[Taylor]

The actual situation adds a level of complexity to the situation, and honestly to give a fair assessment, more information is needed.

You say the average on the test was below 50%. I assume this course is a first- or second-year class, most likely a "weed-out" course. It is not uncommon for tests to be made such that the average would be a 50% - that way the professor gets a nice bell curve out of the results. I'm not saying I agree with this strategy, it's just the way it is. This strategy only works if the class is graded on a curve - is it? If it is graded on a curve, then I'd say help in any fashion beyond the textbook is more harmful. This would be in essence 'wrecking the curve' and would result in an artificially inflated average which further skews the data.
Example of harmfulness: Clair scores a 90% on the test; Cliff, Theo, Vanessa, Denise, and Rudy all score 85%. Secure in the knowledge that she has an A, Clair decides not to do the take-home test, choosing to focus on another class in which she's struggling. Cliff and his friends, however, choose to work together and complete the take-home perfectly, earning themselves 5 percentage points (a third of what they'd missed). This makes them all at 90%, artificially raising the curve and potentially bringing Clair's grade down to a B.

If, on the other hand, the professor administered a test which was more difficult than he intended, the take-home quiz would be bonus points and would benefit those who take the class seriously and are willing to do some extra work, without further penalizing those who choose not to do so (they get the grade they got).

At any rate, the instructions were given - no help other than book or notes. I'd stick to the professor's instructions, and understand he's not only teaching the subject, but also giving the class a life lesson in ethical behavior, and trusting the students to act in what we FIRSTers would call "graciously professional" behavior.

Just because you don't get caught, doesn't make it right.

Do it by yourself. Even if your grade suffers, it's early in the school year, you've got plenty of opportunities to get your grade to where it should be, and you're secure in the knowledge that you made the right decision - that's more important than any grade.