\documentclass[12pt]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{geometry}
%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Version=5.00.0.2570}
%TCIDATA{}
%TCIDATA{Created=Wednesday, September 21, 2005 19:23:38}
%TCIDATA{LastRevised=Friday, June 01, 2007 06:49:10}
%TCIDATA{}
%TCIDATA{}
%TCIDATA{CSTFile=40 LaTeX article.cst}
%TCIDATA{ComputeDefs=
%$f\left( -10\right) +f\left( 15\right) =\left( 12\left( -10\right)
%-42\right) =\allowbreak -162$
%}
\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\geometry{left=0.6in,right=0.6in,top=0.6in,bottom=0.6in}
\input{tcilatex}
\begin{document}
\begin{center}
\bigskip {\Large Puzzles}
\end{center}
\begin{enumerate}
\item Two mathematicians are having a conversation. Mathematician A asks B
about his kids. B answers: "I have three children, the product of their ages
is 36." \ A says: "I still don't know how old your children are." \ Then B
tells A the sum of his three kids' ages. A answers: "I still don't know how
old they are. Then B adds: "The youngest one has red hair." Now A knows how
old the kids are. Do you?
\item Consider a chess board with two corners missing, as indicated on the
picture below. We also have $31$ pieces of domino, each of them can cover
exactly $2$ fields on the chess board. Is it possible to cover the
chessboard with the domino pieces?\FRAME{dtbpF}{1.727in}{1.2073in}{0pt}{}{}{%
insert.bmp}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
1.727in;height 1.2073in;depth 0pt;original-width 1.6994in;original-height
1.1796in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'insert.bmp';file-properties "XNPEU";}}
\item A king has his birthday. So he decides to let go some of his
prisoners. He actually has 100 prisoners at the moment. They are each in a
separate cell, numbered from 1 to 100. Well, he is a high tech king. He can
close or open any prison door by a single click on the cell's number on his
royal laptop. When he clicks at a locked door, it opens. When he clicks at
an open door, it locks. \ At the beginning, every door is locked. \ First
the king clicks on every number from 1 to 100 (therefore opening every
door). \ Then he clicks on every second number from 1 to 100, (i.e.2, 4, 6,
8, 10, . . . ). \ Then he clicks on every third number.(i.e. 3, 6, 9, 12, .
. . ) \ Now he is opening some doors, locking others. \ Then he clicks on
every fourth number. (i.e. 4, 8, 12, 16, \ . . . .) \ Then on every fifth,
every sixth, every seventh, and so on, until every 100th; finally, he only
clicks on the number 100. Then he orders that the prisoners that find their
door open may go free. \ Who gets to go and who has to stay?
\item There is a 5x5 board as the picture below shows. So happens, on each
one of the fields there is a ladybug sitting. Suddenly, each decides to move
to a neighboring field. \ (Two fields are neighbors if they have an edge in
common.) Is it possible that after each have moved there is again exactly
one lady bug sitting on each field?\FRAME{dtbpF}{0.7178in}{0.7178in}{0pt}{}{%
}{insert3.bmp}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
0.7178in;height 0.7178in;depth 0pt;original-width 0.6867in;original-height
0.6867in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'insert3.bmp';file-properties "XNPEU";}}
\item Mr and Mrs Brown are having a party. They invited three other married
couples, so there are eight people present. When greeting each other, some
people shake hands with some people. Of course, nobody shakes hands with
his/her spouse. \ When Mr Brown asks the other seven people: "How many
people did you shake hands with?", he receives seven different answers. How
many people did Mrs. Brown shake hands with?\pagebreak
\item The picture below shows a rectangle (the sides' length are $2$ and $3$
unit long) and four identical squares (all four sides are $1$ unit long).
Determine which area is greater: the yellow or the blue?\FRAME{dtbpF}{%
1.3439in}{1.19in}{0pt}{}{}{insert6.bmp}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file
"F";width 1.3439in;height 1.19in;depth 0pt;original-width
0.9669in;original-height 0.8536in;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'insert6.bmp';file-properties "XNPEU";}}
\item We are at a cross road. One road leads into a dangerous swamp, the
other road leads into a town. There is a pair of identical twins on the
crossing. We know that one twin always tells the truth, the other twin
always lies. We are allowed to ask only one question from only one brother.
Is there a way to find out which road leads to town?
\item Mindreading. Instructions:
Step 1. Think of a 5-6 digit number, that has at least two different digits
in it. My example is $803225$.
Step 2. \ Create a second number by rearranging the digits of the previous
number. My example is $320258$.
Step 3. \ Subtract the smaller number from the larger number. My example is $%
803225-320258=\allowbreak 482\,967$
Step 4. \ Cross out any non-zero digit of the difference. My example is $%
482\,9\NEG{6}7$.
Step 5. \ Announce the number you obtain by omitting the crossed out digit.
My example is $482\,97$.
If you tell the mindreader the last number, s/he can tell what digit you've
crossed out. How?
\item (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%
}{3.1064in}{3.1851in}{0pt}{}{}{insert10a.bmp}{\special{language "Scientific
Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 3.1064in;height
3.1851in;depth 0pt;original-width 3.1142in;original-height 3.6149in;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'insert10a.bmp';file-properties "XNPEU";}}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}{3.1419in}{2.8323in}{0pt}{}{}{insert10b.bmp}{%
\special{language "Scientific Word";type "GRAPHIC";display
"USEDEF";valid_file "F";width 3.1419in;height 2.8323in;depth
0pt;original-width 3.6876in;original-height 3.8856in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'insert10b.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}{2.8764in}{2.4154in}{0pt}{}{}{insert10c.bmp}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "USEDEF";valid_file "F";width 2.8764in;height 2.4154in;depth
0pt;original-width 2.8331in;original-height 2.3748in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'insert10c.bmp';file-properties
"XNPEU";}}
\end{enumerate}
The area of this figure is $7\left( 9\right) -4=\allowbreak 59$ unit$^{2}$.
\ 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
\bigskip
\end{document}