It would be useful if you'd try and understand what was written before you point out where, in your opinion, we 'got lost'.
You two are applying "non clever" problem solving skills to a "clever" problem. This problem is not about neatness and what I mean by that is if only 2 students out of 20 got it right, then maybe 1 or 2 students missed it because of sloppy work. The other 16, 17, or 18 students missed it because they didn't see the trick.
Now, if the teacher gave them the trick, then the students probably did mess it up with sloppiness, but students that go through the trouble of seeing the trick themselves generally don't follow that up with sloppiness. When one solves a problem from prior art one is prone to silly mistakes much more so than when one solves a problem that requires cleverness.