%  Homework assignment 1
%  21-110: Problem Solving in Recreational Mathematics, spring 2010
%  Assigned: Wednesday, January 13, 2010. Due: Wednesday, January 20, 2010.
%  Brian Kell <bkell@cmu.edu>

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\centerline{\titlefont 21-110: Problem Solving in Recreational Mathematics}
\smallskip
\centerline{\bf Homework assignment~1}
\smallskip
\centerline{Assigned Wednesday, January~13, 2010. Due Wednesday, January~20, 2010.}

\bigskip

\noindent Work at least {\bf FOUR} of the following problems. All problems are
of equal weight. If you submit solutions for more than four problems, you will
get credit for your best four.

You are welcome to work with other students, but the solutions you hand~in
should be written in your own words. You are not allowed to see the paper
another student is going to hand~in. If you do collaborate with other students,
list their names. If you use other sources, cite them. Give credit where credit
is due. See the syllabus for more information about academic integrity.

Hints are encrypted with a {\it Caesar cipher}, in which each letter is
replaced by the letter three places ahead in the alphabet, wrapping around to
the beginning if necessary. For example, the letter~{\it A\/} is encrypted
as~{\it D}, and {\it Y\/}~is encrypted as~{\it B}. To decrypt the hints, move
each letter backward three places.

\bigskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem Find my office. Sign your name on the sheet posted outside my office
door.

\hint Vhh wkh vboodexv.

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem ``How many children do you have, and what are their ages?'' asks the
census taker.

The mother answers, ``I have three children. The product of their ages is~36,
and the sum of their ages is the same as my house number.''

The census taker looks at the house number, thinks for a moment, and responds,
``I'm sorry, but I need more information.''

``My oldest child likes chocolate ice cream,'' says the mother.

``Thank you,'' replies the census taker. ``I have all the information I
require.''

How old are the children?

\hint Zkb grhv wkh fhqvxv wdnhu qhhg pruh lqirupdwlrq, dqg lq zkdw zdb lv wkh
prwkhu'v odvw vwdwhphqw khosixo?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem (``Ins and Outs,'' {\it Thinking Mathematically}, page~175.)\enspace
Take a strip of paper and fold it in half several times in the same fashion as
in {\it Paper Strip\/} (page~4). Unfold it and observe that some of the creases
are IN and some are OUT. For example, three folds produce the sequence
$$\rm in\quad in\quad out\quad in\quad in\quad out\quad out$$
What sequence would arise from 10~folds (if that many were possible)?

\hint Lw pdb eh ehwwhu wr ghvfuleh wkh dqvzhu wkurxjk d sdwwhuq wkdq wr zulwh
lw doo rxw hasolflwob. Wkh errn kdv vrph vxjjhvwlrqv li brx duh vwxfn.

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem (Exercise~1.5 from {\it Problem Solving Through Recreational
Mathematics}, page~27.)\enspace Six players---Pietrovich, Cavelli, St.~Jacques,
Smith, Lord Bottomly, and Fernandez---are competing in a chess tournament over
a period of five days. Each player plays each of the others once. Three matches
are played simultaneously during each of the five days. The first day, Cavelli
beat Pietrovich after 36~moves. The second day, Cavelli was again victorious
when St.~Jacques failed to complete 40~moves within the required time limit.
The third day had the most exciting match of all when St.~Jacques declared that
he would checkmate Lord Bottomly in 8~moves and succeeded in doing so. On the
fourth day, Pietrovich defeated Smith.

Who played against Fernandez on the fifth day?

\hint Wkh ``K'' qhaw wr wkh sureohp lq wkh errn lqglfdwhv wkdw wkhuh lv d klqw
lq wkh Klqwv dqg Vroxwlrqv vhfwlrq rq sdjh~380.

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem A sculpture in an art gallery consists of a wooden cube suspended from
the ceiling by a thin wire. The wire is attached to the cube at one of its
corners, so that the cube hangs at an angle. A fly lands on the top corner of
the cube (the one at which the wire is attached) at 12:00~noon. Every five
minutes thereafter (at 12:05,~12:10, 12:15, and so on) the fly moves along one
of the edges of the cube to reach a neighboring corner. The path of the fly
around the cube is random (subject to the restriction that it can only move
along edges of the cube). What is the probability that the fly will be at the
bottom corner of the cube (the corner nearest the floor) at~12:31?

\hint Krz pdqb wlphv zloo wkh iob kdyh pryhg eb wkhq? Wub frorulqj wkh fruqhuv
ri wkh fxeh vrphkrz.

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem Three guests check into a hotel and ask what the nightly rate is. The
clerk says a room costs 30~dollars a night, so each guest gives the clerk
ten dollars, and they head up to the room.

A while later, the clerk realizes he overcharged the guests; the room they are
staying in is only 25~dollars a night. So he takes five one-dollar bills from
the cash box and hands them to the bellhop with instructions to return the
money.

On the way up to the room, the bellhop realizes that five dollars cannot be
split evenly among three guests, so he pockets two dollars and returns three
dollars to the guests.

Now, each of the guests initially paid ten dollars for the room, but later
received a dollar back, so effectively each guest paid nine dollars. In total,
then, the room cost the guests 27~dollars. With the two dollars the bellhop
kept, this comes to 29~dollars. What happened to the missing dollar?

\hint Vrphwklqj ilvkb lv jrlqj rq khuh. Lv wkhuh uhdoob d groodu plvvlqj?

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem (Note: The two parts of this question are not meant to be related to
each other, apart from the fact that they are both about arranging coins.)

\nobreak\medskip

\item{(a)} Place 10~coins in five straight lines so that each line contains
exactly four coins.

\nobreak\medskip

\item{(b)} Can you arrange four coins so that if you choose {\it any\/} three
of them (i.e., no matter which three of the four you pick), the three coins
you chose form the corners of an equilateral triangle?

\itemhint Zdwfk rxw iru klgghq dvvxpswlrqv brx pdb eh pdnlqj. Wklqn rxwvlgh wkh
era.

\medbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem (From the article ``Number Games and Other Mathematical Recreations''
in the 15th~edition of the Encyclop\ae dia Britannica.)\enspace Three travelers
were aboard a train that had just emerged from a tunnel, leaving a smudge of
soot on the forehead of each. While they were laughing at each other, and
before they could look into a mirror, a neighboring passenger suggested that
although no one of the three knew whether he himself was smudged, there was a
way of finding out without using a mirror.

He suggested: ``Each of the three of you look at the other two; if you see at
least one whose forehead is smudged, raise your hand.'' Each raised his hand at
once. ``Now,'' said the neighbor, ``as soon as one of you knows for sure
whether his own forehead is smudged or not, he should drop his hand, but not
before.''

After a moment or two, one of the men dropped his hand with a smile of
satisfaction, saying: ``I know.''

How did that man know that his forehead was smudged?

\hint Lpdjlqh brx duh rqh ri wkh phq; zkdw gr brx vhh? Zkdw gr brx nqrz derxw
zkdw wkh rwkhu phq vhh? Wkh sdxvh ehiruh wkh pdq dqvzhuv lv lpsruwdqw. Li brx
duh vwxfn, pdnh dq dvvxpswlrq derxw brxu rzq iruhkhdg dqg vhh zkhuh wkdw ohdgv
brx.

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\bye
