1. Do a Simpler Question, Find a Pattern!

2. 1. Michael has the numbers a and b written on a sheet of paper
1. Michael has the numbers a and b written on a sheet of paper. He calculates the average of all the numbers on the paper and writes it onto the page. He finds the new average and adds it. What will be the 100th average he writes?

3. 2. If you can only go right or up, how many paths are there from A to C in the grid below?

4. 3. Henry has an n by n grid. He wants to cover the board entirely in 1 by 2 dominoes that cover 2 grid squares. For what values of n can he do this?

5. 4. Lisa has a circular birthday cake
4. Lisa has a circular birthday cake. She wants to make as many pieces as possible for the kids at Math Club. She can make N vertical cuts through the cake. What is the most pieces she can make?

6. 5. How many paths are there from the bottom left corner to the upper right corner in a 1 by N grid if you can go in any direction, not going any edge twice? (N=3 show below)

7. 6. How many diagonals are there in a regular N-gon
6. How many diagonals are there in a regular N-gon? (A polygon with N equal sides and N equal angles)

8. 7. Sean has a chocolate bar in the shape of an N by M grid of chocolate squares. In one “move”, he takes a piece of chocolate and splits it down one of the lines, to create two pieces. What is the least amount of moves Sean needs to break the chocolate bar into individual squares?

9. 8. Mr. Cheng walks up a staircase of N steps
8. Mr. Cheng walks up a staircase of N steps. Each time he either goes up 1 step or 2 steps. How many different ways are there for him to climb up the staircase?