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\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} | ||
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\reversemarginpar | ||
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\fancyhf{} | ||
\pagestyle{fancyplain} | ||
\fancyfoot[R]{\footnotesize vXXX} | ||
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\setlength{\parindent}{0em} | ||
\setlength{\parskip}{0.2em} | ||
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\titlespacing*{\section}{0pt}{1em}{0.5em} | ||
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\pretitle{\begin{center}\LARGE} | ||
\posttitle{\par\end{center}} | ||
\predate{} | ||
\date{\vspace{-1em}} | ||
\postdate{} | ||
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\title{How to Solve It} | ||
\author{George P\'{o}lya} | ||
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\begin{document} | ||
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\maketitle | ||
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\section*{Understanding the Problem} | ||
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\emph{What is the unknown?} | ||
\marginpar{ | ||
\textbf{First}. | ||
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You have to \emph{understand} the problem. | ||
} | ||
\emph{What are the data?} | ||
\emph{What is the condition?} | ||
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Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or Contradictory? | ||
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Draw a figure. | ||
Introduce suitable notation. | ||
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Separate the various parts of the condition. | ||
Can you write them down? | ||
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\section*{Devising a plan} | ||
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Have you seen it before? | ||
\marginpar{ | ||
\textbf{Second}. | ||
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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? | ||
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\emph{Do you know a related problem?} | ||
Do you know a theorem that could be useful? | ||
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\emph{Look at the unknown!} And try to think of a familiar problem having the same or a similar unknown. | ||
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\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? | ||
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Could you restate the problem? | ||
Could you restate it still differently? | ||
Go back to definitions. | ||
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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? | ||
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Did you use all the data? | ||
Did you use the whole condition? | ||
Have you taken into account all essential notions involved in the problem? | ||
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\section*{Carrying out the Plan} | ||
Carrying out your plan of the solution, \emph{check each step}. | ||
\marginpar{ | ||
\textbf{Third}. | ||
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\emph{Carry out} yout plan. | ||
} | ||
Can you see clearly that the step is correct? | ||
Can you prove that it is correct? | ||
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\section*{Looking Back} | ||
Can you \emph{Check the result}? | ||
\marginpar{ | ||
\textbf{Fourth}. | ||
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\emph{Examine} the solution obtained. | ||
} | ||
Can you check the argument? | ||
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Can you derive the result differently? | ||
Can you see it at a glance? | ||
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Can you use the result, or the method, for some other problem? | ||
\end{document} |