PROBLEM OF THE MONTH

March 2004




Consider a polygon, P, drawn in the Euclidean plane such that all of the vertices of P have integer coordinates. That is, if we think of the plane as a giant sheet of graph paper, the vertices of the polygon all lie on the lattice points of the paper. The area of the polygon can be computed by counting the number of lattice points inside the polygon and the number of lattice points on the edge of the polygon. Let

I = the number of lattice points inside P
and
E = the number of lattice points along the edges of P, including the vertices.

Then the area, A, of P is given by the formula: A = I + E/2 - 1

For example, for the irregular hexagon below, I=13 and E=9. So our formula predicts that the area of the polygon should be:

13 + 9/2 -1 = 16 1/2.

We can check our result by computing the area of the polygon. Breaking P into triangles and rectangles,

we see that the area is:

2 + 4 + 1 + 3 + 4 + 1 + 3/2 = 16 1/2.

PROBLEM:

  1. Prove that the formula for the area of P is correct for any triangle.
  2. Prove that the formula for the area of P is correct for any polygon, regardless of the number of sides.

Submit your solution to the Mathematics Undergraduate Office (Math P-142) or electronically to Prof. Kudzin at problem@math.sunysb.edu by the due date. Acceptable electronic formats are: PDF, Postscript, DVI, (La)TeX, or just plain text. Please include your name and phone number, or preferably your email address.

Closing date: April 15th at 12 pm.