The cake is a liar! Part 4

Shorter solution: Emma says it was Dave and it was Alan. Therefore those are the only suspects, as one of her statements must be right.

Therefore the others, including Chris, are out of question.

Ben mentions that Chris did not do it, and Alan did. Since we have already proved that Chris has not done it, Ben must be lying when he says Alan did it.

This leaves only Dave.

We assume who ate the last piece of the cake one by one and check whether each of the two statements of each person is true or false. The truth table of the statements is as follows:

Only if $\boxed{\text{Dave}}$ ate the last piece of cake that each person's statements were one true one false.

In this problem people are grouped into pairs- Emma + Ben, and Chris + Alan. Each pair is the subject of statements from 2 people who contradict each other.

The statements by Alan and Chris about the pair "Emma + Ben" contradict each other, and so do the statements by Ben and Dave about the pair "Chris + Alan".

If it is one of the members of the two pairs: Emma + Ben, or Chris + Alan- somebody of the whole group is not telling the truth at all.

Which leaves only one candidate for the thief- Dave.

For shorter solution we start with Emma she said it was Dave , It was Alan because one of them is a truth so who ate the cake is Alan Or Dave by using the truth table we find that Dave ate the cake

Ben and Dave accused and defended the same two friends but in opposite order, so both of the accused, Chris and Alan, must be innocent (the two must have the same guilty state, but there's only one cake thief taking that last slice). By Emma's statements, it must be either Dave or Alan, and we already know that Alan is innocent. Dave is the thief (even though he defended Alan).

The most simplest way to get answer is

All the wasn't statements are TRUE

How I solved it within ten seconds of reading it; If one statement is a truth and one is a lie, then every time it says "it was x, it wasn't y", then the "it was" part must be the lie, or else there'd be two persons who committed the crime. Hence, for the first for sentences, every person mentioned must be innocent. Only one sentence gives two positives; Dave and Alan. Since Alan is already mentioned in one of the above sentences, he's innocent. Dave is the only one never mentioned before, and must thus be guilty.

If Alan's first statement was true and second statement was false them both Emma and Ben would have done it. Since this cannot be true Alan's first statement was false and his second statement was true, and this means neither of them did it. The same reasoning can be applied to Ben's statements. This leaves Dave left as the only one who could have done it.

Dave was the only one nobody vouched for XD

According to Emma : It was Dave OR Alan.

If it was Alan, let's look at Dave's statement.

"It was Chris, It wasn't Alan."

If it was Alan, he'd basically be saying "it was Chris". Because one sentence is a lie, and the other the truth. And a cake can't be shared, in this case. I would not share my cake with anyone, too. It's mine!

Hence, we know, it wasn't Alan. It was Dave all along!

u r right ujjwala ananth