Galvanize

This week we were introduced to halt. It is strange how that a recursive function call assume it halts implies that it doesn't halt based on the halt function. But after tracing through the code, it does imply that. Another topic covered was reductions. Seems to be some terminology so it wasn't hard to grasp. And finally onto counting which is a concept I thought it was already mastered in kindergarten... I was wrong. At first the idea that the number of odd integers which has the same number of even integers is also the same number of integers overall. At first this confused me but seeing that if you were to arrange every integer to its odd integer, then the one to one correspondence of them is true for all integers.

On the topic of the last assignment, the first two proofs didn't seem that difficult after seeing the various examples and my partners working on it. The proofs with the definition of the limit seem very strange to me but after working on it, it shouldn't be a challenge.

We must accept finite disappointment, but never lose infinite hope. -  Martin Luther King Jr.

Strain

Got back results from the second term test. Definitely could do better. After realizing that the input I wrote down for the second disproof of a modulo statement did not actually disprove it, I lost a significant amount of marks. Nevertheless, I didn't do terrible. This week we ave learned about Big Oh proofs. After many trials of proofs before, the proof doesn't seem that troublesome. The only difficulty I see in the proof is working backwards to show that the antecedent implies the consequent. With a few weeks left of school, it's about time to study the curriculum from the beginning to be prepared for the final exam although catching a fever during the weekend isn't helping...

Problem Solving Episode: Pascal's Method Part 2

Since we cannot make it to the red squares, I imagine them being 0. I then simple add the moves as normal:

And finally to answer the question, simply add the numbers in the top row:

3 + 6 + 6 + 3 = 18 paths to the top of the board.

Review/Extend:

This was a fairly easy question, by remembering to follow the rules of the question and the theory itself, the path was calculated perfectly.

Endurance

With a few weeks of class left, I hope to continue to do better until the exams are finished. The term test was actually not that bad as I expected. Having practicing and understanding the proofs from the lectures, tutorials, and the internet, these proofs are not impossible. At first I was really worried that I could not even understand the most simplest proof such as the ones in the assignment. But once I started redoing them, it started to make more sense. A colleague I met through my astronomy tutorial also had the same predicaments as I had with proofs. He had told me that one of his classmates asked how the professor was able to do these proofs like it's nothing and he responded "hours, hours, and hours, or practice." I feel more confident doing proofs and now which is a relief. Now onto the material covered this week, the proofs of Big O and Omega don't seem that hard but once again, practice makes perfect. The topic of algorithm analysis does seem strange in how people have calculated the steps in the while loop. The loop guards, 'initialize' variable and the codes under the while loop are understandable but the while loop will take some time to fully understand why the formula is calculated like it. In the end, I feel more confident and motivated to work on the material covered in this course than to brush it off to the side in hopes of not doing this material again (which I know I have to do since the specialist degree requires 3 courses of these.)

Every calamity is to be overcome by endurance. -Virgil

Problem Solving Episode: Pascal's Method Part 1

In paying homage to my dedicated data management teacher who unfortunately got laid off in his first time teaching at my school, I present you the following problem:

A checker is placed on a checkerboard as shown. The checker may move diagonally upward. Although it cannot move into a square with a red square, the checker may jump over the X into the diagonally opposite square. How many path are there to the top of the board?

Understanding the Problem:
I am asked to determine the number of paths at the top of the board. This is calculated by determining the number of steps at each square.

Devise a Plan:

Pascal's method: Created by an iterative process, each term is equal to the term of the two terms immediately above it.


Carry out the plan:

Using Pascal's method, I will start by putting down the patterns that can only be made ONCE and considering the red squares as well which I cannot make a move:

Now, time to calculate the terms above by simply summing them:

Unpredictable

With Assignment 2 finished, looking over it and understanding each step has calm my nerves about the concept. Initially before I thought it was impossible to do any sort of proof at all but just reading over notes in lecture and other sources in the internet has prepared me or at least given me the scope of what kinds of proofs to encounter in this course. On with algorithm analysis, at first writing down the steps beside each line of code helps in the sense to collect all the familiar steps and to finally make an equation for it. What strikes me difficult is the algebraic expressions in a loop (while) since you may get different expressions than what the solution provides (n....n-1) As always I end up reading the notes after the entire lecture so I may be putting on more pressure than necessary to understand the course material. Having heard that 3 proofs to do in the next term test I feel both relieved and stressed at the same time. Relieved that it only consists of proofs but also stressed because of the fact that I have 3 midterms on the week (they don't call this month march madness for no reason!) It's time to redeem myself after having that horrendous mark on my last term-test.

Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.
 - Albert Einstein

The Endeavor

With a few weeks left until the next Term test, I'm nervous. Proofs are definitely hard to master if one doesn't practice. Just doing or even attempting to read the material will give me more confidence to do some proof even at the basic level. Such as the one in the tutorial, I should keep note of a few important proofs so I can figure out to do some proofs with some facts I can use such as k and the natural numbers in the lecture. I guess what causes me such confusion in this concept is the actual proof itself; I'm amazed how people can think of some steps in a proof where in the end it nicely goes through the process. Doing the outline is normal and does make sense since we start from a vague idea and then start breaking it down in steps which the professor did a good job displaying in the lecture. Currently on the curriculum of complexity, it actually is an interesting concept to look at by following up through code. Just making equations to determine the run time of code is interesting.

The firm, the enduring, the simple, and the modest are near to virtue. -Confucius