Five Mathematical Problems You Can Try on the Back of a Napkin.

presented by Fred Stevenson
September 22, 2011

We were interrupted right at the beginning with a forced evacuation of the building. Once we returned to the room, Fred ensured that we all would have plenty to explore. He gave eight different (yet some related) explorations. I will pose the problems as best I can from my scribbled notes:

1. Start with a tic-tac-toe looking grid. Using curves or straight lines, how many of the 12 segments can your cross without crossing any of the same segments twice?
2. Start with a 3 by 3 square grid. Using curves or straight lines, how many of the 24 segments can your cross without crossing any of the same segments twice?
3. Start with a 3 by 4 grid. How many routes can be taken from the top left vertex to the bottom right vertex if all paths must move only to the right or down?
4. Start with a 3 by 4 grid. Count how many interiors of squares are crossed when a segment is drawn from the top left vertex to the bottom right vertex. Explore other sizes of grids to determine the maximum number of square interiors that will be crossed in an m by n grid.
5. Start with a 4 by 4 dot grid. How many squares can be drawn so that each vertex of the square is a dot in the grid? What if you start with a 5 by 5 dot grid? ... an m by m grid?
Pool Table Problems:

6. Start with a 2 by 3 grid. Assume that there is a pocket in each corner. If a ball starts in the bottom left corner and is hit at a 45 ° angle, determine the number of caroms (bounces) the ball makes, how far it travelled, and which pocket it enters.
   Now explore a large 1 by 1 square. Assume that there is a pocket in each corner. If a ball starts in the bottom left corner and is hit to a spot 1/3 of the way up the opposite side, determine the number of caroms (bounces) the ball makes, how far it travelled,and which pocket it enters.
   What if it is hit 2/3 up the opposite side? How many caroms will the ball make, how far did it travel, and which pocket will it enter?
   Continue exploring until you can determine a path that will be 25 units long.
7. Start with a 4 by 4 dot grid. Those are 16 points of an affine plane. There's a problem here that I didn't get written.



Fred introduced the Number of Squares on a Grid Problem	Roz explored the number of segments problem.	A group shared solution ideas.






A couple of the AIMS Standards addressed include:
4.1.2 (HS) Visualize solids and surfaces in 3-dimensional space when given 2-dimensional representations and create 2-dimensional representations for the surfaces of 3-dimensional objects.
5.2.2 (HS) . Solve problems by formulating one or more strategies, applying the strategies, verifying the solution(s), and communicating the reasoning used to obtain the solution(s).