1. Dylan has a $100 \times 100$ square, and wants to cut it into pieces of area at least 1 . Each cut must be a straight line (not a line segment) and must intersect the interior of the square. What is the largest number of cuts he can make?
Answer: 9999
Since each piece has area at least 1 and the original square has area 10000, Dylan can end up with at most 10000 pieces. There is initially 1 piece, so the number of pieces can increase by at most 9999 . Each cut increases the number of pieces by at least 1, so Dylan can make at most 9999 cuts. Notice that this is achievable if Dylan makes 9999 vertical cuts spaced at increments of $\frac{1}{100}$ units.