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| 1 | +\documentclass[11pt]{article} |
| 2 | +\usepackage[utf8]{inputenc} |
| 3 | +\usepackage[top=2.0cm, bottom=1.5cm, inner=5.9cm, outer=1.5cm, marginparwidth=3.9cm, marginparsep=0.5cm,footskip=0.5cm]{geometry} |
| 4 | +\usepackage{fancyhdr} |
| 5 | +\usepackage{titlesec} |
| 6 | +\usepackage{titling} |
| 7 | + |
| 8 | +\reversemarginpar |
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| 10 | +\fancyhf{} |
| 11 | +\pagestyle{fancyplain} |
| 12 | +\fancyfoot[R]{\footnotesize vXXX} |
| 13 | + |
| 14 | +\setlength{\parindent}{0em} |
| 15 | +\setlength{\parskip}{0.2em} |
| 16 | + |
| 17 | +\titlespacing*{\section}{0pt}{1em}{0.5em} |
| 18 | + |
| 19 | + |
| 20 | +\pretitle{\begin{center}\LARGE} |
| 21 | +\posttitle{\par\end{center}} |
| 22 | +\predate{} |
| 23 | +\date{\vspace{-1em}} |
| 24 | +\postdate{} |
| 25 | + |
| 26 | + |
| 27 | +\title{How to Solve It} |
| 28 | +\author{George P\'{o}lya} |
| 29 | + |
| 30 | +\begin{document} |
| 31 | + |
| 32 | +\maketitle |
| 33 | + |
| 34 | +\section*{Understanding the Problem} |
| 35 | + |
| 36 | +\emph{What is the unknown?} |
| 37 | +\marginpar{ |
| 38 | + \textbf{First}. |
| 39 | + |
| 40 | + You have to \emph{understand} the problem. |
| 41 | +} |
| 42 | +\emph{What are the data?} |
| 43 | +\emph{What is the condition?} |
| 44 | + |
| 45 | +Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or Contradictory? |
| 46 | + |
| 47 | +Draw a figure. |
| 48 | +Introduce suitable notation. |
| 49 | + |
| 50 | +Separate the various parts of the condition. |
| 51 | +Can you write them down? |
| 52 | + |
| 53 | +\section*{Devising a plan} |
| 54 | + |
| 55 | +Have you seen it before? |
| 56 | +\marginpar{ |
| 57 | + \textbf{Second}. |
| 58 | + |
| 59 | + Find the connection between the data and the unknown. |
| 60 | + You may be obliged to consider auxiliary problems if an immediate connection cannot be found. |
| 61 | + You should obtain eventually a \emph{plan} of the solution. |
| 62 | +} |
| 63 | +Or have you seen the same problem in a slightly different form? |
| 64 | + |
| 65 | +\emph{Do you know a related problem?} |
| 66 | +Do you know a theorem that could be useful? |
| 67 | + |
| 68 | +\emph{Look at the unknown!} And try to think of a familiar problem having the same or a similar unknown. |
| 69 | + |
| 70 | +\emph{Here is a problem related to yours and solved before. Could you use it?} |
| 71 | +Could you use its result? |
| 72 | +Could you use its method? |
| 73 | +Should you introduce some auxiliary element in order to make its use possible? |
| 74 | + |
| 75 | +Could you restate the problem? |
| 76 | +Could you restate it still differently? |
| 77 | +Go back to definitions. |
| 78 | + |
| 79 | +If you cannot solve the proposed problem try to solve first some related problem. |
| 80 | +Could you imagine a more accessible related problem? |
| 81 | +A more general problem? |
| 82 | +A more special problem? |
| 83 | +An analogous problem? |
| 84 | +Could you solve a part of the problem? |
| 85 | +Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? |
| 86 | +Could you derive something useful from the data? |
| 87 | +Could you think of other data appropriate to determine the unknown? |
| 88 | +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? |
| 89 | + |
| 90 | +Did you use all the data? |
| 91 | +Did you use the whole condition? |
| 92 | +Have you taken into account all essential notions involved in the problem? |
| 93 | + |
| 94 | +\section*{Carrying out the Plan} |
| 95 | +Carrying out your plan of the solution, \emph{check each step}. |
| 96 | +\marginpar{ |
| 97 | + \textbf{Third}. |
| 98 | + |
| 99 | + \emph{Carry out} yout plan. |
| 100 | +} |
| 101 | +Can you see clearly that the step is correct? |
| 102 | +Can you prove that it is correct? |
| 103 | + |
| 104 | +\section*{Looking Back} |
| 105 | +Can you \emph{Check the result}? |
| 106 | +\marginpar{ |
| 107 | + \textbf{Fourth}. |
| 108 | + |
| 109 | + \emph{Examine} the solution obtained. |
| 110 | +} |
| 111 | +Can you check the argument? |
| 112 | + |
| 113 | +Can you derive the result differently? |
| 114 | +Can you see it at a glance? |
| 115 | + |
| 116 | +Can you use the result, or the method, for some other problem? |
| 117 | +\end{document} |
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