Better Late Than Never... Problem Solving on Diagonals

It seemed simple when I first received this problem in February, but I found out from a couple of other students that it's not as simple as it seemed. Might as well get this out before the end of the course.

Problem: Draw a diagonal line in a rectangle that consists of m rows and n columns and determine how many grid squares are passed through by the diagonal line.

Process: Using graph paper in class, I drew some examples down on paper and found some patterns.
This is what I found comparing m and n (before I ran out of space).

    n 
m       1  2  3  4  5  6  7  8

    1    1  2  3  4  5  6  7  8              RED: Whenever n = m
    2    2  2  4  4  6  6  8  8              GREEN: Whenever n = m are odd
    3    3  4  3  6  7  6  9 10             YELLOW: Whenever n and m are odd
    4    4  4  6  4  8  8 10 8
    5    5  6  7  8  5 101112
    6    6  6  6  8 10 6 1212
    7    7  8  9 101112 7 14
    8    8  8 10 8 121214 8

I wasn't so sure about the pattern that I found, because it seemed to look like it was all over the place.
I initially thought that there was a pattern whenever n or m are odd or whenever n or m are even, but then I had to look at whenever n = m, where the pattern would break. There also seems to be a pattern when looking diagonally across this chart.

Overall, I'm not entirely sure of what to make of this, even though "answer" were already given. I felt that if possible, I'd want to discuss this with Danny about this, but it's already the end of the course, so oh well.

The final stretch...

So here we are, at the last week of the school year.
I'm feeling a rush from a sense of urgency trying to get my assignments finished before Friday and I'm not going to let up until they are done. Meanwhile, the 3rd assignment is going surprisingly well with the concepts of Big-O, Big-Omega and limit notations becoming more familiar to me. However, the concept I can't seem to grasp seems to be the halting principle which is the basis of the last question of the assignment and a major part of chapter 5 in the course notes.
I hope that the last tutorial can help me with grasping the halting principle so I can finish the assignment and get focused on the final exam later in April.
Things are going fine for me these days, the ideas of diagonalization seem to be simple, yet very abstract in its description. Hopefully, I can grasp this concept before next class then.

Overall, this class wasn't really as bad as I had heard last term. It's probably because the teacher was very clear in his descriptions, posted everything on his website and was all around, very helpful when we had a question ask about something we didn't understand.

About the the problem-solving progress, I had a vague description for a problem based on what we looked at in class, but I'll need another post to describe what I got.