Tutorial 2: Wrangling with postage stamps and inequalities

I'll be honest: I do not like those questions about postage stamps.

I understand complete induction well. I also understand the proofs for the postage stamps questions when I read them. But when I have to prove claims about combinations of stamps myself, I need way more than 10 minutes (which was all that was offered for the quiz) to come up with a solid proof. I feel that part of my confusion derives from the fact that my TA proved the claim differently from the textbook. Mind you, I did not walk into the tutorial feeling marvellous about the postage stamp question, but I did believe that I could write a proof following the textbook's style, given a bit of time. After my TA's explanation, I had two conflicting styles of proofs screaming for my attention, and that definitely did not help with the quiz that followed five minutes later. I think I'll need to try to prove a few more claims about postage stamps on my own later, to see if I can fixate clearly on the idea behind these kinds of proofs.

Another thing: I also do not like inequalities. Actually, that is not true; I have a love-hate relationship with inequalities. They're straightforward when you find a link, but are terrible when you can't. Often times, I seem to run out of terms to prove that X > Y. Where are all these pieces I need?

This is why I really loved the inequalities question this tutorial: prove that for all natural numbers, n^4 <= 4^n + 17. I hated trying to solve it, but the solution (found on the course website) was so clever. I already knew to break up things like 4n to n + n + n + n to maximize the pieces I had. But the solution did things like decreasing the exponents in such a way that it was easy to prove that X > Y. I'll have to make sure I keep the techniques this proof used when I prove inequalities in the future. Maybe one day I'll develop a positive relationship with them.

Today: I'm at home. I'm taking a break from travelling. And would you know it, there's a place called "Home" in Japan (although it'd be pronounced "ho-meh").