Draw a line across a 4 x 5 checkerboard so that the maximum number of regions are intersected. Extend to an m x n checkerboard.

We can observe that the most intersections you can get with horizontal lines is5

The most intersections you can get with a vertical line is 4.

Taking a diagonal through the diagonals of the squares give 4 as the most intersections

Using a GSP sketch, I have come up with 8 going inbetween two diagonals of the board, alternating sides.

How many squares of any size are in an 8 x 8 checkerboard? Extend to an m x n checkerboard.

For an 8x8 checkerboard there are obviously 64 (1 x 1) squares. Then there are 16 (2 x 2) squares. Then there are 4 (3 x 3 ) squares. There are 4 (4 x 4 ) squares. There is 1 (5 x 5 )square, 1 (6 x 6) square, and 1 (7x7 )square, and 1 (8 x8 ) square. So, 92 square total.

If m = n then an m x m checkerboard has m^2 square

If m is not equal to n, then an m x n checkerboard would have m*n squares.

This topic can be extended to a discussion of area.

How many rectangles of any size on an m x n checkerboard?

Since a square is a rectangle there are at least m*n rectangles + m + n + (m-1)+ (n-1) +

Given only the lattice points (points where four of the squares meet), how many squares of any size can be found on a 5 x 5 grid?

It appears to be squares for a 5 x 5 square

Extend to an m x n grid.

Given only the lattice points of an n x m grid where n and m are very large, can you draw a linethrough the grid
which intersects no lattice points, only one lattice point, only two lattice points? Answer the same question for m
and n both infinite.