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solucionProblemas.lyx
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#LyX 2.2 created this file. For more info see http://www.lyx.org/
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\begin_body
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
\SpecialChar LyX
source for textbook
\begin_inset Quotes eld
\end_inset
How to think like a computer scientist''
\end_layout
\begin_layout Plain Layout
Copyright (c) 2001 Allen B.
Downey, Jeffrey Elkner, and Chris Meyers.
\end_layout
\begin_layout Plain Layout
Permission is granted to copy, distribute and/or modify this document under
the terms of the GNU Free Documentation License, Version 1.1 or any later
version published by the Free Software Foundation; with the Invariant Sections
being "Contributor List", with no Front-Cover Texts, and with no Back-Cover
Texts.
A copy of the license is included in the section entitled "GNU Free Documentati
on License".
\end_layout
\begin_layout Plain Layout
This distribution includes a file named fdl.lyx that contains the text of
the GNU Free Documentation License.
If it is missing, you can obtain it from www.gnu.org or by writing to the
Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA
02111-1307, USA.
\end_layout
\end_inset
\end_layout
\begin_layout Chapter
Preludio: Solución de problemas
\end_layout
\begin_layout Standard
\begin_inset Index idx
status collapsed
\begin_layout Plain Layout
solución de problemas
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\end_inset
\end_layout
\begin_layout Section
Solución de acertijos
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\begin_layout Standard
\begin_inset Index idx
status collapsed
\begin_layout Plain Layout
acertijo
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Todos nos hemos topado con acertijos como el siguiente.
Disponga los dígitos del 1 al 9 en el recuadro siguiente, de manera que
la suma de cada fila, cada columna y las dos diagonales dé el mismo resultado:
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\begin_layout Standard
Este acertijo se denomina construcción de un
\series bold
cuadrado mágico.
\series default
Un acertijo, normalmente, es un enigma o adivinanza que se propone como
pasatiempo.
Otros ejemplos de acertijo son un crucigrama, una sopa de letras y un sudoku.
\end_layout
\begin_layout Standard
Los acertijos pueden tener varias soluciones, por ejemplo, la siguiente
es una solución propuesta al acertijo anterior:
\begin_inset Separator latexpar
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\begin_layout Standard
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1
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2
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3
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</cell>
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4
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5
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\begin_inset Text
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6
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<row>
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\begin_inset Text
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7
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<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
8
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<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
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9
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\begin_layout Standard
Usted puede notar que esta solución candidata no es correcta.
Si tomamos la suma por filas, obtenemos valores distintos:
\end_layout
\begin_layout Itemize
En la fila 1: 1+2+3=6
\end_layout
\begin_layout Itemize
En la fila 2: 4+5+6=15
\end_layout
\begin_layout Itemize
En la fila 3: 7+8+9=24
\end_layout
\begin_layout Standard
Si tomamos las columnas, tampoco obtenemos el mismo resultado en cada suma:
\end_layout
\begin_layout Itemize
En la columna 1: 1+4+7=12
\end_layout
\begin_layout Itemize
En la columna 2: 2+5+8=15
\end_layout
\begin_layout Itemize
En la columna 3: 3+6+9=18
\end_layout
\begin_layout Standard
A pesar de que las diagonales sí suman lo mismo:
\end_layout
\begin_layout Itemize
En la diagonal 1: 1+5+9=15
\end_layout
\begin_layout Itemize
En la diagonal 2: 7+5+3=15
\end_layout
\begin_layout Standard
Tómese un par de minutos para resolver este acertijo, es decir, para construir
un cuadrado mágico y regrese a la lectura cuando obtenga la solución.
\end_layout
\begin_layout Standard
Ahora, responda para sí mismo las siguientes preguntas:
\end_layout
\begin_layout Itemize
¿Cuál es la solución que encontró?
\end_layout
\begin_layout Itemize
¿Es correcta su solución?
\end_layout
\begin_layout Itemize
¿Cómo le demuestra a alguien que su solución es correcta?
\end_layout
\begin_layout Itemize
¿Cuál fue el proceso de solución que llevo a cabo en su mente?
\end_layout
\begin_layout Itemize
¿Cómo le explicaría a alguien el proceso de solución que llevó a cabo?
\end_layout
\begin_layout Itemize
¿Puede poner por escrito el proceso de solución que llevó a cabo?
\end_layout
\begin_layout Standard
El reflexionar seriamente sobre estas preguntas es muy importante, tenga
la seguridad de que esta actividad será muy importante para continuar con
la lectura.
\end_layout
\begin_layout Standard
Vamos a ir contestando las preguntas desde una solución particular, y desde
un proceso de solución particular, el de los autores.
Su solución y su proceso de solución son igualmente valiosos, el nuestro
solo es otra alternativa; es más, puede que hayamos descubierto la misma:
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4
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9
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2
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3
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5
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7
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8
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1
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\begin_layout Standard
Esta solución es correcta, porque la suma por filas, columnas y de las dos
diagonales da el mismo valor, 15.
Ahora, para demostrarle a alguien este hecho podemos revisar las sumas
por filas, columnas y diagonales detalladamente.
\end_layout
\begin_layout Itemize
En la fila 1: 4+9+2=15%
\end_layout
\begin_layout Itemize
En la fila 2: 3+5+7=15
\end_layout
\begin_layout Itemize
En la fila 3: 8+1+6=15
\end_layout
\begin_layout Itemize
En la columna 1: 4+3+8=15
\end_layout
\begin_layout Itemize
En la columna 2: 9+5+1=15
\end_layout
\begin_layout Itemize
En la columna 3: 2+7+6=15
\end_layout
\begin_layout Itemize
En la diagonal 1: 4+5+6=15
\end_layout
\begin_layout Itemize
En la diagonal 2: 8+5+2=15
\end_layout
\begin_layout Standard
El proceso de solución que llevamos a cabo fue el siguiente:
\end_layout
\begin_layout Itemize
Sospechábamos que el 5 debía estar en la casilla central, ya que es el número
medio de los 9: 1 2 3 4
\series bold
5
\series default
6 7 8 9.
\end_layout
\begin_layout Itemize
Observamos un patrón interesante de la primera solución propuesta: las diagonale
s sumaban igual, 15:
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\series bold
1
\series default
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\series bold
3
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
4
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\series bold
5
\series default
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
6
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\series bold
7
\series default
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
8
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\series bold
9
\series default
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\end_deeper
\begin_layout Itemize
La observación anterior, 1+5+9=7+5+3, también permite deducir otro hecho
interesante.
Como el 5 está en las dos sumas, podemos deducir que 1+9=7+3, y esto es
10.
\end_layout
\begin_layout Itemize
Una posible estrategia consiste en colocar parejas de números que sumen
10, dejando al 5
\begin_inset Quotes eld
\end_inset
emparedado
\begin_inset Quotes erd
\end_inset
, por ejemplo, poner la pareja 6,4:
\begin_inset Separator latexpar
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\series bold
6
\series default
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\series bold
5
\series default
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\series bold
4
\series default
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\end_deeper
\begin_layout Itemize
Para agilizar ésta estrategia es conveniente enumerar todas las parejas
de números entre 1 y 9 que suman 10, excluyendo al 5: (1,9),(2,8),(3,7),(4,6)
\end_layout
\begin_layout Itemize
Ahora, podemos probar colocando éstas parejas en filas, columnas y diagonales.
\end_layout
\begin_layout Itemize
Un primer ensayo:
\begin_inset Separator latexpar
\end_inset
\end_layout
\begin_deeper
\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="3" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
7
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\series bold
6
\series default
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
2
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\series bold
5
\series default
\end_layout
\end_inset
</cell>
<cell alignment="none" valignment="top" topline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout