Broken Vase

Let's run through the outcomes of each child's first and second statements:

Alex broke the vase:

Brian broke the vase:

Charlie broke the vase:

Daniel broke the vase:

From this we can easily see that, only in the case when the vase is broken by Charlie, each child tells one lie and one truth.

$\boxed{\text{Charlie broke the vase}}$

Since one of Brian's statements is true the culprit must be one of either Charlie or Daniel, and since one of Daniel's statements is true the culprit must be one of Charlie or Brian. The intersection is Charlie. Alex and Charlie, whose statements are logically identical, with Charlie as the culprit, each make one true statement, ("It wasn't Brian"), and one false statement, ("It was Daniel"). Thus all their replies are consistent with $\boxed{\text{Charlie}}$ being the one who broke the vase.

You only need Daniel and Brian's statements to solve this one. Start with Daniel's statement: "It was Charlie. It was Brian." One is true, so we know Alex and Daniel are innocent. Now, Brian's statement: "It was not Charlie. It was not Daniel." We already know Daniel is innocent, so the second part of Brian's statement is true. Therefore, "It was not Charlie" is false- so it WAS charlie. You can double check Alex and Charlie's statements. They check out.

This can be solved from Alex and Brian's statements:

If it was Daniel, then both of Alex's statements are true. So it can't be Daniel. Brian says it wasn't Daniel and it wasn't Charlie. We know the first is true so the 2nd must be false. It was Charlie.

A combination of any two statements can be used to solve this problem, except a combination of Alex's and Charlie's statements (because they both confirm the same two innocent children - which leaves one innocent child and one guilty child in question).

My first instinct was to use Daniel's statement as my lynchpin for this problem, because it directly implicates the guilty child. From there, any other statement confirms that Charlie is guilty.

Since we know that everybody says one true and one false statement. If you say a combo as: "it was A", "it wasn't B". We know that "it was A" must be the lie. If "it was A" was the truth it means it "it wasn't B" was the lie so that statement would become "it was A", "it was B" which can't be

Thus we know for a fact from Alex & Charlie: It wasn't neither Brian, nor Daniel

By looking at either Charlie or Daniel we can correct their statement to

Brian: "It was Charlie. It wasn't Daniel." (we know Daniel can't be per above)

Daniel: "It was Charlie. It was Brian." (We know Brian can't be per above)

Brian says it wasn't Charlie and It wasn't Daniel, one of which is a lie. So it is either Charlie or Daniel.

If it was Daniel, both of Alex's statements become true which is impossible. Therefore it has to be Charlie

Simple trick! was wasn't Daniel Brian Brian +1 Charlie Charlie Daniel +1

No plus in charlie so he must be guilty.

Daniel's answer implies that it must've been either Brian or Charlie. Brian's answer implies that it must've been either Charlie or Daniel. The only way for both of these to be true is for Charlie to have done it.

No need for big complicated charts or long explanations. You can get the correct answer from just the last two (Charlie's and Daniel's) statements, no need for the other two. Daniel's statement means that it had to be either Charlie or Brian. If it were Brian, then both of Charlie's statements would be true, so it had to be Charlie.

Just focus on the Daniel's statement.It is clear that it has to be either Charlie or Brian.Now coming on to Charlie's 1st statement,which says "It wasn't Daniel" has to be true because it is either Charlie or Brain.Now Charlie's 2nd statement "It was Brain" has to be false.So the answer is Charlie.

Alex's 1st statement can't be false because if it is false, then his 2nd statement will be true. It means both Brian & Daniel broke the vase which is not possible at all. So, Brian's 1st & 2nd statements are false & true respectively and hence Charlie broke the vase.

First look at Daniel and you will find that it is either Charlie or Brian, which means Alex and Daniel are off the list. We know from Alex that it wasn't Daniel as his false and it wasn't Brian being true. We get Charlie then. It all fits in as well.

I added them all together with + and - and charlie came out with 2 strikes.