A "dot" (commonly also called a "lattice point") is a point with integer coordinates.

In the plane, what is the total number of dots inside or on the boundary of the triangle with vertices (0,0), (x,0), (x,y) where x and y are positive integers?

In the event that it is not utterly obvious from the form of your answer that a whole number is being specified, give an independent argument to show this.

What total do you get if you count the three vertex dots together as just half a dot and any other boundary dots as half a dot each?

Using Pick's Theorem: Area = I + B/2 - 1

Points in the interior

  I

add one-half for points on border (not vertices)

     + (B - 3)/2

add one-half for vertices

                  + 1/2

gives

  I + (B - 3)/2 + 1/2 =

  [Area - B/2 + 1] + (B - 3)/2 + 1/2 =

  Area =

  x*y/2

Edited on June 9, 2005, 9:43 pm

Edited on June 15, 2005, 2:42 am

Posted by Bractals on 2005-06-09 21:17:31