144
MATHEMATICSMAGAZINE
andtheirtangentlines, whichis a special
meansfromfunctionsf
case of the meansin Horwitz[6]. Underthe assumptionthat f"(x) is nonzeroand
continuous,a mean is given by the x-coordinateof the intersectionof the tangent
lines to y = f(x) at x = a and x = b. In this case, takingf(x) to be x2, .,1 and
1/x generatesA, G, and H, respectively.The slope meanalso belongsto this family,
generatedby f (x) = N/x2
Some familiesof meansdo not containthe slope meanbecausethe meansare homogeneous,andyet thereare still otherfamilies [10] whereit is not clearwhetheror
not the slope meanis a member.
In their study of the three classic means, Bullen, Mitrinovi6,and Vasi` implicitly characterizedthese means througha family of functions as follows. Let
be a family of functionsindexed by such'thatf21l(x) = f;(x)
{fx(x) o E I1R}
thatfor everypair (a, b) E (0, oc) x (0, oc),
and suppose
X.thereexists a uniqueindex
) =_ .(a, b) suchthatfx(a) = b. Thenm = M(a, b) canbe definedby fX(m) = m. It
can be seen thatourcharacterization
of meansby two sets of functionsleadsto sucha
characterization
usinga single familyof functions.
Acknowledgment. We would like to thank all the anonymous referees for their comments and suggestions,
which certainlyimprovedthe presentationof the paper.
REFERENCES
1. J. M. Borwein and P. B. Borwein, The way of all means,Amer Math.Monthly94 (1987), 519-522.
2. P.S. Bullen, D. S. Mitrinovic,andP.M. Vasi6,Means and TheirInequalities,D. Reidel PublishingCompany,
1988.
76 (2003), 52-61.
3. B. C. Dietel and R. A. Gordon,Using tangent lines to definemeans, this MAGAZINE
4. H. Eves, An Introductionto the History of Mathematics,6th ed., SaundersCollege Publishing,1992.
5. G. H. Hardy,J. E. Littlewood,andG. P6lya, Inequalities,Cambridge,1952.
6. A. Horwitz,Means and Taylorpolynomials,J. Math.Anal. Appl. 149 (1990), 220-235.
7. M. E. Mays, Functionswhich parameterizemeans,Amer Math. Monthly90 (1983), 677-683.
8. D. Moskovitz, An alignmentchartfor variousmeans,Amer Math.Monthly40 (1933), 592-596.
9. P. Sprent,Models in Regressionand Related Topics,Methuenand Co., LTD, 1969.
48 (1975), 87-92.
10. K. B. Stolarsky,Generalizationsof the logarithmicmean, this MAGAZINE
A Carpenter'sRule of Thumb
ROBERT FAKLER
University of Michigan-Dearborn
Dearborn, MI 48128
[email protected]
In an episodeof the PBS televisionseries "TheNew YankeeWorkshop,"
host and
mastercarpenterNormAbramneededto constructa rectangular
woodenframeas part
of a piece of furniturehe was building.Aftergluingandclampingfourpieces of wood
togetherto forma rectangle,he checkedthe rectanglefor squarenessby measuringthe
two diagonalsto determinewhetheror not they were of equal length.Upon finding
a small differencein the two measurements,he announcedthathe would "splitthe
difference."He proceededto carefullynudgethe top cornerof the frameat the end of
the longerdiagonaluntilhis measuringtapeindicatedthatits lengthwas the averageof
his two originaldiagonalmeasurements.He then said he was satisfiedthatthe frame
Mathematical Association of America
is collaborating with JSTOR to digitize, preserve, and extend access to
Mathematics Magazine
®
www.jstor.org
145
VOL. 78, NO. 2, APRIL2005
was square.After hearingthis, I wonderedif the framewas indeed square,that is,
whethertherewas a rightangleat each of the fourcomersof the frame.
To answerthe abovequestion,we needto solve the followingproblem:Supposewe
withsidesof lengthsa andb. Letx andy be the lengthsof thetwo
havea parallelogram
it into a rectanglewith
If
we
by transforming
squareup this parallelogram
diagonals.
side lengthsa andb, whatis the commonlengthd for the two diagonals?FIGURE1
andits squaredup version.
showsourparallelogram
a
a
x
y
d
b
b
o
FigureI
A wooden frame before and afterstraightening
Fromthe Law of Cosines,we see that
and
x2=a2+b2-2abcosO
y2=a2+b2-2abcos(r_-0).
Therefore
x2+y2=2a2-+-2b2
andso d2=a2+b2=(x2+y2)/2.
Thus
d=
This is evidentlynot the same as the averageof the two originaldiagonalmeasurements, (x +- y)/2, the lengththatAbramrecommendedfor the new diagonal.His algorithmdoes not exactly producethe lengthneededto turnthe parallelograminto a
rectangle,butwe will see thatit is a good approximation.
What is the mathematicalbasis for this approximation?Considerthe function
wherex > 0 and y > 0. SupposeL(x, y) is the linearizad(x,
tionof d at thepoint(x0,xO),wherexO> 0 is the correctmeasurementof thediagonal.
(Note thatd(xo, x0) = x0.) Then
L(x,
Since dx
dy(xo, x0) = 1/2. Also, d (xo, xO) = xO.Therefore
we have dx(xo,xo) -
L(x,
In the "TheNew YankeeWorkshop"episode, Abramtook the diagonalmeasurementsx andy of the woodenframe.Insteadof the exactdiagonallengthd(x, y) that
would squareup the frame,which is difficultto computementally,he used the linL(x, y), whichis easy to calculate.In practice,the framewouldbe
ear approximation
before
its diagonalsweremeasuredso thatx andy wouldbe nearly
squaredup by eye
and
to
equal
nearlyequal the exactdiagonalxO.In this case, the linearizationL(x, y)
is a quitegood approximation
for d(x, y), as the followingexampleshows.
146
MATHEMATICSMAGAZINE
arex = 36 inchesandy = 37 inches.
Example Supposeourdiagonalmeasurements
Then d(36, 37) = tt
= 36.503424 and L(36, 37) = 36.5.
The readermay wish to estimatethe errorin this linearapproximation
using Taylor's theoremin two variables[1]. If y is the longerdiagonalandx the shorter,andif
bothIx - x0IandIy - x0Iareknownto be less thanh, the errorcan be seen to be less
thany2h2/(2x3).
REFERENCE
1. JerroldE. Marsdenand Michael J. Hoffman,ElementaryClassical Analysis, 2nd ed., W. H. FreemanandCo.,
New York, 1993.
Chess:A Cover-Up
ERIC K. HENDERSON
DOUGLAS M. CAMPBELL
DOUGLAS COOK
ERIK TEN NANT
Computer Science Department
Brigham Young University
Provo, Utah 84602-6576
The gameof chess has alwaysproveda rich sourceof interestingcombinatorial
problems to challengemathematicians,
logicians,andcomputerscientists.Apartfromplaying strategiesand end-games,manychess-basedproblemshave been posed over the
centuriesthattax the limitsof symbolicreasoning,suchas the n-queensandre-entrant
knight'stourproblems.However,the modemcomputerhas enablednew approaches
to thesetypesof problems,andsome of thesequestionshavebeenexplored(andeven
decided)in ways not previouslypossible.
One such problemhas been attributedto JosephKling [8], a music producerwho
operateda chess-orientedcoffee housein Londonfrom 1852. Kling,who migratedto
EnglandfromGermany,is describedas "apioneerof the modemstyle of chess"[5];
he publishedseveralstudieson the gameincludingthe popularbutshort-livedjournal
Chess Player, co-editedwith the chess professionalBernhardHorwitz.Kling posed
the followingquestion:using a player'seight majorchess pieces and no pawns,can
all squareson the chessboardbe covered(attacked)?At firstglancethe problemlooks
easy-the eightpieces collectivelyhavemorethanenoughattackpower.Determining
whetherthereis a solution,however,is nontrivial.No simplelogical argumenthas yet
beendiscovered.
Becausea chessboardis small,the combinatorialsize of chess problemsis theoretically tractable.However,only the recentadvancesin computingpowerhavemadethis
truein practice.In 1989,Robison,Hafner,andSkiena[8] appliedanexhaustivesearch
to prove that no solutions exist to the Kling cover problem. To accomplish this using
the computing power available at the time, they developed a novel technique for reducing, or "pinning," the number of solutions that need to be searched. Their approach,
although not immediately intuitive, reduced the search space by more than 99.9%.
With the computing power available today, problems such as this can now be exhaustively searched in a reasonable time with no need for creative pinning of the search
space. However, there will always be larger problems pushing the limits of computing
power, and methods for reducing the search space help put more of these within reach.