PicksTheorem 2002-06-12 06:55:50 2002-06-12 06:55:50 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) \usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\Prob}[2]{\mathbb{P}_{#1}\left\{#2\right\}} \newcommand{\Expect}{\mathbb{E}} \newcommand{\norm}[1]{\left\|#1\right\|} % Some sets \newcommand{\Nats}{\mathbb{N}} \newcommand{\Ints}{\mathbb{Z}} \newcommand{\Reals}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} %%%%%% END OF SAVED PREAMBLE %%%%%% Let $P\subset\Reals^2$ be a polygon with all vertices on lattice points on the grid $\Ints^2$. Let $I$ be the number of lattice points that lie \emph{inside} $P$, and let $O$ be the number of lattice points that lie on the boundary of $P$. Then the area of $P$ is $$ A(P) = I + \frac{1}{2}O - 1. $$ \begin{center} \includegraphics{pick} \end{center} In the above example, we have $I=5$ and $O=13$, so the area is $A=10\frac{1}{2}$; inspection shows this is true.