Copyright © University of Cambridge. All rights reserved.
'Pick's Quadratics' printed from http://nrich.maths.org/

Consider a rectangle with coordinates $(0,0), (3,0), (3,2),
(0,2)$ and count the number of lattice points (points with whole
number coordinates, coloured yellow in the diagram) on the
perimeter and inside the rectangle.
Define $k$points as points with coordinates $({a\over k},
{b\over k})$ where $a, b$ and $k$ are integers. For example, for
the rectangle in the diagram the yellow points are the lattice
points, $k=1$, and the 2points are the red and yellow points taken
together.

It is known that for any polygon in the plane which has vertices at
lattice points the number of $k$points in the interior of the
polygon is $Ak^2  Bk + C$ and the number of $k$points in the
closed polygon, including the perimeter and the interior, is $Ak^2
+ Bk + C$. Verify that these quadratic formulae hold for the given
rectangle and find $A$, $B$ and $C$.
Suggest a connection between the coefficients $A$, $B$ and $C$ and
the area of the rectangle and the number of $k$points on the
perimeter.
Assume that for any plane polygon there is a quadratic formula for
the number of $k$points inside the polygon given by $Ak^2 Bk +
C$. Explain why, for large $k$, the area of the polygon is given by
$$\lim_{k\to \infty} \frac {{\rm number of interior} k{\rm
points}}{k^2}= A.$$