Switching Between Simple and Complete Induction

It’s been about a month since the start of CSC236 and I can honestly say it’s gone by a lot more smoothly for me than the first month of CSC165. I struggled with the equivalences portion of the assignments in the previous course, as well as on the test, which I did very poorly on. However, once we began the actual mathematical proofs in CSC165 I greatly improved, and since the material right now in CSC236 is building on that section of the previous course, I’m finding that I understand the concepts more readily and I can apply them with greater ease.

That said - I ran into some trouble this week with complete induction, specifically the stamp combination problem. When Professor Heap asked us in class whether we felt confident in utilizing both simple and complete induction, I raised my hand for both. Later in tutorial I realized I wasn't completely correct. 

I had worked my way through the mathematical induction solution for it the other week and felt that with simple induction, the solution was very intuitive. However, when it came to reorganizing the problem into complete induction on the quiz, I found myself constantly reverting to the simple induction method. When the handout was first posted, I recognized the question and thought that I wouldn't have to write it out with complete induction because I understood the initial solution so well that just listening to the TA's solution would be enough for me to grasp the complete solution. I was very wrong. For questions involving trees, I can very easy visualize the problem and apply complete induction to solve it, but with the stamp problem it was different. My TA had gone over the solution for the stamp problem very briefly and I wasn't sure how detailed my answer on the quiz was supposed to have been (and, of course, as I'm writing this entry, the Tutorial 2 solutions PDF from the course website repeatedly refuse to load). I had no problem establishing the base cases. I understood that the formula for any amount n was the combination for the amount (n - 3), plus another three, but when it came to actually elaborating on the reasoning behind that formula on the quiz, I found myself struggling. My mind kept jumping back to the more intuitive-for-me logic of the simple induction solution. So, even though simple induction and complete are equivalent theoretically, diverting my mind from one to the other proved difficult, at least with this problem. I haven't really had trouble with complete induction on any other questions, so I'm hoping that it was just my having cemented the earlier solution to the stamp problem to memory that caused my confusion.

I'm definitely going to go over some more complete induction examples before the test and try them out on my own - I'm not as confident with it as I thought I was. I might try to go over question 4 from the assignment again, which gave us the option of using either simple or complete induction. I handed in the assignment using simple induction on that question, but I'll give it a try with complete and see if I have any issues.