First, a response to your last comment: I do apologize that some of the questions are not well phrased. Generally exercises are not moderated at all. However, exam questions are moderated. They are inspected carefully by half a dozen people, and still errors and bad phrasings occur --- and as you probably know, they occur frequently. The reason for this is simply that it is extremely hard to write perfect questions (although we do try). The way that exercises get better is by making slight changes from year to year improving them from feedback each year until all the possible ambiguities are removed. 6.13 was an exam question (clearly with some problems) and I've been trying to turn it into a good exercise for some time -- but I agree it is not there yet. The main issue in this case is that part of the challenge of the question is to figure out that one is supposed to think of this as an antiferromagnet. If the problem stated this directly, then the problem would be identical to the antiferromagnet problem from problem set 5 -- and thus is would not be a very useful question. On the other hand, if one does not say this, then students can run into exactly the type of problems that you have run into. So each year I've tried to tweak it to see what works and what doesn't. Many of the other exercises have converged to something useful (indeed, I think most of them are very good), but this one is still troublesome.
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