I've heard from a Japanese native speaker friend of mine that both 間違う and 間違える are correct usage, but he wasn't able to explain the difference in nuance between them. Is there a difference, and if so, what is it?
When we say that someone makes a mistake about something, we can use both 間違う and 間違える and there is no difference in meaning or nuance. For example, 計算を間違えた and 計算を間違った mean the same thing.
[Added remark: As repecmps pointed out in a comment, some people consider this usage of 間違う as incorrect. Someone states that this usage of 間違う was originally incorrect, but I do not know if this statement is true or not.]
When we say that something is in a wrong state, we can only use 間違う. Examples from Daijisen with my English translations:
○ この手紙は住所が間違っている。 (このてがみはじゅうしょがまちがっている。) This letter has an incorrect address on it.
○ 間違った考え方 (まちがったかんがえかた) a wrong way of thinking
The answer is a simple point of grammar:
間違う is an intransitive verb. With no direct object, the particle を cannot be used.
間違える is a transitive verb, with a direct object (use of を possible as well as other particles)
According to the chat discussion with Tsuyoshi I understand that recent dictionaries allow the use of 間違う with a direct object, making all his examples correct.
More details in this link.
But as a grammar enthusiast, I think that using the rule above, you will always be correct.