Quantum Volume 1 Issue 2

68 Pages·078.116 MB·English

by  National Science Teachers Association

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Siberian Dogs in the Snow |Lg}g-l9lllbyEranzMarc
fff hen we decided to go to exffemes in this issue oI Quantum (see A.L. Rosenthal's article on page 8), all kinds of extremes sprang
W to mind. As we enter the cold season in the Northem Hemisphere/ extremes of temperature were at the forefront. And when
we're almost blinded by a field of snow, we perceive white as one extreme of a continuum of color (whether we think of it as all colors
combined or removed, depending on the medium). Franz Marc {1880-1916) touches on these extremes in his expressive portrait of
his sheepdog Russi, seen from two angles.
Marc spent much o{ his brief career painting animals. (His life was cut short near Verdun during World War I.) He developed a
profound nature mysticism that, combined with an tuge toward abstraction and a symbolism of colors, tended to produce intensely
colored canvases o{ animal and vegetable life. Marc believed this was the best way to express the conflicts and resolutions of natural
forces that civilization shields from us or teaches us to ignore.
Some elements of his later style are absent from Siberian Dogs in the Snow-mlost obviously, the color symbolism. At this point
in his development Marc was more concemed with the interaction o{ color and light. h a letter to a {ellow artist, Marc described
how the painting arose out of an experiment in the use of a prism to clao.lfu tonal relationships.
UA
O TU
NOVEMBER/DECEMBER 1 990 VOLUME 1, NUMB=R2
FEATURES
4
Cutting-Edge Physics
Tomaltawk fit'owing lnade sa$y
byV. A. Davydov
Math to the Max
Eoing to erFelnss
byA L Rosenthal
12
Front-Line Physics
LiUlr[tinU in a cl'ysNal
byYuryR Nosov
2l
For the Scottish engineer and instrument Math to the Min
maker fames Watt, the 1780s were a very
productive decade. Fifteen years earlier, while tulafting fie cl'oofted straiglrt
working on a Newcomen steam engine, he by Yury Solovyov
greatly improved its efficiency by adding a
separate condenser chamber. But in l78l a
business partner urged him to invent a rotary
DEPARTMENTS
steam engine for use in com, malt/ and cotton
mills, and Watt went to work. In that vear he
devised the sun-and-planet gear, which al- 2 0
lowed a shaft to produce two revolutions for Publishel''s Pa0e 0 lookinu Baolr
ptehuaecll ehd dos tuarosb klwee-e aolclf tatihns egp ueesnnhggeiinndee., . T iInhn i sw1 7he8nic2gh ihn eeth perea qpteuinsirtteeoddn ll7 0Bul.aaninlluemas e$mrsiles 42 AGet ntefiaelo Bgilcaafil[ otahlr'edes
a new method of rigidly connecting the pis- An incident on the train
ton, engaged in linear motioq to another part, Physics for fools 48 ln
uepng wagitehd jtnh er oretaqruyi rmedo tliionne.a rSizoi ning 1d7e8v4i chee. cWamatet 24 llow Do You FiUul'e? WhYy oaurt'c l ltehaed cheese holes
considered this "one of the most ingenious, 20 Gettinu to l(now... roundl
simple pieces of mechanism I have contrived,, The natural logarithm 51 lla[penings
aSntrda iigt'hs t"t hoen s puabgiee c2t 0o. f I"nM 1 a7k88in hge tahded eCdr oao ckeend- 30 Itllalfiematical $ul'Irises llth AtHouSmMaE.m-AenIMt oEf- towns
trifugal govemor to automatically control the Play it again USAMO-IMO... Bulletin
speed of the engine, and with his invention of 32
l(aleidoscole Board
the pressure gauge in 17 9O, theW att engine 5l
was all but ready to make its dramatic contri- What's new in the solar Solutions
bution to the Industrial Revolution. system!
04
For a look at a cleaner, quieter device at the 34 Conlest [hecImate!
forefront of modem technology, tum to ,,Liglrt- Rook versus knight
ning in a Cryst al" onpage 12. Shapes and sizes ...
Neutrinos and
supernovas
0lJAirrlJltl/[0ilrilrtrs
PUBLISHER'S PAGE
llloles ul a lrauellel,
And news of a partnership
FtrSE ARE EXCITING TIMES we live rr, and more of that nurture highly talented students, wherever they canbe formd; reform-
us than ever before are finding ourselves on planes ers seek to "raise the water table" by improving mathematics education Ior
heading to or from Mosc ow. Quantum staff will be everybody.
In one important area, however, there is a striking contrast between the
spending a week in the Soviet caprtal, planning future US practice and the Soviet tradition: testing. US students go through
issues and working to improve the logistics of ourbihem- sixteen years o{ short-answer, multiple-choice tests in mathematics,
ispheric production. beginning with number facts in prirnary school and continuing right
One recent visitor to the USSR was Lynn Arthur Steen, through a multiple-choice Graduate Remrd Exam administered to college
who teaches mathematics at St. Olaf College in Minne- seniors. Lr the USS\ mathematics tests are often given in oral or written
(essay) form, smulating the type oI environment in which mathematical
sota and sewes as a member of the Mathematical Sciences ideas are used in &e working world.
Education Board of the National Research Council. Dur- Bite-sized test items eviscerate education as surely as TV sound bites
inghis week-long stay he investigated the Soviet approach trivialize politics. In contrast, open-ended tests requiring holistic re-
to math education. What follows are Professor Steen's sponses encurage higher-order thinking and creative problem solving.
Students in the USSR leam from their experience with school tests to
impressions and thoughts about math and science educa-
think before answering. US students instead train {or rapid response, leam-
tion in the two countries, which both students and ing how to take tests rather than how to solve problems. [r Soviet schools
teachers will no doubt find of interest. tests are used as an intrinsic part o{ the curriculum, and the teacher's re-
sponses focus on each individual student in order to prevent failure.
In the past few years Americans have learned quite a bit about the The mathematics curriculum in the USSR is, for the most paft, more
Sorriet Union. We know that their economy is deteriorating, national stri{e forrnal and traditional than that becoming common irr the United States.
is increasing and their mfitary empire is crumbling. We can also see that The mathematical tools of academia predominate; those oI the state or
the USSR is seeking to integrate itself into the world economy and expand
business ({or instance, statistics, discrete mathematics) are almost invis-
contacts in all areas. ible. So in this respect US schools appear better attuned to the real needs o{
One thing America did not learn from news coverage o{ recent US-
sociery.
Soviet relations, however, is how the USSR has managed to produce so But we have a lot to learn from the USSR in the area oI testing. Tests
many talented mathematicians and scientists, who shocked us with should be part oI the ctmiculum-an opportuniry to leam and be taught-
Spumik and continue to impress us in oll,mpiads and scientfic exchanges. not separate from it. They should enable students to reveal what they can
The answer lies in one of the Soviet Union's best-kqlt secrets: a system of dq oot merely what they don't know or can't quickly recall. ff we are to be
mathematics education that produced a radition of excellence in research number one in mathematics and science, as President Bush has urged we
that is as good as any produced in Westem countries. need tests that measure what's important, not iust what's easy and cheap
Even as Gorbachev was touring the United States this past spring, a to grade.
small delegation of US mathematicians visited Moscow at the invitation of
Yevgeny Velikhov of the Soviet Academy of Sciences to explore means of As part of NSTA's efforts to reform the scopg sequence/
cooperation in mathematics education. The invitation was especially and coordination of secondary science education in this
timely, since math and science education in the United States is currendy count4/, we are developing a prototype interactive digital
under siege.
Many parallels between mathematics education in the Soviet Union video disk teaching system for high-level ability assess-
and in the US can be seeq but the differences are more striking. The US can ment. Rather than requiring students to recall isolated
leam much from both the similarities and the differences. facts about phenomena, this exciting technology will
)ust as President Bush has laid out national goals for mathematics and allow measurement of a student's understanding of scien-
science education for the United States, so Gorbachev has established a tific concepts. The interactive optical disk may prove to be
commission to improve mathematics education in the Soviet Union. The
emphasis irr the USSR is to increase the role of computers in education at an important element in a new approach to teaching and
all Ievels. testing.
In the Soviet Union, iust as in the United States, tlere is great uneven-
ness from school to school, and from teacher to teacher, in the quality of I'M HAPPY TO ANNOUNCE that Quantum has en-
mathematics education. Both nations have responded with similar inter- tered into an agreement with the intemational publisher
ventions: special high schools for math and science and university-based Springer-Verlag which is based in Heidelberg, Germany,
enrichment programs for students who can benefit Irom greater chal- and has offices in New York, Tokyo, London, and else-
lenges. where. The National Science Teachers Association will
Both countries debate how best to deploy limited resources {or math
education. Consewatives (mostly university professors) prefer programs retain editorial control over Quantumt and our working
il0llr]IlBrR/[rIrllllBrR lgg0
relationship with Qriantum Bureau
in Moscow will remain the same.
Springer will handle our printing, sub-
scriptions, and mailing; NSTA and
Springer will both engage in promo-
tion and solicitation of advertisements. THE STUDENT MAGAZINE OF MATH AND SCIENCE
As part of the agreement/ Quan-
tum wlll be published bimonthly A pubkcation of the National Science Teachers Association NSZA)
throughout the year beginning with e) Quantum Bureau of the USSR Academy of Sciences
the September/October 1991 issue, sb in coniunctionwith
those who have subscribed at full the American Association of Physics Teachers (AAPT)
price (as opposed to the introductory d the National Council of Teachers of Mathematics (NCTM)
price of $9.95) will receive six issues,
not four. Those who renew will, of
Publisher
course/ receive six issues peryear.
Bill G. Aldridge, Executive Director, NSTA
Wewelcome Springerto the Quan-
tumventlxe. We are confident that USSR editor in chief Managing editor
the impressive resources of Springer- Yuri Ossipyan Timothy Weber
Verlag will help make Quantum avatT- Vice President, USSR Academy of Sciences Production editor
able wherever English is spoken or Elisabeth Tobia
taught in schools. With that kind of US editor in chief for physics
Intern ation al consultant
exposure/ Quantum is more Likely to Sheldon Lee Glashow Edward Lozansky
attract hrgh-quality submissions, and Nobel Larreate, Harvard University Advertising dfuector
our readers will share in the excite- US editor in chief for mathematics Paul Kuntzler
ment of being part of an intemational William P. Thurston Dfuector of NSTApublications
experience. Fields Medalist, Princeton University Phyllis Marcuccio
Aldridge
-BillG.
Be a lactol' in lhe US aduisoryboad
Bernard V. Khoury, Executive Officer, AAPT
OUANTUM |ames D. Gates, Executive Director, NCTM
Lida K. Barrett, Dean, College of Arts and Sciences, Mississippi State University, MS
George Berzsenyi, Pro{essor of Mathematics, Rose-Hulman Institute of Techlology, IN
squalioll!
Arthur Eisenkraft, Science Department Chair, Fox Lane High School, NY
lrdy Eranz, Professor of Physics, West Virginia University, WV
Have you written an article that Donald F. Holcomb, Professor of Physics, Comell Universit, NIY
you think belongs in Quantum! Margaret ). Kenney, Associate Professor of Mathematics, Boston College, MA
Do you have an unusual topic that Larry D. Kirkpatrick, Professor of Physics, Montana State University, MT
students would find fun and chal- Robert Resnick, Pro{essor of Physics, Rensselaer Polytechnic krstitute, NY
lenging? Do you know of anyone Mark E. Saul, Computer Consultant/Coordinator, Bronxville School, NY
who would make a great Quan- Barbara I. Stott, Mathematics Teacher, Riverdale High Schoof LA
tum author? Write to us andwe'll
USSR aduisoryboad
send you the editorial guidelines
Sergey Krotov, Chairman, Quantum Bureau
for prospective Quantum contribu- Victor Borovishki, Depury Editor in Chie{, Kvant magazine
tors. Scientists and teachers in
Alexander Buzdin, Professor of Physics, Moscow State University
any country are invited to submit Alexey Sosinsky, Professor o{ Mathematics, Moscow Electronic Machine Design lnstitute
material, but it must be written in
colloquial English and at a level Quantum (ISSN 1 048-8820) contains authorized English-language transla-
appropriate for Quantum's pre- tions from Kvant, a physics and mathematics magazine published by the
dominantly high school reader- Academy of Sciences of the USSR and the Academy of Pedagogical Sciences
ship. of the USSR, as well as original material in English. Copyright O 1990
National Science Teachers Association. Subscription price for one year (six
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Quantum by the
1W7 4a2s hCinogntnoenc, tDicuCt A2vOeOnOue9 -Nll7Wl mtuamtterts 1s7h4ou2l dC boen andedrcetsisceu dt tAo vQeunaune- NatOipoinnionas le xSprscssgied anrec theos gF oo{ thue naudthaorstion
NW, Washington, DC 20009-ll7l. and not ngcessarily those ol the Foundation
OUAIITUltil
Tomahaullr lhl'ouuinU made
r'ii
No, it's not "all in the wrist",
byV, A. Davydov
<<-:_s_6E
HEN I WAS A BOY, BACK
in the 1960s, my friends and I
were fascinated by the novels
of |ames Fennimore Cooper
and others. We dreamed of the adven-
turous life among the Indian tribes of
North America. Making a lasso out of
a clothesline, we tried to catch a bush, thrown against the wall of his bunga-
a tree branch/ or even an unfortunate low, and each Indian throws his toma-
cat chancing to emerge from the base- hawk at him. The last tomahawk
ment to doze in the sun. But most of cuts the rope, and when the uncon-
all we envied the skill with which scious victim slides to the ground the
Indians wielded their menacing moviegoers see the outline of a hu-
weapon-the tomahawk. man body formed by tomahawks stuck
|ust about every author of "krdian" in the wall. After the film was over
novels devotes some pages'to the everyone was eager if not to master
wonderful art of tomahawk throwing. hatchet throwing (we realized that it
Our interest in the problem was kept was beyond the capacity of a paleface
at a fever pitch by the movies. At that boy) then at least to understand the
time Indian films were very popular, technique Native Americans used to
and their heroes never missed a chance throw tomahawks.
to throw a tomahawk. Take the fol- The younger generation isn't as
lowing scene. An Indian tribe decides interested anymore in "Indian ques-
to punish a paleface. He's tied up and tions." My own childrerl for instance,
l, i{s 2.30
4 1w 3. 30 !.f"> L 2n'
//
/r, *L q.8C u.ev 7 50 8.50
t4
"The axe cleaved the air in front of
Heyward, and cutting some of flowing
ringlets of Alice, buried itself and quivered
in the tree above her head." l>
o
Fennimore Cooper, The Last of
-James Uo
the Mohicans @o
o
ol
il0lltilBtR/[tcr[l0rR
1$90
can't tell a Huron from a Comanche Lr this modei we also assume (and valid even if angle o( isn't n/2. In spite
and hardly know who Osceola was. this is veryimportantlthat thehand of its simplicity, this result is very im-
Nevertheless, they're still impressed doesn't give the hatchet any addi- portant. It means that the ratio of the
by the fanustic ability of NortJr American tional rotation. Try it yourself and translational velocity of the toma-
natives in rnanipulating their tradi- you'l1 see that it's practically impos- hawk's center of mass to its angular
tional weapon. sible to add rotation to the toma- velocity of rotation doesn't depend on
The basic idea behind the theory of hawk's motion by moving your palm. the "force of the throw" (the momen-
hatchet throwing was bom when we You can only release the tomahawk tum transferred to the hatchet at the
started hiking regularly. Finding a dry and let it move freely. moment of release) and equals
tree trunk near our campsite (and Let's introduce the following para- ,r=Vt..)
there are lots of dead trees in our meters (see the figure on the facing o'+ l'
forests), we'd try to hit it with a hatchet. page): l is the distance from the arm's (r)
We immediately discovered an inter- center of rotation (the elbow) to point This means that the distance I cov-
esting fact: i{ the person throwing the B on the handle where we hold the ered by a flying hatchet afterlfull
hatchet stands at a certain distance tomahawk; a is the distance from rotations also doesn't depend on the
from the tree, the probability that the point B topoint 71,[ which is the center t}rowing force. Since the time needed
hatchet will stick in the tree (and not of the hatchet's mass; o( is the angle {or n rotations is equal to Znnf a, we
fali back after the hitting the tree with between the arm and the handle. The get the following fistance:
the butt or handle) suddenly increases. angle o can have different values, but
Only a little practice was needed to it's easier to *rrow wh en a = nl2. The L,=2finv/ a2 + 12 .
ensure that you'd hit the target tree, velocity of the hatchet's center of
say, ahundredtimes out of ahundred- mass is then described by the follow- This is realTy aremarkable conclu-
provided, of course, you were stand- ing equation: sion: there's a range of distances 2,,
ing at the proper distance. My at- and to make a successful throw you
tempts to understand this phenome- ,=o\fo\1. have to position yourself at the fol-
non led me to formulate a model, lowing distance from the target:
which I'11now try to explain. You'll Now let's define the angular veloc-
see that in order to master tomahawk ity of rotation of a thrown tomahawk. L,+ arctan(*)J ,'.V
throwing, you don't have to be an The simplest way to do that is to shift
Indian. What's really needed is skill to a reference system that moves with (the second term arisingbecause the
in estimating distances. Once you the hatchet's center of mass with hatchet's handle makes an angle of
know how to do that, the rest is a velocity v. Point M of this reference xc tan(l1 al a the vertical at the moment
cinch. system (the center of mass) doesn't of release).
So let's take a look at the model. move, whereas point B (like every Now let's estimate the magnitude
The problem is obviously divided into other point of the tomahawk) rotates of the elementary "euantum" Ir-
two parts. First, you have to be able to around Mi the velocity u of point B is that is, the distance covered by the
at least hit apole or a tree trunkwith at any moment directed perpendicu- hatchet after one full turn. Let 1 :
a hatchet; second, you have to hit it larly to the handle and equals rora, 33 cm, a : 20 cm (measurements
with the cutting edge and not the butt where rrru is the angular velocity of taken from my own arm and my own
or handle. I'11 assume you can man- rotation o{ the flying tomahawk. hr a ca4renter's hatchet). The calculation
age the first problem on your own. stationary reference system the ve- givesusl, :2.42m. Soif Ithrowmy
While throwing, you move your locity roi of point B at the moment of hatchet from a distance of 2.82 m
hand in the following way. The arm release is directed perpendicularly to (don't forget to add the arc tangent
holding the hatchet rotates at the /-that is, along the handle. So term to l,), it hits the target after one
elbowwith an angularvelocity rrl and full tum.
the throw takes place when the veloc- Lt=0) a= - (co/)-)
ity of the hatchet's center of mass is B
dirested.lrDri zontally. Strictly speak- Substituting
ing, i{ we want the hatchet to hit a
,=r\/
certain spot on the pole, its direction ,r?*P,
at the moment of release might be we get
something other than horizontal. But
> ABa = (Da ,
we've posed a more modest problem:
g 0u=0
how to embed the tomahawk in a
q
vertical pole at any spot whatsoever. -that is, a fiying tomahawk rotates
E [r this case we can ignore the effect of with an angular velocity ro equal to
E- gravity. In an actual experiment the the angularvelocity of the hand dur- My experience has shown that,
E
point of impact would be lower. ing the throw. This conclusion is with hardly any practice, you can hit
OUAITITUlli|/IIATURI
the target from a distarice of Lrand Lr. Let's estimate the handle lengh b for rather large. That's why hrdian toma-
Mastering the subsequent "quantum which the nth level I, equais the hawks have such long handles!
levels" is more difficult, but many (n + I )th level corresponding to the Actually, though, even shorter handles
friends of mine were able to hit a tree minimum possible distance a*n from will do: the hatchet can hit the tree
trunk {rom a'distance of In (more than point B to the center of mass M. This trunk at either the upper or lower part
10 meters). condition gives us the following equa- of the cutting edge, which brings the
"This is all very nice," you may be tion: boundaries of the zones still closer.
thinking. "But an Indian can hit his My experience suggests that a handle
target from other distances as well, znnt/ P * t] =2n(n+D17 * length of about 50 cm is quite suffi-
not just from those equal to 1,. How cient.
My experience has shown that it's
do you explain that?" Our model shows that there's no
hard to throw the hatchet if a is less
apaItr asmeeemtesr tozr ibne oquurimte osdimelp tlhea. tT chaenr eb'se tehqauna lsl0 1c0m c.m S.o Sleotl'vsi nags stuhme el atsht aetqau-a,.- hdaiffwicku ltthy roinw minags.te Yrinogu tjhues ta hrta ovfe t otom bae- "
easily altered: all you have to do is able to judge the distance to the target
tion, we get
shift the palm of your hand to another and hold the hatchet at the right place.
position on the tomahawk's handle. ,l A good idea is to cut marks on the
(2n+ l) + (n+ 1)'oi,,,
This shift modifies the whole range of n 12 handle showing the respective target
throwing distances. It also changes An examination of this equation shows distances.
the location of point B, which results that the longest handle is needed to But how do ieal Indians throw
in a "blurrin g" of Ln"levels" and the ensure the overlap of the first and tomahawks? It's quite possible they
appearance of a "zonal structure." klside second zones-that is, for n = I. do it just the way I've described. Or
each of the zones we can adiust point Substitutingl:33 cfl, d*in: l0 cm, maybe they know how to give the
:
B (that is, the value of the parameter a) andn 1, we get ahandle length of b tomahawk an additional rotation with
to a position ensuring a successful = 50 cm. Overlap of the second and a flip of the hand? I have no idea, since
throw. But we can do more than that. third zones (n:2lris achieved if the I've never met a single Native Ameri-
If the tomahawk shandle is longenouglr, tomahawk's handle is 40 cm long, can. I certainly hope that I will some-
we can get the zones colresponding to and so on. So to be able to hit any day, and that I won't pass up the
adjacent leveis I, and I,*, to overlap. target from any distance, b has to be opportunity to learn more secrets of
this remarkable art.
Tomahawk throwing is an excit-
ing sport. Maybe in the not-so-distant
to the bottom future its practitioners will organize
lourney
an association and sponsor tourna-
{ -,----.a. of the ments. And-who knows-maybe
sea one day this sport will even be in-
cluded in the Olympic Gamesl O
Does your library have
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tures, black smokers, and See page 55 ior subscription infor-
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ltl0l,illilBrR/[rct1lllItR
BRAINTEASERS
il
Just lol' IhE lun ol
Problems offered for your enjoyment
by G A. Halperin, V.V. Proizvolov,'N.n. Rodina,
L.M. Salakhov, and L.A. Steingraz
816
81B
Is the pattern shown in figure 1 symmetrical? Move a single match in each row of figure 2 to get a true
equality.
Vsflf, fVfflJ
,
\/fl.=$f,+VflflJ,
V[flryflJ+Vfru
FisureZ
819
A square is cut into a number of rectangles in such a
way that nopoint of the square is a commonvertex o{
four rectangles. Prove that the number of points of the
square that are the vertices of rectangles is even.
817
You have two red balls, two blue ones, two green/ two
yellow, and two white. A number of balls of different
colors areplaced on the leftpan of abalance while the
otherballs of the same colors are put on the right one.
The balance tips to the left. If you exchange any pair of
balls of the same color, however, the balance either tips
to the right or stays even. How many balls are there on
the balance?
o
o 820
N
ZG A steel ball floats in mercury. Will the depth of immer-
E
6 sion increase or decrease as the temperature dses? O
EU=
a:
SOLUTIONS ON PAGE 59
IlJAilruuirnAIilTrASrR$
As withproblemZ, thete's another
approach here. Consider the set of dis-
tances between pairs of points of M. II
set M is finite, there is only a finite
number of paired distances, so that
the largest among them can be found.
Let it be the fistance between points
0oinUIo exll'elno$
A and B. But point B is the center of an
interval CD whose ends, according to
our assumption, belong to M (fig. 1).
Now it's easy to prove that either AD
Sometimes an "end run" is more direct than
orAC is longer than,4-B (do it yourself,
a "dive up the middle" making use of the fact that the me-
dian m drawn to one of the triangle's
sides is less than half the sum of the
byA. L. Rosenthal two other sides).
4,
I F YOU WANT TO ACQUIRE Let's assume that M is a finite set
! some skill in solvingmathemati- and apply the extremity rule. If Mis
! cal problems, you should try to finite, it has extreme points-the ex-
I master the more or iess common treme ldt and the exteme rigfrr Consider
approaches, techniques, and methods one of them-for instance, the left
of mathematical reasoning. Here's one-and denote it by A. Point A is an
one verygeneral approach, which we'Il extreme one and, consequently, can't
call the "extremity rule." lie inside the interval connecting two
The extremity rule can be suc- other points of the set M. The contra- A_D
cinctly stated in {our words: "Con- diction proves that M isn't a finite set. Figure :
1 --
sider the exffeme case!" This is actu- There's another solution to the same
ally a recommendation to consider an problem, also based on the extremity Problem 3. The squarcs of aninfi-
object having extreme-or as mathe- rule. Assuming again that M is a nite chessboard arc markedby naw- -:=
maticians say, "extremal"-proper- finite set, consider the lengths of inter- rul numberc in such a way that each -=:-
ties. If we're considering a set of vals connectingpairs of points inM. number is equal to the arithmetic
points on a straight line, the rule tells This set of numbers is finite. Apply- mean of the four adi acent numberc-
us to focus our attention on the ex- ing our n-rlg consider the longest inter- the uppe4 loweL rigfit, and left ones.
reme left or extreme right point of the vd. BC. Clearly, there are no points of Prove that all the numberc written on
-
set. If the problem concerns a set of Moutside BC; otherwisg therewould the chessboard are ec1ual.
numbers, the extremity rule recom- be longer intervals. Therefore, all the The extremity rule is helpful here -*7
mends that we consider its maxi- points of M lie on the intervai BC, in one of its variations: "Consider the
mum and minimum. Here are some which implies that neither B nor C smallest number!" Among the num-
examples.l satisfies the above condition-again a bers written on the chessboard there's ,/
Problem l. A set of points Mis contradiction. the smallest one. This is easy to
given in a plane such that each point Now let's return to problem 1. prove. Let k be one of the numbers. If
in M is the center of an interval con- Assuming that M is a finite seg apply I is one of the numbers on the chess-
necting a pair of points in M. Prove the extremity rule this way. Fix an board, then I is the minimum num-
that setMis infinite. orientation of the plane and consider ber (since there are no natural num- /
A good way to statr is to consider a the extreme left point of set M. If there bers less than 1). ff 1 isn't on the
simpler but similar problem. So be- are several "extremeleft" points, choose chessboard, see whether 2 is on it. If it
fore doingproblem I let's try this one. the lowest one. You can easily see is, then 2 is the smallest number. _--7*
Problem 2. A set of points M is that this point (denotedby A) can't lie Otherwisg look for 3, and so on. [r no
given on a straightline suchthat each within an interval connecting two more than k steps the smallest num-
point in M is the center of an interval points of M. Lrdeed, i{ such aninterval ber will be iound. Denote it by m arrd i
P.
corurccting two other points belong- exists one of its ends is either to the the square in which it's written by
E
ing to M. Prcve that set M is infinite. left of point A or on the same vertical Denote the numbers in the adjacent :
line with A but below it. Both situ- squares by a, b, c, and d (tig.2l. Ac- !
lOther examples of applying the rule ations contradict the choice of point cordingtoourcondition,m=(a+b + i
can be found in recent issues of A. c+dll4,ora+b+c+d=4m.Because i
Quantum-for instance, in problems
MiO and Mls.-Ed.
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