Problem Solving Episode

For my problem, I have selected the "DiagonalCount" problem.

Step 1: Understand
The problem asks for a formula that will return the numbers of squares intersected by the diagonal of a rectangle made of m rows and n columns.

Step 2: Devise a plan
Let the formula be D(m, n). I will start by finding D(m, n) for a lot of rectangles, and finding a formula that fits them. If I can explain WHY this formula works, then it should be true.

Step 3: Carry out the plan
I found that a lot of rectangles had D(m, n) = ceil(max(m, n) / min(m, n)) * max(m, n). However, this did not work for rectangles where all rows did not have the same number of intersected squares, like 3*5 and 5*8. My rationale was that ceil(max(m, n) / min(m, n)) would give the number of squares intersected on each row (or column, whichever is larger), and multiplying it by max(m, n) gives the total. However, this doesn't work for rectangles  where all rows did not have the same number of intersected squares.

Trying another method, I found that for some rectangles D(m, n) = m + n - 1. This was because there are m  intersected squares, and on every row but the first, there is a sort of "extra" square, caused by the diagonal coming through the bottom of the above square.

This still doesn't work for 2*2, 4*6, and infinite other rectangles. However, the (m, n) pairs of these rectangles all have common factors. Dividing out the GCF from m & n, calculating m + n - 1, and multiplying the result by the GCF gives D(m, n).

Therefore:
GCF(m, n) * ((m + n ) / GCF(m, n) - 1) =
m + n - GCF(m, n) =
D(m, n)

Week...WHAT IS THE DATE, MAN!?!

AAAAAAAAnd I forgot again.

While proofs were fairly easy, I've been having trouble with complexity. Maybe it's the material, maybe its my constant sleepiness, or maybe I just have so much stuff in my head more won't fit, but it just isn't as intuitive to me as proofs and other things.

We had the term test yesterday, and I'm fairly certain I screwed up on the floor problem. I had been doing problems throughout the week, but there weren't any more office hours by the time I got to the ceiling question from the previous test, and  there was no solution posted for it, so I was doomed (I spent over an hour trying to solve it, and still no luck).

I think I aced everything else, though, so at least there's that.

Week...Something

AAAAAAAAAnd I forgot about this.

Class is still going great though! I still find most material very easy to grasp.  Our work on proofs over the last week has been going very well, and I feel that I did well on the term test.

I actually find myself wishing the course had a faster pace...maybe that's just me.

See y'all next time!

Week 2

Hello again!

We've been focusing a lot on implication and negation so far this week. This week's tutorial seemed like a bit of a curve-ball, as it had a predicate with multiple arguments. I had a lot of trouble solving those questions before the tutorial. However, I feel that the tutorial helped me understand it better, and I got the right answer on the quiz. The course is shaping up to be an informative one!

See you next time!

Week 1

Well, here it is; the beginning of my SLOG!

It's been a fun week so far of learning about sets, universal and existential claims, and implications. I like that we're learning how to express a boolean expression using only symbols. Nothing seems difficult so far, but this IS only the first week, so that's liable to change. I feel I did well on the quiz. I finished really early, so I had time so study for other courses.

I wish I had more to write about, but there's not really any problems I'm having in the course right now!

See you next time!