diff --git a/main.tex b/main.tex new file mode 100644 index 0000000..7521d39 --- /dev/null +++ b/main.tex @@ -0,0 +1,117 @@ +\documentclass[11pt]{article} +\usepackage[utf8]{inputenc} +\usepackage[top=2.0cm, bottom=1.5cm, inner=5.9cm, outer=1.5cm, marginparwidth=3.9cm, marginparsep=0.5cm,footskip=0.5cm]{geometry} +\usepackage{fancyhdr} +\usepackage{titlesec} +\usepackage{titling} + +\reversemarginpar + +\fancyhf{} +\pagestyle{fancyplain} +\fancyfoot[R]{\footnotesize vXXX} + +\setlength{\parindent}{0em} +\setlength{\parskip}{0.2em} + +\titlespacing*{\section}{0pt}{1em}{0.5em} + + +\pretitle{\begin{center}\LARGE} +\posttitle{\par\end{center}} +\predate{} +\date{\vspace{-1em}} +\postdate{} + + +\title{How to Solve It} +\author{George P\'{o}lya} + +\begin{document} + +\maketitle + +\section*{Understanding the Problem} + +\emph{What is the unknown?} +\marginpar{ + \textbf{First}. + + You have to \emph{understand} the problem. +} +\emph{What are the data?} +\emph{What is the condition?} + +Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or Contradictory? + +Draw a figure. +Introduce suitable notation. + +Separate the various parts of the condition. +Can you write them down? + +\section*{Devising a plan} + +Have you seen it before? +\marginpar{ + \textbf{Second}. + + Find the connection between the data and the unknown. + You may be obliged to consider auxiliary problems if an immediate connection cannot be found. + You should obtain eventually a \emph{plan} of the solution. +} +Or have you seen the same problem in a slightly different form? + +\emph{Do you know a related problem?} +Do you know a theorem that could be useful? + +\emph{Look at the unknown!} And try to think of a familiar problem having the same or a similar unknown. + +\emph{Here is a problem related to yours and solved before. Could you use it?} +Could you use its result? +Could you use its method? +Should you introduce some auxiliary element in order to make its use possible? + +Could you restate the problem? +Could you restate it still differently? +Go back to definitions. + +If you cannot solve the proposed problem try to solve first some related problem. +Could you imagine a more accessible related problem? +A more general problem? +A more special problem? +An analogous problem? +Could you solve a part of the problem? +Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? +Could you derive something useful from the data? +Could you think of other data appropriate to determine the unknown? +Could you change the unknown or the data, or both if necessary, so that the new unknown and the new data are nearer to each other? + +Did you use all the data? +Did you use the whole condition? +Have you taken into account all essential notions involved in the problem? + +\section*{Carrying out the Plan} +Carrying out your plan of the solution, \emph{check each step}. +\marginpar{ + \textbf{Third}. + + \emph{Carry out} yout plan. +} +Can you see clearly that the step is correct? +Can you prove that it is correct? + +\section*{Looking Back} +Can you \emph{Check the result}? +\marginpar{ + \textbf{Fourth}. + + \emph{Examine} the solution obtained. +} +Can you check the argument? + +Can you derive the result differently? +Can you see it at a glance? + +Can you use the result, or the method, for some other problem? +\end{document}