Since 2 and 3 say the same thing, neither of them is speaking the truth bcuz only 1 of them speaks the truth Therefore statements of 2 and 3 are false. Now if 1 says the truth then 2 and 3 become true when they actually are telling lies. Thus 4 speaks the truth as only he is left. Therefore 3 stole the wallet.

In this problem as it is given that three of them lied and only one of them say the truth so we will try to violate this condition and find the answer.

Case 1 : If 1 said truth, i.e. 2 stole the wallet now going on to statements of 2 and 3 both becomes true which is not possible.

Case 2 : If 2 said the truth, then again statement of 2 and 3 becomes true, again which is not possible.

Case 3 : If 3 said the truth, then again statement of 2 and 3 becomes true, again which is not possible.

Case 4 : If 4 said the truth, here we found that statements y 1,2 and 3 are false therefore its the right solution.

If 1 tell the truth, then

1 : 2 stole (truth)

2 : 3 didn't steal (lie) then 3 stole [not the same as 1]

3 : 3 didn't steal (lie) then 3 stole [not the same as 1]

4 : 3 stole (lie) then 1 or 2 or 4 stole [same as 1]

2 and 3 statement not the same as 1's statement, then 1 didn't tell the truth

If 2 tell the truth, then

1 : 2 stole (lie) then 1 or 3 or 4 stole [not the same as 2]

2 : 3 didn't steal (truth) then 1 or 2 or 4 stole (truth)

3 : 3 didn't steal (lie) then 3 stole [not the same as 2]

4 : 3 stole (lie) then 1 or 2 or 4 stole [same as 2]

1 and 3 statement not the same as 2's statement, then 2 didn't tell the truth

If 3 tell the truth, then

1 : 2 stole (lie) then 1 or 3 or 4 stole [same as 3]

2 : 3 didn't steal (lie) then 3 stole [not the same as 3]

3 : 3 didn't steal (truth) then 1 or 2 or 4 stole

4 : 3 stole (lie) then 1 or 2 or 4 stole [same as 3]

2's statement not the same as 3's statement, then 3 didn't tell the truth

If 4 tell the truth, then

1 : 2 stole (lie) then 1 or 3 or 4 stole [same as 4]

2 : 3 didn't steal (lie) then 3 stole [same as 4]

3 : 3 didn't steal (lie) then 3 stole [same as 4]

4 : 3 stole (truth)

1, 2, and 3 statement same as 4 statement, then 4 tell the truth

Easiest way (IMO) to figure this out, and subsequently any problem similar, is to find the two points which reference the same variable but have different evaluations. 2 and 4 both refer to 3 and they have differing opinions. Thus one must be correct and one must be false since the two events can't happen simultaneously. Since you know this, then you know that one of the two is telling the truth which also means the other two options (1 and 3) are false. Since 3's statement points to itself you can then use that to prove 3 did it. Any problem like this is predicated on a logic gate which requires two hypothetical outcomes. So look for either a source which makes two statements (a switch) or two sources which refer to the same object.

Solution

Let's assume that 1 , 2 and 3 are lying, and 4 is telling the truth.

Here are the people's statements:

1 - 2 stole the wallet.

2 - 3 didn't steal.

3 - I didn't steal.

4 - 3 stole your wallet.

Assuming that 1 is lying, 2 didn't steal the wallet. So we can cross out 2 as stealing the wallet.

1 (?), 2 (X), 3 (?), 4 (?)

Assuming that 2 is lying, 3 stole the wallet. So we can put a tentative O (for a tick) next to 3, and a definite no for 2 .

1 (?), 2 (X), 3 (O), 4 (?)

Assuming that 3 is lying, 3 actually stole the wallet. So we'll leave the O there, except we're almost sure that he stole it now! We can also cross out 1, because if 4 is telling the truth then 1 didn't steal the wallet.

1 (X), 2 (X), 3 (O), 4 (?)

Finally, assuming that 4 is telling the truth, 3 stole the wallet. We can cross out 4 !

1 (X), 2 (X), 3 (O), 4 (X)

Now we're completely sure that 3 stole the wallet, so thus....

drumroll

3 stole the wallet.

2 & 3 said that 3 isnt the culprit. Yet only 1 can be true. So between these 2, only one of them can be true. So their statement must be a lie because both of them said it isnt 3. So 3 is the culprit

Follow the T Table

Case 1 2 3 4 Result Valid Reason

1 T T {2} No Only One Truth Possible

2 T T {2} No Only One Truth Possible

3 T T {2} No Only One Truth Possible

4 T T {2} No Only One Truth Possible

5 T F {2} ? {3} No Empty Intersection

6 T F {2} ? {3} No Empty Intersection

7 T F {2} ? {3} No Empty Intersection

8 T F {2} ? {3} No Empty Intersection

9 F T T "{1,3,4}" No Only One Truth Possible

10 F T T "{1,3,4}" No Only One Truth Possible

11 F T F "{1,3,4} ? {1,2,4} ? {3}" No Empty Intersection

12 F T F "{1,3,4} ? {1,2,4} ? {3}" No Empty Intersection

13 F F T "{1,3,4} ? {3} ? {1,2,4}" No Empty Intersection

14 F F T "{1,3,4} ? {3} ? {1,2,4}" No Empty Intersection

15 F F F T "{1,3,4} ? {3} ? {3} ? {3}" Yes Only Consistent Solution

16 F F F F "{1,3,4} ? {3} ? {3}" No There has to be one Truth

Either Person 2 or 4 is telling the truth, and all the others are lying. So Person 3 is lying, he stole your wallet!

Statements 2 and 4 are the opposite of each other, so one of them is the only true statement. Evaluate the statements when they are False, True, False, False and when they are False, False, False, True. The FTFF sequence is inconsistent and the FFFT is. So the answer is Person 3 (evaluate the "sequences" yourself).

If person 4 is correct, three is lying because he stole it. 2 is lying because 3 stole it. 1 is lying because because 2 did not steal it. If more than one solution is correct, then it would not work. Thus, since this solution is correct, then no other is correct. 3 stole the wallet.

2 and 3 say the same thing; and since only one person is telling the truth, they are both lying. If Person 4 is telling the truth, then Person 1 is lying. If Person 1 told the truth, then Person 3 is telling the truth, and only one person can be telling the truth at a time. So, Person 4 tells the truth, leaving the other 3 as liars, which fulfills the conditions of the problem ☺☺☺☺

Let's discuss the truths and lies depending on who did it. Remember 1 person tells the truth and the other 3 tell lies.

Person 1: 1 lied, 2 told the truth, 3 told the truth, 4 lied (2 truths, 2 lies)

Person 2: 1 told the truth, 2 told the truth, 3 told the truth, 4 lied (3 truths, 1 lie)

Person 3: 1 lied, 2 lied, 3 lied,4 told the truth (1 truth, 3 lies, which fulfills the brief)

Person 4 (to prove it): 1 lied, 2 told the truth, 3 told the truth, 4 lied (2 truths, 2 lies)

Thus person 3 stole your wallet.

The easy way is that here two people talk about person 3. So two people can't truth as question. Therefore one people must be true and stolen people is person 3.

Person 2 and Person 3 both say "3 did not steal the wallet". Since we know they can't both be telling the truth (we're told there is exactly one truth teller), they must both be lying. This means that in fact, 3 stole the wallet. We need not even consider person 1 and 4!

Person 2 and 4 give you contradictory information. They can not both be false, and neither can they both be true. Thus, either 2 or 4 speaks the truth. Assume 2 speaks the truth, then 3 also speaks the truth, which is not possible. Thus 4 speaks the truth. Therefore, 3 stole your wallet.

2 and 3 say the same thing and therefore it is false, thus 3 stole the wallet.

All comes down to 2&3 are agreeing with each other, and as only one of the four can tell the truth, therefore they both must be lying.

This problem has a simpler solution than most.

Because statement 2 and statement 3 agree with each other (both agree that 3 didn't steal), and there can only be one true statement among the four, then both 2 and 3 must be lies.

Given that the lie told is that 3 didn't steal, then 3 must have stolen.

You actually don't need to examine statements 1 or 4 for the solution.

If 2 stole the wallet: Statements 1, 2, and 3 would be true so therefore it cannot be 1 If 3 stole the wallet: Statements 1, 2, and 3 would be false and 4 would be true. Therefore we can stop here as there are 3 lies, meaning it was 2 who stole it

2 and 3 are saying the same thing therefore both are lying so not"3 didn't steal it" which means 3 stole it.

If Person 3 is truthful, then Person 2 is, too, contradicting the 3 lies rule. So Person 3 must be lying, so he stole the wallet.

You don't need to examine all of them, just one of the statements needs to be looked at to solve the question, the others can be checked for verification

Case 1 :-Person 2 is the only one telling the truth. In that case Person 3 would also be telling the truth, which is impossible according to the question.

So Case 1 fails.

Thus Case 2 must be true.

Case 2 :-Person 2 is lying.

This is true, as Case 1 is not (if one case is false, the other must be true)

If Person 2 is lying, then Person 3 did take the wallet.

You don't need to look at the other statements (unless you want to check if all the statements were consistent)

if 1 is telling the truth 1 and 2's statements would clash

if 2 is telling the truth 2 and 4's statements would clash

if 3 is telling the truth 3 and 4's statements would clash

So 4 is telling the truth, and to make that straight-forward, let's convert all the lies into true statements:

- Person 1 said, "2 didn't steal the wallet."

- Person 2 said, "3 steal it."

- Person 3 said, "I steal it."

- Person 4 said, "3 stole your wallet."

- And none of these will clash.

*Thus we can see that Person 3 stole the wallet * Viola!

Since 2 says 3 didn't steal it, 3 says 3 didn't steal it & 4 says 3 stole it, only 1 of them is true which means 4 says the truth that 3 is the thief.

Even though the logic would be the same, I think the better question would've been to determine who was telling the truth since it would take one more step to solve.

You have to let someone telling the truth and start to connect every statements. For exp, if you let 1 telling the truth, and based on the statement given, we have to refer to the next statement which refers to 2. But based on the given statements, 2 also telling the truth. Hence, this will contradict with the given information in which exactly 3 are lying. So if you try to let 4 telling the truth, the other 3 will tell the lies.

So there is only one guy telling the truth.

If we suppose that Person 4 said the truth, then Person 3 stole the wallet. Then Person 1,2 and 3 lie. This is it.

2,3 both contradicts with 4. Why is 1 even there?

Since there are exactly 3 lies and one truth 3 must have stolen it because if three didn't then statements 2 and 3 would be true.

Second and third statements can not be true at the same time as they say same thing so they are lies.......next if first statement is true then second and third will be true again contradiction....so only possibility is statement 4 is true ....so 3 stole wallet

It states that 3 of them lie. Among 1-4...3 lie. That means number 3 of the four people is indeed the liar

In short, assume that 4 told trust, then 1, 2, and 3 lied. So, 3 did stolen