The Robbery
There are 4 suspects in a robbery: A, B, C , and D. Each makes a statement, but only one speaks the truth:
A. B did it.
B. C did it.
C. I did it.
D. Either A or C is the guilty one.
Who is the robber?
A. B did it.
B. C did it.
C. I did it.
D. Either A or C is the guilty one.
Who is the robber?
Labels: logic, SharedPuzzle
posted by Zaux at
5:45 PM
48 Comments:
c says "i did it", so obviously it was him!
Only A is saying the truth, therefore B did it.
B is the robber
MK
I am pretty sure Robber B commited the foul deed.
(Although, technically, I guess these guys are all guilty because they are all defined as "robbers" in the problem.)
By running through each statement made by a robber and assume it is true creates a dilemma between statements made by Robbers A and D.
If Robber's A statement is true, then the other three statements are false. Meets thr criteria.
If Robber B speaks the truth, then the statement from Robber C is also true -- fails the criteria.
Likewise, if Robber C speaks the truth, the statement by Robber B is also true -- agains fails the criteria.
Which leaves us with the most difficult statement made by Robber D. This statement can be true if Robber A committed the deed making all the other statements false but this creates a conflict with part of the criteria -- "only one speaks the truth".
Since this is an "either/or" statement, and we have another possible fit, we have to assume it is false leaving us with the statement made by Robber A being true -- "B did it".
Richard
Hi Richard ... they are called suspects.
The Robbery
Given:
-4 suspects
-1 truthful statement
Call statements:
*1) B did it
*2) C did it
*3)I did it
*4) Either A OR C is the guilty one
if A tells truth, *1=true, *2,*3,*4 = false
(*1)=B guilty
!(*2)=C innocent
!(*3)=C innocent
!(*4)=A+C are innocent
All statements compatible. B is guilty.
if B tells truth, *2=true, *1,*3,*4 = false
!(*1)=B innocent
(*2)=C guilty
!(*3)=C innocent
!(*4)=A+C are innocent
(*2) and !(*3) are incompatible thus B cannot be telling truth
if C tells truth, *3=true, *1,*2,*4 = false
!(*1)=B innocent
!(*2)=C innocent
(*3)=C guilty
!(*4)=A+C are innocent
!(*2) and (*3) are incompatible thus C cannot be telling truth
if D tells truth, *4=true, *1,*2,*3 = false
!(*1)=B innocent
!(*2)=C innocent
!(*3)=C innocent
(*4)=A OR C are guilty
All statements compatible, if and only if C is innocent and A is guilty. Thus A is guilty.
Either A or D could be the one telling the truth.
If A tells the truth then B is guilty,
If D tells the truth A is guilty.
Cam
I came to the same conclusion as Cam. But I assume that there is a unique answer.
So I think D is a liar because because if A was the robber, he shouldn't be mentioning C and vice versa. At least, D would not be telling the whole truth and nothing but the truth. So that takes us back to only A is telling the truth and so B is the robber.
There again, what is truth? It's not the same thing as fact. e.g. any one of them could be making a correct statement whilst believing themselves to be lying. This subtlety couldn't apply to C as it probable that he would actually know if he was the robber. In which case C is the truthful robber and A, B and D are not telling the truth - they may not have the faintest idea who the robber actually is. B (and D) just got lucky when naming C as the robber. Unfortunately, a flaw there is that D may actually be a logician, in which case he meant logical or, and so (believe himself to be) telling a logical truth. So I'll go for C as he's probably hoping for leniency by being helpful.
Enough of my silliness.
Everyone wants to toss away D's statement....
However the statement by D can't be tossed away as false.
If one statement in an OR expression is true, regardless of the other statement it will be true:
(2=2 OR Purple unicorns are growing out of my nostrils) = TRUE
(2=2 OR Paris Hilton is really smart) =TRUE
(2=2 OR 5=6) = TRUE
(TRUE OR FALSE) = TRUE
So if A is guilty:
A OR C is guilty is true
Cam
P.S. purple unicorns would hurt
A is the nasty culprit ...
Why?
The only way that I can see it is rather twisted. When D uses the word guilty, he simply means guilty of something (not necessarily the robbing). In particular his whole statement is logically true because he has told a truth about C (being guilty of something other than the robbery). C can't be the robber though, otherwise there would be more than one truth teller). But then, we could argue that D's reference to A need not have to be the truth (only the whole statement has to be the truth). That leaves us with the possibility that A or D could be the robber.
But if we deem that D is lying, and he was referring to being guilty of the robbery, then he could be lying, so we're back to A tells the truth and B is the robber.
The only thing I'm reasonably sure of is that C isn't the robber and he isn't telling the truth, so B is also a liar.
the answer is....
letter M
letter E
SIMPLE b,coz
its ME
im the robber
naaaaaaa!
its B,
A. Is the correct answer since only one is correct and B,C,and D have all commented somehow that C. did it. Therefore B is guilty. Since all three can't be right!
One more time ... A did it.
Zaux, you haven't explained the logic. You can't just insist that A did it.
Patricia. I'm not sure if you're asking a question or providing a solution.
Because B and C can't both be telling the truth, C cannot be the robber. But that doesn't make D be a liar because, if A is the robber, then D would be telling the truth about A but not C, but altogether he's telling the truth. See Cam's last response for the reason. So A could be the robber and D telling the truth. A, B and C would be the three liars. But I see no reason why B can't be the robber, A the truth teller and B, C and D be the liars.
So as far as I can determine, A or B could be the robber.
I just hope Zaux explains why it's A soon.
Chris-i agree
in a real life case,knowing that only 1 person told the truth,a cop would not know if it was A or B.
Solution ...
For a moment, consider that the statement by D is false. If it is, in fact, false, then A and C are not the robber. By assuming D's statement to be false, we still cannot solve the mystery. Knowing that the solution should be apparent from the given information, D must be telling the truth. From that, we conclude the other three comments to be false. C is not the robber ... therefore it is A.
Hi Zaux
your facts might work out on paper, but the fact is that if a cop hear this statements,and he knew only 1 to be truth,he wouldn't know if it was A or B.
thats not in riddle mode
thats in real life mode
We know that the problem should have a solution based on the given information.
If we consider the validity of each suspect's statement, D is the only statement which offers additional information. By considering D's statement as false, we still can not determine the robber. Yet, by considering it true, we know the others are lying. Now we know it's A or C. C is lying, and therefore not guilty... our robber is A.
Additional info:
this is just a fun logic problem ... not reality
Why is this so hard everyone???? Its all about C and him saying hes the one. the answer is B he is the robber.
explanation: C cant be telling the truth because then B would be too and only one is telling the truth.
B cant be telling the truth because then C would be too so hes lying. and D cant be telling the truth because he said C did it and then he would be telling the truth too. so by process of elimination A is telling the truth so B is the robber.
So why can't it be A that is the robber?
If A is the robber, the only one telling the truth is D.
Saying either A or C did it would be the truth.
This post has been removed by the author.
Zaux. Your explanation is unsatisfactory.
Reminder: we know that C can't be the robber and that B and C are necessarily lying.
If A is telling the truth then B is the robber - provided that D's statement being false is consistent with that. If D is using "guilty" to mean guilty of being the robber, then as neither A or C is the robber, D's statement is false, as required. Your solution asserts that this isn't the case, that's wrong, so your conclusion can't rest on that.
So that seems to leave us with no unique solution. Taking it for granted that the problem isn't flawed and there is a uniue solution, we have no choice other than to assume that D is not making a statement about who the robber is. He might for instance be making a statement about who's lying. If so, then his statement is true because C is definitely lying. So A's statement is false and so B can't be the robber. That causes no difficulty with any of the statements if either A or D is the robber. Again, not a unique solution.
So D has to have meant guilty of something else (D's certainly guilty of wasting police time). Again A or D could be the robber.
Altogether, I can see no satisfactory solution to the problem.
Anonymous 2:15 AM, D said that "either A or C is the guilty one" - he did not say that C is the robber. Most of the posting are because of that.
Really posting this to kick my prevous post through.
Hi Chris ... the solution came from the problem source ... I am leaving for most of the day (to rework the circuitry in a Fender strat guitar, my first passion). when I return, I will take look at it. You are probably right!
Hi Zaux. Don't worry about it. I know it's not your fault that the source is probably at fault.
I'm only surprised that you didn't notice the difficulty sooner.
I'm sure we'll all survive the experience. I was just hoping there was a good solution, I no longer believe there is one.
The robber has returned the stolen goods ... all charges were dropped.
oh yea sorry guys i thought it was easy last night at 2.... not in the right state of mind i guess. Maybe it comes down to the way D worded his answer... is he really speaking "Truth" if he says either of these two did it. Other than that its impossible.... like that hear a tree logic BS. Its all about how you define sound. In this case its whether of not D's answer counts as truth.
D did it
If D did it, who was telling the truth?
Im sure B did it...or atleast I'm sure that is what the intended answer is...
Why do you think B is the robber?
The problem was intended to make A be the robber and D be the truth teller.
Unfortunately, the wording of the problem doesn't permit a unique solution. So B being the robber and A being the truth teller also works. This has been covered extensively in the posts.
Its B, and A is telling the truth...
Nobody except A is pointing B, the others(C n D) are protecting B.
.
.
.
.
B is the robber Cuff him....
This post has been removed by the author.
This post has been removed by the author.
And apologies to Zaux for having accused him of posting a bad problem. In fact it seems to be a cracker. Thanks for making all my algebra of statements training pay off.
I just hope I haven't goofed.
As it's nearly 7 am I'm going to turn in. But if any of you feel brave enough to wade through my hieroglyphics the following may help: "." is the logical AND operator. "~" is the logical inversion or NOT operator. "+" is the logical OR operator.
I've used "obvious simplificatons" to mean e.g.
X.X = X and ~X.X = 0.
All the statements have the truth value of 1 (except for three of the original statements made by the suspects). So, e.g. A.~B means A is true AND B is false.
I'm replacing my earlier post with this one as it clarifies and amplifies the explanation and may make the solution accessible to a wider range of readers (I hope it doesn't alienate them further).
Let A, B, C, D denote the guilt logic value of A, B, C and D. e.g. A being true means that A is the robber.
The given statements are
Sa: B
Sb: C
Sc: C
Sd: A.~C + ~A.C
Sd => only one of A or C is the robber, but not both. If D had meant A or C or both, then he would have been telling a porky as suspects A and C cannot both be the robber. Put that another way, A+C => A could be the robber, C could be the robber, or both A and C could be robbers - the last alternative is not true, so his whole statement would not be true.
All the following formal statements are constructed to be true. i.e. their logic value is TRUE.
Only one is telling the truth =>
Sa.~Sb.~Sc.~Sd + ~Sa.Sb.~Sc.~Sd + ~Sa.~Sb.Sc.~Sd + ~Sa.~Sb.~Sc.Sd
Read as Sa is true AND statement Sb is false etc. If a statement is true, then the assertion made in that statement must be true also. If the statement is false, then the assertion must be false also. i.e the statements and their assertions imply each other and are therefore equivalent. Boy, am I milking this ;)
Now replace the statements with their assertions.
B.~C.~C.~(A.~C + ~A.C)+ ~B.C.~C.~(A.~C+ ~A.C)
+ ~B.~C.C.~(A.~C + ~A.C) + ~B.~C.~C.(A.~C + ~A.C)
Make obvious simplifications
B.~C.~(A.~C + ~A.C) + ~B.~C.(A.~C + ~A.C)
De-Morgan => ~(a+b) -> ~a.~b, so
B.~C.~(A.~C).~(~A.C) + ~B.~C.A.~C + ~B.~C.~A.C
=> B.~C.~A.C.A.~C + ~B.~C.A
=> ~B.~C.A
So A did it and it follows (but I won't prove it) that D told the truth.
I MUCKED IT UP
I'M SHOUTING THIS IN THE HOPE THAT YOU'LL NOTICE IT
I made a small error (of no consequence) the De-Morgan's law should have read ~(a+b) = ~a.~b
Unfortunately, I found a critical error.
I have assumed that e.g. ~(~A.C) = A.~C.
That is quite wrong. ~(~A.C) = A + ~C. I'll spare you the excuses.
In consequence we end up where we started. It's still either A or B wot dun it and the problem is flawed after all.
I'll leave the last post up as a testimony to my stupidity.
Heh, ho, it was fun doing it all the same :-(
Despite my deep disappointment at not having shown that A was the robber, I am intrigued by the distinction between the two interpretations of D's statement.
In the ordinary sense, we'd say that as long as a least one of A or C was the robber, that D had told the truth. But (I believe) that a logician would take a different view. It's a matter of interpretation. I believe that for a statement to be true, it must produce the right result all the time. So both versions cannot be (independently) true in that sense.
I wonder if there is a deep philosophical division out there in the logician's world, or if I simply haven't properly understood this stuff.
I know that there are at least two divisions in the world of statistics regarding the meaning of confidence. One lot say it is a probability, the other lot say it isn't. I think I'll leave the to battle it out amongst themselves.
Enough of my witterings.
This is really bugging me. So I'm going to resort to what may actually turn out to be sophistry. The statement of the problem says that it is not possible for both A and C to be the robber(s). By interpreting D's either/or as the usual logical or, D has asserted that it is possible that both A and C are the robber(s). Therefore D's statement is a false logical statement. Moreover, that directly means that neither A or C can be the robber (OK I might be pushing that a bit far, but I'm not depending on it to conclude).
It should now be clear that B is the robber and A is the truth teller.
And that's my last word about it (unless...^^). I'll probably be another twifty years before I've got the hang of it.
After a few days break from this conumdrum, I took a fresh look.
I refuse to spend additional time typing an elaborate analysis because I, too, now believe there is no unique solution.
It seems to me, if A tells the truth, B or D could be guilty. (no solution)
If B is the truth teller, there is a conflict in the statements (no solution).
If C tells the truth, there is also a conflict (no solution).
It D is the truth teller, our culprit could be A or B.
Therefore, I do agree there is no unique solution to this problem. I would love think I could depend on a problem posted by a respected author in a published book to be accurate. For me to question and solve every problem before posting is tedious and time consuming. I refuse to do that. Therefore, I will continue to rely upon the pre-determined logic of the porblems I post as being accurate. In the event, this assumption is wrong, then all the more fun. We should take a certain joy in finding errors in the logic and boasting our accomplishment. Hooray! I am now done with this one.
Done except for my stupid typos.
Last paragraph, 2nd line, of the preceding post should say "love to think".
"porblems" is obviously "problems"
Zaux. Your first conclusion doesn't make sense. If A is telling the truth then B is the robber, no conflicts or otherwise. I agree with the rest of your conclusions (except that I prefer my last answer).
I only made a noise because I really thought that I'd proven that A was the robber (until I found my error).
ooops, I also disagree with your last conclusion. If D is telling the truth, then A is the robber, not B.
My last argument (Jan 12, 2:13 PM) is rubbish. A or B could have done it.
OMG
I did it !
Karys
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