Who Ate This Cake?

Indeed, this is harder as only the total number of true statements are known. We do not know for sure the individual breakdown of truths and lies. We have to do trial and error one by one:

• Alfred was guilty: 5 true statements in all.

• Brenda was guilty: 4 true statements in all.

• Charles was guilty: 3 true statements in all.

• Darius was guilty: 6 true statements in all. We have our answer.

We can see that there were 3 statements about Darius. Let's use Capital Letters for accusations and lowercase letters for denials / defences. So we have DD vs d (2-1 scoring). There were 2 statements about Alfred, and now it's become a 3-2 scoring, because one was an accusation against Alfred and the other defended him (not sure whether it's { DDA vs da } or { Dda vs dA } , but the numbers are fixed). Now, there were 2 about Charles, but both were in defense of him, so both must occupy the same side. Thus, cc must join d with 2 available slots (we were told that the true-false scoring must be 4-4), but as c and d can't both be false or we'd have more than one cake thief, then c and d must have been the two truths, of Charles' and Darius' innocence. Since the truths side of the statements are already full house, the the last statement about Brenda can only be false. If "It wasn't Brenda" is false, then "It REALLY WAS Brenda" must have been the final truth.

Let's use only first letters A, B ,C and D. We need to explore 4 cases: First - If A ate the cake, then it would be true when: 1-A said that C didn't, 2-B said that A did and 3-D didn't, 4-C said that B didn't and 5-D said that C didn't eat the cake - total 5 true statements. Second - If B ate the cake, then true statements are: 1-what A said about C, 2-what B said about D, 3-what D said about A and 4-what D said about C - total 4 true statements. At this point we can stop and give the answer that Brenda ate the cake, because we found the first and only case with 4 true statements. But it is a very good practice to explore the Third and Fourth cases using the same method and make sure they do not by any chance give 4 true statements. Indeed Third case - C ate the cake leads to total of 3 true statements (what B said about D, C - about B and D - about A) and Fourth case - D ate the cake - to 6 true statements (what A said about C and D, C - about B and D and D - about A and C). End.