Who took the cake?

We just need to observe that all four siblings were being talked about except for Darius. The reason why we can do this is because every sibling mentioned someone they thought was innocent and another person they thought was guilty. In other words, suppose one of the people deemed guilty was really guilty, then the statement that this person was guilty is true. At the same time, the other person deemed innocent is really innocent since there is only one culprit. This means at least one person made two true statements which contradicts with what we know that each told a lie.

If you still do not understand, take for example Alfred is guilty. Then looking at Brenda's statements, both would be true as if Alfred was the culprit, Brenda must be innocent as there was only one culprit. You can do the same for Brenda and Charles and you will run into the same issue. In sum, since Darius is the only one not mentioned, he is the only possible culprit.

Everyone made defensive statements for 3 people first, either in self-defence or their sibling's, and their second statements are all accusations towards another siblings, also on the same 3 people. As each of the accusations has a contradictory defensive statements made to counteract them, then we can either have all the accusations to be true for at least a total of 3 guilty siblings, or have all the defensive statements to be true for a total of 3 innocent siblings. Since we're told that the parents only have ONE suspect and by the given options, it must be the latter with Darius implied guiltiness (being the only siblings that didn't get accused).

All you have to do is look at the statements made by Charles and Darius. Both of them say it was Brenda, this is either their true statement or their false one. If it is their true one, then their other one is false, but their negations of the other statements put the blame on himself and Alfred, respectively, which is a contradiction. So the statement about Brenda must be false. Also, their other statements are true, which means that it wasn't Charles or Alfred. Also, since the statement "It was Brenda" was false, it isn't Brenda either. The only person left to eat the cake was Darius .