%  Homework assignment 3
%  21-110: Problem Solving in Recreational Mathematics, spring 2010
%  Assigned: Friday, February 5, 2010. Due: Friday, February 12, 2010.
%  Brian Kell <bkell@cmu.edu>

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\centerline{\titlefont 21-110: Problem Solving in Recreational Mathematics}
\smallskip
\centerline{\bf Homework assignment~3}
\smallskip
\centerline{Assigned Friday, February~5, 2010. Due Friday, February~12, 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.

\bigbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem (``A Swimmer and a Hat,'' from {\it The Moscow Puzzles\/} by Boris~A.
Kordemsky, edited by Martin Gardner.)\enspace A boat is being carried away by a
current. A man jumps out and swims against the current for a while, then turns
around and catches up with the boat. Did he spend more time swimming against
the current or catching up with the boat? (We assume his muscular efforts never
change in strength.)

The answer is: Both times were the same. The current carries man and boat
downstream at the same speed. It does not affect the distance between the
swimmer and the boat.

Now imagine that a sportsman jumps off a bridge and begins to swim against the
current. The same moment a hat blows off a man's head on the bridge and begins
to float downstream. After 10~minutes the swimmer turns back, reaches the
bridge, and is asked to swim on until he catches up with the hat. He does,
under a second bridge 1,000~yards from the first.

The swimmer does not vary his effort. What is the speed of the current?

\hint Wkhuh lv d zdb wr vroyh wklv sureohp zlwk doprvw qr dojheud.

\bigbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem International paper sizes, standardized in ISO~216, are used in nearly
every country in the world except the United States and Canada. The largest
size in the so-called ``A~series'' is called~A0, and has an area of one square
meter. The next size is called~A1 and is formed by cutting a sheet of A0~paper
in half. Then A2~paper is formed by cutting a sheet of A1~paper in half, and so
on. This continues through the smallest size,~A10. Additionally, the various
sizes are defined so that all of the paper sizes are similar (that~is, they
have the same shape, but different sizes). This makes it easy to scale
documents from one size of paper to another. What size of paper corresponds
most closely to the ``letter'' size paper common in the United States
($8{1\over2}$~inches by 11~inches)? What are its dimensions, rounded to the
nearest millimeter? [There are 25.4~millimeters in one inch, and
1000~millimeters in one meter.]

\hint Gudz d slfwxuh. Wklqn derxw wkh {\it sursruwlrqv\/} ri wkh sdshu. Zkdw
grhv lw phdq iru wzr uhfwdqjohv wr eh vlplodu?

\bigbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem (Exercise~3.27 from {\it Problem Solving Through Recreational
Mathematics}.)\enspace When Erica was two years old, Leroy was four times as
old as Miriam. When Miriam was twice as old as Erica, Leroy was three times as
old as Miriam. How old was Erica when Leroy was twice as old as Miriam?

\hint Wklv sureohp uhihuv wr wkuhh gliihuhqw srlqwv lq wlph. Lw khoshg ph wr
gudz d wlpholqh zlwk wkuhh srlqwv pdunhg vr L frxog odeho zkdw zdv wuxh dqg
zkhq. Li brx lqwurgxfh yduldeohv, eh vxuh wr fodulib hadfwob zkdw brx lqwhqg
iru wkhp wr phdq.

\bigbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem (Exercise~3.19 from {\it Problem Solving Through Recreational
Mathematics}.)\enspace The silver currency of the Kingdom of Bonoria consists
of glomeks, nindars, and morms. Four glomeks are equal in value to seven
nindars; and one glomek and one nindar together are worth thirty-three morms.

On my last visit to Bonoria, I entered a bank, handed the teller some glomeks
and nindars, and asked him to change them into morms.

``Do you think that I am a magician?'' he replied. (Bonorians are noted for
their warped sense of humor.) ``Well, let's see,'' he continued. ``If you had
twice as many glomeks, I could give you 120~morms; and if you had twice as many
nindars I could give you 114~morms.''

How many morms did he give me?

\hint Lw pljkw eh khosixo iluvw wr iljxuh rxw krz pdqb prupv d jorphn lv zruwk,
dqg krz pdqb prupv d qlqgdu lv zruwk. Brx pdb kdyh pdqb gliihuhqw yduldeohv lq
brxu vhwxs; zulwh grzq suhflvhob zkdw hdfk ri wkhp vwdqgv iru. Wkh ``D'' eb wkh
sureohp qxpehu lq wkh errn lqglfdwhv wkdw wkh dqvzhu (exw qrw wkh vroxwlrq) fdq
eh irxqg lq wkh dqvzhuv vhfwlrq ehjlqqlqj rq sdjh~433.

\bigbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem (Exercise~3.63 from {\it Problem Solving Through Recreational
Mathematics\/}; originally from the {\it Greek Anthology}, compiled about
{\smfont A.D.}~500 by Metrodorus.)\enspace I am a brazen lion, a fountain; my
spouts are my two eyes, my mouth, and the flat of my right foot. My right eye
fills a jar in two days [$\hbox{1~day}=\hbox{12~hours}$], my left eye in three,
and my foot in four; my mouth is capable of filling it in six hours. Tell me
how long all four together will take to fill it.

\hint Wklv lv vlplodu wr Vdpsoh Sureohp~3.5.

\bigbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem There is a unique real number~$x$ that can be expressed in the
following form:
$$x=1+{1\over\displaystyle1+{\strut1\over\displaystyle1+
	{\strut1\over\strut1+\cdots\rlap{\thinspace,}}}}$$
where the dots~``$\,\cdots\,$'' mean ``and so on, forever.'' What is this
number~$x$? (This strange-looking kind of infinite fraction is called a
``continued fraction.'')

\hint Zkdw lv rqh soxv wkh uhflsurfdo ri~$a$? Qrwh wkdw $a$ pxvw eh srvlwlyh,
ehfdxvh hyhubwklqj wkdw lv ehlqj dgghg lv srvlwlyh.

\bigbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem In the figures below, there are $n$~points on the circumference of a
circle, and a chord is drawn between every pair of points. This divides the
circle into a number of regions. The points are chosen in such a way that no
three of the chords intersect in a single point, so that the number of regions
is maximized. How many regions would be formed if 20~points were chosen around
the circumference in this way? ({\it Be careful\/}---the ``obvious'' pattern
in the number of regions in the examples below does {\it not\/} hold in
general! You will need to count the number of regions for $n=6$, and probably
$n=7$, in order to find the general pattern.)
$$\vbox{\halign{&\quad\strut\hfil#\hfil\quad\cr
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	\midfig{homework-3-two-points.mps}&
	\midfig{homework-3-three-points.mps}&
	\midfig{homework-3-four-points.mps}&
	\midfig{homework-3-five-points.mps}\cr
	$n=1$&$n=2$&$n=3$&$n=4$&$n=5$\cr
	1~region&2~regions&4~regions&8~regions&16~regions\cr
}}$$

\hint Ilqg d srobqrpldo irupxod iru wkh qxpehu ri uhjlrqv lq whupv ri~$q$. Wub
wkh phwkrg zh hasoruhg lq fodvv, lq zklfk zh orrnhg dw wkh gliihuhqfhv ehwzhhq
vxffhvvlyh qxpehuv lq d vhtxhqfh.

\bigbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem (From {\it Challenging Problems in Algebra\/} by Alfred~S. Posamentier
and Charles~T. Salkind.)\enspace A shopkeeper orders 19~large and 3~small
packets of marbles, all alike. When they arrive at the shop, he finds the
packets broken open with all the marbles loose in the container. Can you help
the shopkeeper make new packets with the proper number of marbles in each, if
the total number of marbles is~224?

\hint Wklqn derxw zkdw nlqg ri qxpehuv zrxog pdnh vhqvh iru wkh vroxwlrq lq wkh
frqwhaw ri wkh sureohp.

\bigbreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem (From a short story, ``Coconuts,'' by Ben Ames Williams.)\enspace
$\ldots$~So at last Wadlin told him. ``Well,'' he explained, ``according to the
way the thing was given to me, five men and a monkey were shipwrecked on a
desert island, and they spent the first day gathering coconuts for food. Piled
them all up together and then went to sleep for the night.

``But when they were all asleep one man woke up, and he thought there might be
a row about dividing the coconuts in the morning, so he decided to take his
share. So he divided the coconuts into five piles. He had one coconut left
over, and he gave that to the monkey, and he hid his pile and put the rest all
back together.''

He looked at Marr; the man was listening attentively.

``So by and by the next man woke up and did the same thing,'' Wadlin continued.
``And he had one left over, and he gave it to the monkey. And all five of the
men did the same thing, one after the other, each one taking a fifth of the
coconuts in the pile when he woke up, and each one having one left over for the
monkey. And in the morning they divided what coconuts were left, and they came
out in five equal shares.''

He added morosely, ``Of course each one must have known there were coconuts
missing; but each one was guilty as the others, so they didn't say anything.''

Marr asked sharply, ``But what's the question?''

``How many coconuts were there in the beginning?'' Wadlin meekly explained.

\hint Wklv lv d kdug sxccoh! Grq'w vshqg wrr pxfk wlph rq lw xqohvv brx kdyh
douhdgb zrunhg rxw irxu ri wkh rwkhu sureohpv. Wkhuh duh dfwxdoob lqilqlwhob
pdqb vroxwlrqv; wkh txhvwlrq lv suredeob dvnlqj iru wkh vpdoohvw srvlwlyh
qxpehu ri frfrqxwv wkdw zrxog vdwlvib wkh frqglwlrqv.

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\bye
