Broken Window

If B were the culprit then he's making a true statement, but since the culprit must always lie we conclude that B is not in fact the culprit. This in turn implies that D is telling the truth, and thus is not the culprit either. This means that either A or C must be the culprit, so at least one of them must be lying. Now if A is telling the truth then C is telling the truth as well, but since at least one of them must be lying it must be the case that A is lying, (with C lying too, for the record), implying that $\boxed{A}$ broke the window, i.e., is the culprit.

Consider the statements told by A , B , C , D organizing them from the kid about each they speak in the statements.

Then B and D speak about B (1) and A and C speak about A (2) anyway.

From B's statement because it is known that the culprit always lies and B accuses himself of breaking it it can be concluded that B is not the culprit since if it would be then he will have told a truth resulting in a contradiction. Because B is not the culprit , the other statement about B can be verified , leading to the conclusion that D speaks the truth when he says B is not guilty and therefore that D is not the culprit either.

Now , remains to check the statements about A and , since it is known that one of the kids broke the window verify which from the remaining suspects , A and C , is actually the culprit. Since A and C tell the same thing their statements are either both true or both false but if they would be both true then both A and C tell the truth meaning that among the suspects which remained to be verified there is no one who lies which would contradict that the culprit always lies meaning therefore that their statements (that A is not guilty) is false and A is guilty anyway.

Let us consider cases Case 1: B is the culprit. Then he must surely lie but he is telling that he is culprit which is the truth , so we conclude B is innocemt. Case 2: C is the culprit. Then he must surely lie but he says A is innocent that in our case is true so C is innocent. Case3: D is the culprit. This implies his statement should be false but he says that B is innocent which is true . So he is innocent. Case4:A is the culprit . He is only left .So he has to be the culprit....:)