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\mode<presentation>{
%\usetheme[numbers]{Warsaw} %Rebordes y n\'{u}meros
\usetheme{Frankfurt} %Rebordes y n\'{u}meros indice con puntos
%\usecolortheme{seahorse} % Rebordes malvas
%\usecolortheme{rose} %Reborde azules m\'{a}s claros en bloques
%\usefonttheme[onlylarge]{structuresmallcapsserif} %\sc en t\'{\i}tulos
%\usefonttheme[onlysmall]{structurebold} % t\'{\i}tulo de arriba en negrita ??
%\setbeamerfont{title}{shape=\itshape,family=\rmfamily} %t\'{\i}tulo en it\'{a}lica
%\setbeamercolor{title}{fg=red!80!black,bg=red!20!white} %t\'{\i}tulo en rojo con algo de negro y fondo rojiblanco
}







\title[Taller de problemas]{\bf Taller de problemas 4º de la ESO + empresa}

\author[Fernando Chamizo]{Fernando Chamizo Lorente\\ {\small http://www.uam.es/fernando.chamizo}}
\date{}
\institute{ICMAT \quad 25 de marzo de 2015}

\begin{document}

\setbeamertemplate{headline}
{%
\begin{beamercolorbox}{section in head/foot}
%\alert{\ \insertframenumber\ de \inserttotalframenumber}
\ \vskip6.5pt\par \insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}

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% TRANSPARENCIA 0  %
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%Esto simplemente crea la página del título
\begin{frame}
\titlepage
\end{frame}


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% TRANSPARENCIA 0+ %
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%Crea el índice
\section*{\'{I}ndice}
\begin{frame}
\frametitle{\'{I}ndice}
%\tableofcontents[pausesections]
\tableofcontents
\end{frame}




%  _____           _     _                        __ 
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% | |__) | __ ___ | |__ | | ___ _ __ ___   __ _   | |
% |  ___/ '__/ _ \| '_ \| |/ _ \ '_ ` _ \ / _` |  | |
% | |   | | | (_) | |_) | |  __/ | | | | | (_| |  | |
% |_|   |_|  \___/|_.__/|_|\___|_| |_| |_|\__,_|  |_|
%                                                    
                                                   

%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\section{Problema 1}
\begin{frame}
\frametitle{Problema 1}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\textbf{¿Cuánto se puede separar de la vertical una pila de libros?} 
\end{block}}

\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.15]{./images/stack1.jpg}
\\[10pt]
\textbf{¡Sin caerse!}
\end{tabular}
\end{center}

\end{frame}



%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{La torre de dos libros}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.45]{./images/torre2.eps}
\\[10pt]
\textbf{Posición límite de equilibrio}
\end{tabular}
\end{center}
\end{block}}

\end{frame}


%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{La torre de tres libros}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.38]{./images/torre3.eps}
\\[10pt]
\textbf{Posición límite de equilibrio}
\end{tabular}
\end{center}
\end{block}}

\end{frame}



%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{La torre de cuatro libros}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.39]{./images/torre4.eps}
\\[10pt]
\textbf{Posición límite de equilibrio}
\end{tabular}
\end{center}
\end{block}}

\end{frame}




%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{La torre general: n libros sobre otro}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.32]{./images/torren.eps}
\end{tabular}
\end{center}
\end{block}}

\end{frame}




%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Foto real $n=4$}

\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.26]{./images/stack1.jpg}
\end{tabular}
\end{center}
\[
 \frac{1}{2}+
 \frac{1}{4}+
 \frac{1}{6}+
 \frac{1}{8}=1.041\dots >1
\]

\end{frame}




%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Foto real $n=4$}

\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.26]{./images/stack2.jpg}
\end{tabular}
\end{center}
\[
 \frac{1}{2}+
 \frac{1}{4}+
 \frac{1}{6}+
 \frac{1}{8}=1.041\dots >1
\]

\end{frame}


%  _____           _     _                        ___  
% |  __ \         | |   | |                      |__ \ 
% | |__) | __ ___ | |__ | | ___ _ __ ___   __ _     ) |
% |  ___/ '__/ _ \| '_ \| |/ _ \ '_ ` _ \ / _` |   / / 
% | |   | | | (_) | |_) | |  __/ | | | | | (_| |  / /_ 
% |_|   |_|  \___/|_.__/|_|\___|_| |_| |_|\__,_| |____|
                                                      

%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\section{Problema 2}
\begin{frame}
\frametitle{Problema 2}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\textbf{¿De dónde sale la relación entre el número de vértices, aristas y caras en los poliedros?} 
\end{block}}

\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.15]{./images/balon_w.png}
\end{tabular}
\begin{tabular}{c}
\includegraphics[scale=0.1]{./images/octaedro_w.jpg}
\end{tabular}
\begin{tabular}{c}
\includegraphics[scale=0.17]{./images/piramide_w.jpg}
\end{tabular}
\end{center}

\end{frame}

               


%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Tetraedro}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.28]{./images/tetraedro_w.jpg}
\end{tabular}
\end{center}
\end{block}}

\end{frame}



%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Cubo}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.25]{./images/cubo_w.jpg}
\end{tabular}
\end{center}
\end{block}}

\end{frame}


%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Cubo}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.25]{./images/octaedro_w.jpg}
\end{tabular}
\end{center}
\end{block}}

\end{frame}


%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Cubo}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.25]{./images/dodecaedro_w.jpg}
\end{tabular}
\end{center}
\end{block}}

\end{frame}


%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Cubo}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.25]{./images/icosaedro_w.jpg}
\end{tabular}
\end{center}
\end{block}}

\end{frame}


%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Tabla}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{l|c|c|c|}
& Vértices  & Caras & Aristas\\ \hline
Tetraedro &4&  4&  6 \\ \hline
Cubo &8&  6&  12 \\ \hline
Octaedro &6&  8&  12 \\ \hline
Dodecaedro &20&  12&  30 \\  \hline
Icosaedro &12&  20&  30  \\  \hline
Pirámide de Egipto &5&  5&  8 \\ \hline
Balón de fútbol &60&  32&  90 \\ \hline
\end{tabular}
\end{center}
\end{block}}

\end{frame}



%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Aplanar}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.47]{./images/cubo.eps}
\end{tabular}
\qquad$\longrightarrow$\qquad
\begin{tabular}{c}
\includegraphics[scale=0.25]{./images/cubop.eps}
\end{tabular}
\end{center}
\end{block}}

\end{frame}




%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Inundar}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.21]{./images/cubop.eps}
\end{tabular}
$\longrightarrow$
\begin{tabular}{c}
\includegraphics[scale=0.21]{./images/cubop2.eps}
\end{tabular}
$\longrightarrow$
\begin{tabular}{c}
\includegraphics[scale=0.21]{./images/cubop3.eps}
\end{tabular}
\end{center}
\end{block}}

\end{frame}



%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Quitar vértices}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\phantom{$\longrightarrow$}
\includegraphics[scale=0.23]{./images/cuboe1.eps}
\end{tabular}
$\longrightarrow$
\begin{tabular}{c}
\includegraphics[scale=0.23]{./images/cuboe2.eps}
\end{tabular}
\\
$\longrightarrow$
\begin{tabular}{c}
\includegraphics[scale=0.23]{./images/cuboe3.eps}
\end{tabular}
$\longrightarrow$
\begin{tabular}{c}
\includegraphics[scale=0.23]{./images/cuboe4.eps}
\end{tabular}
\end{center}
\begin{center}
Se van suprimiendo los vértices exteriores y sus aristas
\end{center}

\end{block}}

\end{frame}



%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Un contraejemplo}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.8]{./images/torop.eps}
\end{tabular}
\qquad
\begin{tabular}{c}
$V=16$, $C=16$, $A=32$
\end{tabular}

\

El fallo en la fórmula se debe a que hay un agujero
\end{center}
\end{block}}

\end{frame}



%  _____           _     _                        ____  
% |  __ \         | |   | |                      |___ \ 
% | |__) | __ ___ | |__ | | ___ _ __ ___   __ _    __) |
% |  ___/ '__/ _ \| '_ \| |/ _ \ '_ ` _ \ / _` |  |__ < 
% | |   | | | (_) | |_) | |  __/ | | | | | (_| |  ___) |
% |_|   |_|  \___/|_.__/|_|\___|_| |_| |_|\__,_| |____/ 
%                                                       
                                                       

%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\section{Problema 3}
\begin{frame}
\frametitle{Problema 3}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\small
\textbf{¿Cuándo un cuadrado y el doble de un cuadrado difieren en~uno?} 
\end{block}}


\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.36]{./images/prob3.jpg}
\end{tabular}
\end{center}

\end{frame}



%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Buscamos soluciones de $2a^2-b^2=\pm 1$}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\textbf{Valores de $a$}
\[
1 ,\  2 ,\  5 ,\  12 ,\  29 ,\  70 ,\  169 ,\  408 ,\  985 ,\  2378 ,\  5741 ,\  13860.
\]
\end{block}}

\

\begin{center}
\Large\bf
¿Qué regla de formación siguen?
\end{center}


\end{frame}




%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Tabla de  soluciones de $2a^2-b^2=\pm 1$}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$a$ &
   1 & 2 & 5 & 12 & 29 & 70 & 169 & 408 & 985 & 2378 & 5741
\\
\hline
$b$ &
1 & 3 & 7 & 17 & 41 & 99 & 239 & 577 & 1393 & 3363 & 8119 
\\
\hline
\end{tabular}
\end{center}
\end{block}}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\textbf{Bloques:}
\begin{center}
\begin{tabular}{|c|c|}
\hline
 1&2
 \\
 \hline
 1&3
 \\
 \hline
\end{tabular}
\quad
\begin{tabular}{|c|c|}
\hline
 2&5
 \\
 \hline
 3&7
 \\
 \hline
\end{tabular}
\quad
\begin{tabular}{|c|c|}
\hline
 5&12
 \\
 \hline
 7&17
 \\
 \hline
\end{tabular}
\quad
\begin{tabular}{|c|c|}
\hline
 12&29
 \\
 \hline
 17&41
 \\
 \hline
\end{tabular}
\quad
\begin{tabular}{|c|c|}
\hline
 29&70
 \\
 \hline
 41&99
 \\
 \hline
\end{tabular}
\end{center}
Parece que el bloque
\begin{center}
\begin{tabular}{|c|c|}
\hline
 $\alpha$&$\gamma$
 \\
 \hline
 $\beta$&$\delta$
 \\
 \hline
\end{tabular}
\quad
siempre cumple \pause \quad $\gamma=\alpha+\beta$ \quad  y \quad  $\delta=2\alpha+\beta$
\end{center}
\end{block}}


\end{frame}



%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Escribir con raíces}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
\begin{center}
\begin{tabular}{|c|c|}
\hline
 $\alpha$&$\gamma$
 \\
 \hline
 $\beta$&$\delta$
 \\
 \hline
\end{tabular}
\quad
siempre cumple  \quad $\gamma=\alpha+\beta$ \quad  y \quad  $\delta=2\alpha+\beta$
\end{center}
\end{block}}


{\setbeamercolor{block title}{bg=white}
\begin{block}{}
Las dos propiedades se escriben como una sola mediante
\[
 \gamma\sqrt{2}+\delta=(\alpha\sqrt{2}+\beta)(\sqrt{2}+1).
\]
Partiendo de la primera solución $a=1$, $b=1$, ¡se obtienen todas de este modo multiplicando por $\sqrt{2}+1$!
\begin{align*}
 2\sqrt{2}+3&= (1\sqrt{2}+1)(\sqrt{2}+1)=(\sqrt{2}+1)^2
\\
 5\sqrt{2}+7&= (2\sqrt{2}+3)(\sqrt{2}+1)=(\sqrt{2}+1)^3
 \\[-1pt]
 \dots\quad&\qquad\dots\qquad\dots\qquad\dots\qquad\dots
\\
 70\sqrt{2}+99&=  (\sqrt{2}+1)^6
\end{align*}
\end{block}}


\end{frame}




%%%%%%%%%%%%%%%%
% DIAPOSITIVA  %
%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Soluciones}

{\setbeamercolor{block title}{bg=white}
\begin{block}{}
La  soluciones de $2a^2-b^2=\pm 1$ cumplen
\[
 a\sqrt{2}+b = (\sqrt{2}+1)^n
\]
\end{block}}

Usando el conjugado $-a\sqrt{2}+b=  (-\sqrt{2}+1)^n$ y sumando y restando:

{\setbeamercolor{block title}{bg=blue}
\begin{block}{}
La  soluciones de $2a^2-b^2=\pm 1$ son:
\[
 a = \frac{(\sqrt{2}+1)^n-(-\sqrt{2}+1)^n}{2\sqrt{2}},
 \qquad
 b = \frac{(\sqrt{2}+1)^n+(-\sqrt{2}+1)^n}{2}
\]
\end{block}}

\end{frame}






\end{document}