Staring at trees...

I am sorry to admit that since the ternary tree problem was postponed to the next assignment, I had not pondered upon it. Now, when my stream of consciousness seems to be ubiquitously defined by a common cold virus, it seems like it could have been a good idea to read the hand out properly a week ago, in the heights of health. But at least I can still plan the steps to be taken over the next few days. Based on Polya's memory-tickling problem-solving approaches, I intend to:

1) Draw trees - ternary trees of n nodes. I have heard of cases where hundreds of tree sketches were needed. I will hopefully stop before my first hundred, if I get there, and approach the problem from an alternative perspective then. Careful tracking of the number of nodes and the number of non-equivalent ternary trees will be crucial at this stage. Perhaps considering the subtrees of the subtrees will help... we'll see.

2) Visual quantification - I usually find patterns more appealing when they can be perceived in some way. It's like unwinding a function - aligning columns of expressions can often make clear what the numerical pattern for a particular case is.

3) Small n results - attempting to find patterns for the lower values of n can often be confusing as a pattern does not always arise immediately, but it can be very valuable to observe how the pattern progresses as the trees bloom deeper and depper.

4) Detecting a pattern!

5) Verifying pattern - for other values of n that are smaller (including, perhaps, a base case or a few base cases) and for values of n that are bigger.

6a) Discard pattern - if the perceived pattern did not work, check steps from number 1) to see if anything went wrong at any stage e.g. trees were not properly drawn, trees to be considered should have been less/more, etc.

6b) Prove pattern - maybe through some flavour of induction?

I will report back once (if?) I figure this problem out. Also note that the problem set 4 might be useful for the 4th problem of the assignment, as it seems to be a smaller version of a similar exercise. I will write about this guess on the next entry as well.

Post-A1

This might be (is) a bit of a late entry, but I'd like to draw from the box of recent memories and write briefly about my experiences with the first two problem sets; then I'll express my frustration with the first assignment.

I found the first problem set to be appropriate to the amount and difficulty of things that we had faced so far in the course, which had not been too much yet. Then came the second problem set, with a similar spirit but with a bit of a higher challenge for me - how to make a proof as "elegant" or "not having to have 5 different cases-ish" as possible. The readings were very helpful in this regard, and even without knowing my mark yet, I feel happy about how it ended up looking.

And finally, the assignment. Mm. It's not that I'm unhappy, but I know that I could be far happier. I hope that you, my dear reader, will understand this dilemma of feeling upon learning that even though I was on the right track to solve the last problem, I erased my the incomplete parts leading to what would have been, with more time, a sound solution. I got rid of them so as to submit a proper assignment without "random" facts about the elements of the third problem that did not seem to lead to anything. If only that final minute of insight had happened moments before... but I guess I'll try not to leave things until the last minute next time. It sucks to learn this way.

But that last paragraph didn't actually say much. I found the problems in assignment 1 to be challenging, giving us a good chance to practice our ninja-like inductive skills. I particularly liked the second question, not only because of its relation to food, but also because I could simplify the solution that I originally found.

And now it's time to review the readings for the first test. Yay!...