Proofs and Puzzles

These past few days, this course has only been concerned with proofs. Although the proofs so far are not difficult, I predict they will eventually become fairly difficult. Proofs have always been one of the hardest aspects of mathematics. As opposed to calculations, which eventually become fairly mechanic, proofs require a high level of intelligence and understanding. It is not simply a matter of following the same algorithm over and over again. Usually, the middle part is the most difficult. It's the part that requires some intuition, some intense reasoning, and (excuse the cliche) the ability to think outside the box.

It's not that proofs are particularly difficult for me. Not to say that I am particularly good at proofs, but I tend to get along on math test. The way I've always studied for math is to do lots of problems. Even when it comes to proofs, to study for those, I would do a lot proof problems. So far, it's working pretty well, but maybe this course can give me more effective methods at tackling proofs.

Anyways, I might as well use this opportunity to record a problem solving scenario. The problem I had to solve was the tutorial 3 office hour puzzle. Since this puzzle was decently easy, in the future, I will probably post another problem solving scenario.

Anyways, the link to the tutorial can be found here: http://www.cdf.toronto.edu/~heap/165/W10/Tutorial/t3/t3.pdf

Understand the problem:

This problem is fairly straightforward to understand. Basically, you are given a bunch of information, and from that information, you have to figure out the order the students came to the office and what each students wants to do.

The second step is to plan a solution, and for this fairly simple problem, there isn't going to be a complex plan of any sort. Basically, I am going to list all the given information one by one and try to deduce further information from the given information. Basically, the plan is to just connect the dots.

So, we know that the last two are not interested in computer games and one of them is interested in designing programs to create music. We also know that Alex came to office hours first, so he could be interested in computer games, but he is not interested in computer generated music. We know that Kim did not arrive second and did not arrive first either, and is not interested in video games, which makes sense given what we know. Since someone came after Lee, we know that Lee could be second or third. And finally, we know that the last person wants to study bioinformatics.

Now, from this, we know that Lee could not have been third, since the person who came after him wants to design databases, and the last person wants to study bioinformatics. Hence, Lee could only have been second. We also know that the last person is female and that Kim is male, and hence, Kim could not have been fourth. Hence, Kim is third and that leaves Robin to be fourth.

Since Robin is fourth, she is interested in bioinformatics. Since the person who came after Lee is interested in databases, that means that Kim is interested in databases. And, since Alex is not interested in computer generated music, that means that Lee is, and that Alex is interested in video games. Thus, we have:

Alex: first, computer games

Lee: second, music

Kim: third, databases

Robin: last, bioinformatics

However, I guess it can be argued that we don't know what Alex is interested in. All the problem says is that the last two are not interested in computer games and that neither is Alex. That does not necessarily mean that anybody is interested in computer games. But, I guess it is implied.

Looking back, I realize that this problem is just a matter of connecting the dots. Next time I'm faced with a similar problem, I'll also begin by listing all the given information and try to deduce further information from that. Now, I get my opiods.