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%TCIDATA{Created=Tuesday, September 21, 2004 06:51:08}
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\begin{document}


\begin{center}
{\Large Puzzle 10}

\bigskip
\end{center}

(Paul Curry) Imagine that we cut the figure below out of paper. The area of
the triangle is $A=\dfrac{10\cdot 12}{2}=60$ unit$^{2}$.\FRAME{dtbpF}{%
4.3301in}{3.851in}{0pt}{}{}{insert1.bmp}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file
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4.3024in;original-height 3.8233in;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'insert1.bmp';file-properties "XNPEU";}}

\bigskip

Suppose we color the other side of the paper, turn the pieces upside down,
and rearrange them to obtain the figure shown below.\FRAME{dtbpF}{3.7152in}{%
3.9133in}{0pt}{}{}{insert2.bmp}{\special{language "Scientific Word";type
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3.8856in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'insert2.bmp';file-properties "XNPEU";}}

Now the area appears to be $58$ unit$^{2}$ since there is a $2$ unit$^{2}$
area of a whole that the triangle developed. To make matters worse, we now
again rearrange the pieces, turning some of them back to the original side. 
\FRAME{dtbpF}{2.8608in}{2.4025in}{0pt}{}{}{insert3.bmp}{\special{language
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0pt;original-width 2.8331in;original-height 2.3748in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'insert3.bmp';file-properties
"XNPEU";}}

The area of this figure is $7\left( 9\right) -4=\allowbreak 59$ unit$^{2}$.

\bigskip

Logic tells us that areas of figures do not change if we rearrange them or
turn them on their other side. So, what is wrong with these pictures, and
how much is really the are of these shapes, $58$ unit$^{2}$, $59$ unit$^{2}$%
, or $60$ unit$^{2}$?

\bigskip 

Solution is:

\qquad 

\end{document}