- Michael Curzan

# Mathematical Exercises and Proofs

In the second image, we have a sketch of a possible tiling of a rectangle using different sizes of squares. We then calculate how long and wide each square is by solving an equation. Then on the next page, we draw a more accurate picture using these side lengths. At the bottom of the first page, we show that 999*2 + 90 equals 2088, and that 999*8 + 90 equals 8082, which have the same digits. At the bottom of the second page, we prove by calculation that the difference between the squares of two sums always equals equals a perfect cube.

The third and fourth images show the same, but with me in them, as well.

In the fifth image, we have a proof that every perfect number that is even must end in 6 or 8. This is because of modular arithmetic.

In the first image, we have a proof of the geometric theorem that the squares of all the sides of a parallelogram add up to the same as the sum of the squares of both diagonals, which is like a Pythagorean Theorem for parallelograms.

Happy New Year 2020!