Assignment Two PROVED to be difficult!

Assignment 2 was, in my opinion at least, a lot harder than the first one. When it comes to proofs, I am often flabbergasted by the sheer number of options I have to approach it. I don't find it as intuitive as some people to know when to factor, or square both sides, or multiply by a constant. It feels to me like I'm grasping around in the dark for something that might have some potential to get me where I want.

That being said, I do think that the way proofs are approached in this course is a lot more my style than, say, high school math. Maybe it's because the course material was effective in introducing the foundations, but I generally feel more warmly towards proofs in this course than ones I previously encountered. Assignment 2 was still pretty hard, though.

In particular, I could not come up with a solution for question #4, so I simply wrote the structure, did the first step that I was sure was right, then wrote "I cannot solve this past this step." Hopefully the "substantial part marks" given to incomplete proofs really are substantial. Looking at the solutions, however, the answer seems almost embarrassingly obvious.

I had managed to get to the step where I squared both sides of the equation n = 11q + 5, and I knew that the only thing I could do from there was expand it into n^2 = 121q^2 + 110q + 25. After that, I couldn't figure out how to manipulate that polynomial into the form 11p + 3. The solution was to separate 25 into 22 + 3 then pull 11 out of each of the terms except 3. Unfortunately, I was just as dense solving this problem as I was during the tutorial quiz on February 26, where I ran into a similar situation and was unable to perform the simple arithmetic required to split an integer into two other integers. At least now I am aware that this is something I often overlook, so I (hopefully) won't make the same mistake in the future. 

Also, interestingly, we were allowed to write all of the proofs for the assignment in plain English! I didn't realize that until I had already completed (er, almost completed) the assignment, but I think if I had written some of the proofs in English it would have helped me get a better grasp on what was required, and maybe I would have finished it sooner and with fewer mistakes. 

Ah well. I will have to make sure I do the next assignment well in advance.