We can also do the above in reverse order; "Not Matt, not Carl" evaluates to Matt or Carl being the killer, which is only compatible with "Matt, Steve" if Matt is the killer.)

Let's have a bit of fun!

I figured it was a bit amusing to evaluate the other statements as well.

Carl: It was Ben, it wasn't Terry

If the first statement here is true, then both Ben and Terry did it! But we know only one person committed the crime from Jacobi John's description, so this can't be right. This means the statement about Ben is false, so Ben didn't do it, while the statement about Terry is true, meaning Terry didn't do it.

This reasoning holds for any statement with Killer || Not killer. If "Killer" is the true statement, "Not killer" is the false one, meaning both are guilty, which isn't possible. If we want to, we can combine the three statements using this form to conclude Matt's guilt!

Not C || S

C || Not T

B || Not T

This gives us Not C, Not S, Not C, Not T, Not B, Not T. Or, to put it in a more readable way, None of S, C, T, or B. A "not" for everyone but Matt.

Matt

Resasoning:

Every suspect tells one truth and one lie, therefore from Ben's testimony, it has to be either Matt or Steve. Steve cannot be the culprit because if he was, then Terry would be telling two truths, which goes against the initial premise that every suspect tells one truth and one lie.

• isn't this what I just wrote 3h ago? Aug 5 '16 at 11:36
– user20
Aug 5 '16 at 20:09

Matt

Reasoning:

Terry : It wasn't Carl, It was Steve (Terry is lying on Steve)

Matt : It was Carl, It wasn't Terry (Mattis lying on Carl)

Steve : It wasn't Matt, It wasn't Carl Since Steve is saying one lie, it must be "Matt or Carl"

It is already proved it was not Carl.

So, it was Matt