Problem Solving Attempt

I decided to tackle the SumProducts problem on the link Prof. Heap gave us in the SLOG handout. The question reads as follows:
The list of positive whole numbers that add up to 1 is simply (1), and you might think of the product of this list as being 1.
There are several lists of positive whole numbers that add up to 2, and the lists (1,1) and (2) have different products if you multiply the list elements: 1 and 2. The lists of positive whole numbers that add up to 3 yield products 1, 2 and 3, corresponding to the lists (1,1,1), (1,2) and (3). If n represents a positive whole number, what is the largest product that can be formed by multiplying the elements of a list of positive whole numbers that sum to n?

Understanding the Problem

Understanding the problem is simple. Given an integer n, what is the largest product of integers that sum to n. We are given the fact that for any integer n, n sums to n and n is the product by itself. So given the integer 3, the possibilites are :
(1,1,1)
(1,2)
(3)
So the largest product here would be '3'.

Devising a Plan

I will try to find a connection or some sort of a pattern given an integer n to find the value of its largest products of sums.

Carry out a Plan

A couple of things I notice. For n<=3, the greatest product will be n itself. For the numbers up to 10, I wrote up a comprehensive list of all the products of sums, avoiding any sum with a '1' as that does not contribute any growth to the product. So for 1 the product is 1, for...
  

2 - 2

3 - 3

4 - 2 * 2

5 - 3 * 2

6 - 3 * 3

7 - 3 * 2 * 2

8 - 3 * 3 * 2

I notice that the pattern arises with powers of 2 and 3, as those are the smallest prime number divisors. The reason we choose powers of 2 and 3 is because we know that exponential function grow more rapidly than any power functions.

Looking Back

I came up with the following solution.When:

n%3 = 0, We will have n/3 '3's.

n%3 = 2, We will have (n-2)/3 '3's  and 1 '2'

n%3 = 1, We will have (n-4)/3 '3's and 2 '2's

I knew that 3*n grows more rapidly than 2*n for all n, so I set up my solution in a way to have as many terms with 3 as possible and fill up the remaining sum with terms of 2 for which any more '3's would contribute a remainer.

Pre/Reading Week

We had received our assignments back, as well as our midterms. The midterms went well, the assignment not so much, but I feel confident I can improve on the next one. Proofs are giving me some problems, not so much the actual proof itself, but the structure and how to organize everything on paper. I'm in MAT137, so I'm familiar with some proofs we had looked at, but they are a lot more formal in this 165 class. Sometimes, after a proof is finished, I still ask myself the question, as I have been doing for a long time, "How does this prove it's True/False??".  I guess I'll just need some more practise.

Reading Week was a good week off, didn't spend it as productively as I could have, but it was nice to have some time off and refocus on everything afterwards. It was healthy. I feel good about the home stretch to finishing off first year.

Second/Third Week

The second week started to get a little more challenging. More and more is being demanded from you, and already from the get-go it is important to be on top of things. I found the tutorial in week two to be a little difficult, but after receiving clarification from the TA I found it rather easy. As for the quiz, it was a piece of cake.

Starting up on Assignment 1 didn't seem too challenging, but I have yet to finish, and the further along I got with it the harder it became. I am comfortable with symbols and notation, and I find truth tables to help me a lot as well. What I find challenging that we touched on today in class during the beginning of week 4 are the proofs. I had a hard time proving the statement which said that if P=>(Q=>R) then Q=>(P=>R). The negation is a little challenging for me as well but I feel that completing A1 and practising/looking over the annotated slides will defenitely prove beneficial. Also, it's nice that the lecture hall in Sid Smith isn't as hot as a sauna as it was during the first day of class, still don't know what was up with that, was the heater turned up too high, or was I simply wearing too many layers ? :)

The First Week

I didn't know what to expect this week. Okay, that's a lie. I was told by my buddy that this course is really challenging, however this same buddy of mine had also nearly failed 108. Besides the room being as hot as a sauna on Monday, the rest of the week was pretty good. I was familiar learning about existential and universal quantifiers from my 137 Math class, so this was all review to me. The quiz today in tutorial was very straightforward, so straightforward that I thought it was a trick. I'm pretty sure it went well.

I believe the little "pop quizzes" in class are good, because they sort of force you to participate in them, even if you don't feel like it, they still somehow get you to tackle them.

I'm not entirely sure what this problem-solving episode is that we have to make up, but I guess I'll save it for later. Overall, I feel confident that the material in this course will be suitable to me, as I have always had a knack for problem solving and logic. Productive week and a half so far. Let's keep it going. :)