Entry #5 - Midterm 2.0

While writing the midterm, I felt fairly confident with my proofs. I knew I had most of the structure correct, and for the actual proving part, I tried to prove everything as explicitly as possible. Where I was unsure of my proof, I made sure to add comments that either explained what I wanted to prove but didn't know how, or further proved something to make it clear.

After receiving my marked midterm, I'm happy to say I did better than I expected. The extra comments I added definitely helped my mark, and I now know that I'll still get a decent amount of marks if I don't know how to prove some element of the proof, but still write out the structure and explain what I want to prove.

And I also found out my partner and I got a 95% on assignment #2, so I guess it's time for the happy dance .gif

Entre #4 - Diagonals

Problem: Find the number of squares that the diagonal line will pass through the interior of.

Plan: 

• Draw examples starting with smallest number of rows/columns
• Try to find a pattern (most obvious ones: m=n & m=1 or n=1)
• Redraw the squares with unknown patterns in a logical order (keep m constant, change n up to 5)
• Try to find a pattern in chart and drawings

Results:
After executing the plan up to 5 m (5 rows), a pattern was found with an even and odd number of m. Exceptions were also found with m = n (square). From this, we wrote out generic formulas for the odd and even number of m. The formulas are as follows:

m is Odd: number of shaded squares = m + n - 1 (as long as m != n)

m is Even: number of shaded squares = m + 2 [[(n-1)/2]] ([[]] is the floor of the function, and as long as m != n)

Looking back:
My partner whom I was working with informed me that she had spoken with Professor Danny who said that a problem occurs with our solution when it comes to multiples. Obviously our solution needs a little more work, and isn't perfected yet. Continuing to graph out more examples and analyzing them would probably help fix any problems that occur.