\hfill \hfill } %} \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}} \input{tcilatex} \geometry{left=0.6in,right=0.7in,top=0.7in,bottom=0.9in} \pagestyle{fancy} \lhead{\color{blue} \Large } \lfoot{\small Last revised: September 3, 2010} \rfoot{Puzzles - page \thepage} \cfoot{} \chead{\LARGE Puzzles} \rhead{\large } \textwidth 7.3in \textheight 9.3in \setlength{\headheight}{30pt} \setlength{\parindent}{0pt} \begin{document} \textit{These problems are mostly from my high school math teacher, \textbf{% Prof. Lajos Posa} who is a well known mathematician/math teacher in Budapest. It is difficult to find words that describe his talent and dedication. So I just let the problems speak for themselves.\bigskip } \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 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 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}{2.3315in}{1.6302in}{0pt}{}{}{% insert2.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.3315in;height 1.6302in;depth 0pt;original-width 1.6994in;original-height 1.1796in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert2.bmp';file-properties "XNPEU";}} \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}{1.2695in}{1.2695in}{0pt}{}{% }{insert3.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.2695in;height 1.2695in;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? \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. And so on. 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 Mind-reading. \ Instructions: 1) \ Think of a 5-6 digit number, that has at least two different digits in it. My example is $803225$.\newline 2) \ Create a second number by rearranging the digits of the previous number. My example is $320258$.\newline 3) \ Subtract the smaller number from the larger number. My example is $% 803225-320258=\allowbreak 482\,967$\newline 4) \ Cross out one non-zero digit of the difference. My example is $482\,9% \NEG{6}7$.\newline 5) \ Announce the number you obtain by omitting the crossed out digit. My example is $482\,97$. The mindreader can tell what digit was crossed out. How? \item The picture below shows a rectangle (the sides' length are 2 and 3 unit long) and four identical squares (sides' length are 1). Determine which area is greater: the yellow or the blue?\FRAME{dtbpF}{2.2139in}{1.6968in}{0pt% }{}{}{insert1.bmp}{\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 2.2139in;height 1.6968in;depth 0pt;original-width 0.9669in;original-height 0.8536in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert1.bmp';file-properties "XNPEU";}} \item (This is a well known topology problem, and I heard it from my high school student, Eric.) Connect three buildings with the three utilities shown on the picture below. Each building has to be connected to all three utilities, in two dimension, and no two utilities pipe can cross each other.% \FRAME{dtbpF}{2.8694in}{1.4555in}{0pt}{}{}{insert4.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.8694in;height 1.4555in;depth 0pt;original-width 2.8262in;original-height 1.42in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert4.bmp';file-properties "XNPEU";}} \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% }{2.2113in}{2.5624in}{0pt}{}{}{insert6.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.2113in;height 2.5624in;depth 0pt;original-width 3.1142in;original-height 3.6149in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert6.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}{2.4336in}{% 2.5616in}{0pt}{}{}{insert7.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.4336in;height 2.5616in;depth 0pt;original-width 3.6876in;original-height 3.8856in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert7.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}{2.175in}{1.8265in}{0pt}{% }{}{insert8.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.175in;height 1.8265in;depth 0pt;original-width 2.8331in;original-height 2.3748in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert8.bmp';file-properties "XNPEU";}}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$^{2}$? \item A bus makes a roundtrip between towns A and B. From A to B, the bus completes the trip with an average speed of $60$ $\dfrac{\text{mi}}{\text{hr}% }$. From B to A, it travels with an average velocity of $40$ $\dfrac{\text{mi% }}{\text{hr}}$. What is the average speed of the bus for the entire roundtrip? (Hint: it is NOT the average of $40$ and $60$.) \item There is a very difficult track in a mountain area, where the world record of driving a lap is $15$ miles per hour average velocity. A race car driver announces that he intends to beat this record and run the course with an average velocity of $30$ miles per hour. As he is driving, his progress is monitored. At the half of the track, his average velocity was $15$ miles per hour. In spite of this, he arrives at the finish just a few seconds too late to have an average velocity of $30$ miles per hour. His funeral was three days later. How did he die? \item This problem was told me by my student, Ramon Gozales. \ The triangle below has angles $50^{\circ }$, $80^{\circ }$, and $80^{\circ }$ as shown on the picture below.\FRAME{dtbpF}{2.8971in}{1.7236in}{0pt}{}{}{insert5.bmp}{% \special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.8971in;height 1.7236in;depth 0pt;original-width 2.853in;original-height 1.6864in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert5.bmp';file-properties "XNPEU";}}We measure up $10^{\circ }$ and $30^{\circ }$ from the base as shown on the picture below. Find the measure of angle shown on the picture.% \FRAME{dtbpF}{3.3425in}{1.9657in}{0pt}{}{}{insert9.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 3.3425in;height 1.9657in;depth 0pt;original-width 3.5665in;original-height 2.0868in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert9.bmp';file-properties "XNPEU";}} \end{enumerate} \bigskip \bigskip Enjoy! If you have any questions, you can ask me at mhidegkuti@ccc.edu \end{document}