EDITORIAL POLICY Mathematics Magazine aims to provide lively and appealing mathematical exposi tion. The Magazine is not a research jour nal, so the terse sty le appropriate for such a journal (lemma-theorem-proof-corollary) is not appropriate for the Magazine. Articles should include examples, applications, his torical background, and illustrations, where appropriate. They should be attractive and accessible to undergraduates and would, ideally, be helpful in supplementing un dergraduate courses or in stimulating stu dent investigations. Manuscripts on history are especially welcome, as are those show ing relationships among various branches of mathematics and between mathematics and other disciplines. A more detailed statement of author guidelines appears in this Magazine, Vol. 74, pp. 75-76, and is available from the Edi tor or at www.maa.org/pubs/mathmag.html. Manuscripts to be submitted should not be concurrently submitted to, accepted for pub lication by, or published by another journal or publisher. Submit new manuscripts to Frank A. Farris, Editor, Mathematics Magazine, Santa Clara University, 500 El Camino Real, Santa Clara, CA 95053-0373. Manuscripts should be laser printed, with wide line spacing, and prepared in a sty le consistent with the format of Mathematics Magazine. Authors should mail three copies and keep one copy. In addition, authors should supply the full five-symbol 2 000 Mathematics Subject Classification number, as described in Math ematical Reviews. Cover image, Emil Post Ponders Tag Sys tems, by Brian Swimme. Number sequences stream down around Emil Post according to the tag system rules, described in Stillwell's article in this issue. The first three columns start with strings that lead to the same peri odic pattern, but the future of the progres sion in the last column is unclear. Does it terminate, does it become periodic? Cover image by Brian Swimme, a student at Santa Clara University, under the super vision of Jason Challas.
AUTHORS John Stillwell is Professor of Mathematics at t h e
U niversity o f San Fran cisco. H e w a s b o r n a n d raised in Melbou rne, Austra l i a, atte n d i n g Mel bou rne High School a n d the University of Mel bo u rn e (M.Sc. 1965). He first came to America in 1965, and received Ph. D. in mathematics from MIT in 1970. At this time his main interests were in logic. From 1970 to 2001 he taught at Monash U niversity in Melbou rne, grad u a l l y changing fiel ds to topol ogy, geometry, a l gebra, and n u mber theory. D u ring this period he wrote a n u mber of books, the best known of which is Mathematics and Its His tory (Sp ringer-Ver l ag 1989, 2nd edition 2001). In 2002 he started a n ew career at the U niversity of San Fran cisco.
Janet Heine Barnett holds a B.S. in mathem at ics and h u manities from Colorado State Univer sity, a n d an M.A. and P h . D. in set theo ry from the U niversity of Colorado. S h e has taught at t h e U n iversity o f Southern Col orado, where s h e has been a fu l l professor since 1990. Her inte rests i n c l ude mathematics history a n d its u s e i n p romot ing mathematica l u nderstan ding; her article in this MAGAZINE was in spired by qu estio n s from stu dents who s h a re these interests. Other proj ects i n c l u d e t h e mathematical history o f Paris (jointly with h e r dance/travel partner/ h u sband George Heine), a n d a study o f proof and intuition in the development of m ath e m atical con cepts, both historical and in today's c l assroom . Daniel J. Velleman received his B.A. from Dart mouth C o l l ege in 1976 and his P h . D. from t h e U nivers ity o f Wisconsin -Mad ison in 1980. H e then taught at the U niversity of Texas before joining the fac u l ty of Amherst Col l ege in 1983. He is t h e a u t h o r of How t o Prove It, coauthor (with Stan Wagon and joseph Kon hau ser) of Which Way Did the Bicycle Go?, and coauthor (with A l exan d e r George) o f Philosophies o f Mathematics. His work on Simpson's ru l e was motivated by a nagging dis satisfaction , over a n u mber of years of teaching Simpson's rule in ca l c u l u s c l asses, with the a lterna tion of coefficients i n the Sim pson's r u l e for m u l a.
Vol. 77, No.1, February 2004
MATHEMATICS MAGAZINE E D I TO R Fra n k A . Farris Santa Clara University ASSOC I AT E E D I TO R S Gl e n n D . A p p l eby Beloit College Arth u r T. Be n ja m i n Harvey Mudd College Pa u l j. Ca m p be l l Beloit College A n n a l i s a Cran n e l l Franklin & Marshall College D av i d M. ja m es Howard University E l gin H . jo h n ston Iowa State University Victor j. Katz University of District of Columbia Occidental College David R. Scott University of Puget Sound S a n ford L . Segal University of Rochester H arry Wa l d m a n MAA, Washington, DC E D ITO RIA L ASSISTANT Martha L . Gia n nini
MATHEMATICS MAGAZINE (ISSN 0025-570X) is pub lished by the Mathematical Association of America at 1529 Eighteenth Street, N.W., Washington, D.C. 20036 and Montpelier, VT, bimonthly except july/August. The annual subscription price for MATHEMATICS MAGAZINE to an individual member of the Associ ation is $131. Student and unemployed members re ceive a 66% dues discount; emeritus members receive a 50% discount; and new members receive a 20% dues discount for the first two years of membership.) Subscription correspondence and notice of change of address should be sent to the Membership/ Subscriptions Department, Mathematical Association of America, 1529 Eighteenth Street, N.W., Washington, D.C. 20036. Microfilmed issues may be obtained from University Microfilms International, Serials Bid Coordi nator, 300 North Zeeb Road, Ann Arbor, Ml 48106. Advertising correspondence should be addressed to Dave Riska (
[email protected]), Advertising Manager, the Mathematical Association of America, 1529 Eighteenth Street, N.W., Washington, D.C. 20036. Copyright© by the Mathematical Association of Amer ica (Incorporated), 2004, including rights to this journal issue as a whole and, except where otherwise noted, rights to each individual contribution. Permission to make copies of individual articles, in paper or elec tronic form, including posting on personal and class web pages, for educational and scientific use is granted without fee provided that copies are not made or dis tributed for profit or commercial advantage and that copies bear the following copyright notice: Copyright the Mathematical Association of America 2004. All rights reserved. Abstracting with credit is permitted. To copy other wise, or to republish, requires specific permission of the MAA's Director of Publication and possibly a fee. Periodicals postage paid at Washington, ditional mailing offices.
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3
VOL. 7 7, NO. 1, F E B R U A RY 2 004
Emil Post and His Anticipation of Godel and Turing JOH N S TILLWELL U n iversity of San Fran cisco San Francisco, CA 94117 sti//well@ usfca . edu
Emi l Post is known to special ists in mathemati cal logic for several i deas in logic and computability theory : the structure theory of recursively enumerabl e sets, degrees of unsolvab i l i ty, and the Post "correspondence problem." However, he should be known to a much wider audience. In the 1920s he discovered the i ncompleteness and unsolv ability theorems that later made Giidcl and Turing famous. Post missed out on the credit becau se he fai led to publ i sh his results soon enough, or in enough detai l . H i s achievements were known t o most o f h i s contemporaries i n logic, but th i s was sel dom acknowledged i n pri nt, and he now seems to be slipping i nto oblivion. Recent comprehensive publications, such as Gi:idel ' s coll ected works and the popular h i story of computation by Martin Davis l3] contai n only a few words about Post, mostly in footnotes . In this article I hope to redress the balance a little by tel ling Post's side of the story and presenting the gist of h i s ideas. This i s not merely t o g i v e Post his due; it gives the opportuni ty to present Post's approach to Giidcl ' s incompleteness theore m , which i s not only more general than CHidel's hut als 0
lim
sin x
--
X
= 1.
A natural general ization of thi s l i m it i s sin x + sin y
lim
(I)
x+y
.r +_qeO
( _t , .\' ) --> ( 0 , 0 )
The assignment I g a ve to m y c a l c u l u s c l a s s was to determ ine if ( I ) exi sts and to e v a l uate the l i m i t if it doe s . The purpose o f thi s note i s t o d i s c u s s w h y i s ( I ) an i nteresting l i m i t , a n d t o sug gest that a broader range of problems can be i nvesti gated by students at vari ous leve l s . Limits are o n e o f the staples of single- variable calculus courses, yet the treatment of limits in multivariable calculus tends to be rather mi n i m al . When considering s i ngu l arities, many standard texts such as Stewart [6] deal almost excl usively with rational function s . Thi s is also the case with many advanced cal c u l u s texts [2, 4, 8] . Undoubt edly, the i ncreased complexity of l i mits in m ultiple dimensions partiall y accounts for the sparse treatment. However, there are many interestin g mathematical questions re garding multivariable l i m its suitable for exploration by undergraduates . How does o n e evaluate the limit i n ( 1 ) , provi ded, o f course, that i t even exi sts? Some of my students s uggested that a graph such as F I G U R E l (a) , produced by Maple, suffices. On the other hand, another software package, Matlab, produced the image in F I G U R E l (b) (the contours of the graph are plotted in the xy plane) . This definitely gives a different view of the l i mit. Given the discrepancies in the images , it isn ' t clear that either fi gure suffices to show, even at an intuitive level , that ( 1 ) exi sts. There are several rigorous ways to evaluate the limit. One of the most elegant is to make the change of variables x = u + v and y = u - v . Then lim
(x , y) --> (0 , 0) x +y ;FO
sin x + sin y x +y
lim
(u , v) --> (0,0) u ;FO
lim
(u , v ) --> (0 , 0) u ;FO
sin (u + v) + sin (u - v )
2u 2 sin ( u ) cos ( v )
------
2u
- 1. -
(2)
62
MAT H EMAT I C S MAGAZ I N E I 0.9 0.8 0.7
i
i i
0.6 �
-I I
(a)
I
(b)
Figure 1
Two re n d e r i ngs of the s a m e fu n c t i o n gra p h
Another possibi lity i s t o l e t z
=
-y, so
si n x + si n y
lim
x+y
l .r . r ) --> ( 0 . 0 ) .r + r ,CO
sin x - sin ( - y )
lim
X - (-y)
(.r . r ) --> ( 0 . 0 ) .r + r ,CO
sin x - sin z
lim
=
(.r . : ) --> (0. 0 ) .r ,C:
X-
=
cos(O)
sin ' (0)
Z
I.
=
(3)
Equati ng the limit with the derivative is natural , but does require some j ustifi cation (see Theorem 3 and Example 3 ) . It may be noted that the method of (2) employs the addition formula for the sine, while the method of (3) uses only the fact that the sine function is odd and a definition of the derivative. B ased on the results of (2) and (3), one might suppose that ( I ) could be generalized in a fairly obvious way.
Generalizing t h e result
A
Q uestw . n:
1·
/ (.r . 1 m , ...., 0 -.•. If )
·
=
a,
lim
does
(.r . r ) --> ( 0 . 0 ) .r + r ,CO
·
f (x ) + f (y) x +y
"-----=----=---
=
a?
(4)
We can observe that both of the iterated l imits are equal to a : . 1. .f (x ) + f (y ) I 1m 1 m X+y
.r --> 0 r --> 0
=
1.
1m
.r --> 0
.f (x ) + 0 X+0
=
1.
1m
r --> 0
f (y ) + 0 y+0
=
1.
1.
1m 1m
r --> O .r --> 0
f (x ) + f ( y ) . X+y .
This might suggest that it would be rel atively si mple to prove that the limit i s a . How ever, if f (x ) x 2 , the limit i n ( 4) does not exist, as the fol lowing counterexample shows.
=
COUNTEREXAMPLE.
. .r� + r2 h m cr . r l --> ( 0 . 0 ) � · .r + r ,CO
.
does not eX ISt.
63
VO L . 7 7 , N O . 1 , FEB R U A RY 2 004
Proof Consider t h e case when x
v-ax i s . Then l i m .r .... o
�
= 0, so that o n e approaches t h e origin along the
i.\" = 0. 2 Now let x = t + t and y = t 2 - t . Then, provided 1 f=. 0,
Since l i m�--> 0
� [(I +
I )2 +
(I
- I ) 2]
= I f=. 0, this establi shes the counterexample .
•
Observe that a graph of z =
>
From (8) it follows that lim�---+ o +
[(
2::: , a 2k g2k (t) = 0.
However, if a
]
= 2k0 ,
azko 2ko 2ko + "' a g "' L..., a zk gzk (t) = 2 t 2k0 - I + 1 ) + ( t 2k0 - I - 1 ) L..., zk zk (t) . (9) k=ko + l k= l For t > 0, 0 < g 2k (t) � ((t + ta) 2k ) /ta . Since lim�---+ o + ( (t + ta) 2k )/ta = 0 uniformly in k for k k0 , the absolute convergence of :E: ko + I a zk x 2k in a neighborhood of 0 00
>
00
VOL. 77, NO. 1 , F E B R U A RY 2 004
65
implies 00
L
lim
k=ko + I
t --> O+
From (9), we may conclude that
. . a zko � azk gzk (t) = hm hm """' t --> O+ t - -> O + 2 k= l 00
-
This, however, contradicts (8). Thus, E X A M P L E 1 . I f f (x )
=
[(t 2ko a2k
+ 1
) 2ko
+
= 0 for all k
sin x + sin y
lim
=
1
( t 2k 0
-1
- 1
]
) 2ko = a zko ·
E N.
sin (x ) , then Theorem 1 immediately yields
X
(x , y ) -+ ( 0 , 0 ) x +y ;fO
EXAMPLE 2 . If f (x )
azk g zk (t) = 0.
e
x
+y
- 1 , then f (O)
lim
=
=
•
1.
0, l i m f (x ) jx x ---> 0
=
1 , but
x +y
(x , v ) ---> ( 0 . 0) x + y ;fO
doesn 't exi st, because f i s not odd . The strong derivative
potheses hold, then
Theorem 1 states that if f is real analytic and the other hy
lim (x . y ) -+ ( 0 , 0 ) x +y ;fO
f (x ) + f ( y ) x +y
exists only if f is odd. If f is an odd function, can the condition that f is real analytic be relaxed? We shall address this question in Theorem 3 below, which employs an alternative definition of the derivative. Suppose that f is an odd function. As in (3), let z = - y . Then lim
(x , y ) -+ (0, 0) x + y ;f O
f (x ) + f ( y ) x+y
This leads to the definition of the j * (xo)
lim
f (x ) - f ( z )
(x , z ) -+ ( 0 , 0 ) x ;fz
x -z
strong derivative f * (x0) by
=
lim
(x , z ) --> (xo , xo) x ;fz
f (x ) - f ( z ) x -z
when the limit exists . Bruckner and Leonard [1] attribute the definition of the strong derivative to Peano [5] , who is well known for his axioms for the natural numbers . Esser and S hisha [3] show that if f* (x ) exists, then f* (x ) = f ' (x ) ; that f* (x ) is continuous on its domain of definition ; and provide necessary and sufficient conditions for the existence of the strong derivative . They also give an easily checked sufficient condition for the existence of the strong derivative [3] :
66
THEOREM 2 .
at x0 .
MATHEMA T I C S MAGAZ I N E
/f f' is continuous at a point
x0 ,
then f is strongly differentiable
The defi nition of the strong derivative and Theorem 2 immediately yield Theorem 3 , which extends the results o f Theorem I i n one direction.
THEOREM 3 . at 0, then
If f is an oddfunction, f (O) = 0, f ' (O) = a and f' (x ) is continuous lim
(r . r ) --> ( 0 . 0 1 .r + r ;ioO
f (x ) + j ( y ) X +y
= /* (0) = f ' (O) = a .
E X A M P L E 3 . If f (x ) = s i n (x ) then (2) fol lows immediately from Theorem 3 . The orem 2 may also be applied at points of the form (x0 , -x0 ) , yielding lim
(r . .'" ) --> (ro . -.ro I .r + r ;loO
Hence, the function g (x , y ) =
l
s i n (x ) + sin ( y )
------
x +y
s i n ( x ) + sin ( y ) x +y cos (x ) ,
= cos (x0) .
if x =/= - y if X = - y
i s a continuous extension t o all o f JR 2 o f z. = s i n l x:!:i n (n _ The function g (x , y ) can be shown to be ditferentiable, since the partial derivatives exist and are conti nuous . Thus, F I G U R E l (b) i s a more accurate depiction of the behav ior of the function than F I G U R E l (a) . E X A M P L E 4 . If .f ( x ) = cos (x ) , then Theorem 2 yields lim
( .r . r ) --> ( 0 . 0 ) x ;lo -'
cos (x ) - cos ( y ) X -y
. = - sm (O) =
0.
Thi s i s n ' t exactly obvious from a graph such a s F I G U R E 3 .
0.5 0 -0. 5 -I -4
-4
Figure
3
An u n i n formative i m age
VOL. 77, NO. 1 , F E B R U A RY 2 004
67
Higher dimensions Since w e can evaluate ( 1 ) a s a limit i n two dimensions , i t is natural to inquire whether similar results hold in higher dimensions . For example, does
sin(x ) + sin(y) + sin(z)
. hm
(x,y,z)---* ( 0,0,0) x+y+z#O
X
+
Y
+
Z
=
1?
( 1 0)
Interestingly, the limit on the left-hand side of ( 1 0) does not exist, as the following theorem shows. THEOREM 4. If limx4o
f (x) X
= a 1 , and f is real analytic at 0, then
lim
(x,y,Z)---* ( 0,0,0) x+y+z#O
f (x) + f (y) + f (z) = a1 x+y+z
(1 1 )
if and only if f (x) = a 1 x. Much of the proof of Theorem 4 is analogous to Theorem 1 , so only a brief sketch of the proof will be provided. First, if f (x) = a 1 x , then ( 1 1) follows trivially. Now assume that ( 1 1 ) holds, and that f (x ) = 2:: : 1 a11 X 11 in a neighborhood of 0. If z = 0, then ( 1 1 ) reduces to ( 5 ) , which allows us to conclude that a 2k = 0 for all k E N. Let x = y = t 3 - t , and z = t 3 + 2t , for t > 0. Then
f
a f (x (t)) + f (y (t)) + f (z (t ) ) a zk+ J h zk+ l (t ) , = a l + 3 ( 2 (t 2 - 1 ) 3 + (t z + 2 ) 3 ) + 3 x (t ) + y (t) + z (t ) k=Z where
h 2k + l (t ) =
2 (t 3 - t ) 2k + l + (t 3 + 2t ) 2k+ l 3t3
As i n the proof o f Theorem 1 , it may b e shown that limt-+o+ 2:: : 2 a 2k+ 1 h zk+ l (t) = 0, leaving lim �-+o+
f (x (t)) + f (y (t)) + f (z (t)) x (t) + y (t) + z (t)
=
a l + 2a 3 .
However, if ( 1 1 ) holds, then a 3 = 0. A similar argument, changing the highest power • in the parameterization of x, y and z , shows a5 0, and so on. =
From Theorem 4, we immediately obtain
THEOREM 5 . If limH o
n :=::: 3,
f (x ) / x = a1 and f is real analytic at 0, then, for integer
lim
(X j , Xz , . . . ,Xn ) ---*
(0,0, ... ,0) x1 +xz+, ... , #O xn
X1
+ x z +, . . . , X 11
if and only if f (x ) = a 1 x . Limits in JR 2 and lR 11 can be a source of interesting and sometimes counterintuitive problems . Many of the results of this note can be read ily used at the undergraduate level. Students in a multivariable calculus course may Conclusions and suggestions
MAT H EMAT I C S MAGAZ I N E
68
certainly b e asked to evaluate ( 1 ), possibly with aspects of (2) and (3) given as hints . Using the counterexample as a model, students may be guided to conjecture Theo rem 1 , although the proof of Theorem 1 would be more appropriate for a course in advanced calculus or elementary analysis. The strong derivative can be used to revisit the definition of the derivative, and build on earlier concepts of the meaning of the derivative. To show that Theorem 1 cannot be extended to higher dimensions, the pa rameterization given in the outline of the proof of Theorem 4 may be used to show that the limit of the left-hand side of ( 1 0) doesn' t exist. Acknowledgments. The author would like to thank Peter Jarvis, Warren Johnson, and the referees for their many
useful and helpful suggestions.
REFERENCES A. M. Bruckner and J. L. Leonard, Derivatives, Amer. Math. Monthly 73 ( 1 966), 24--56. R. C. Buck with E. F. Buck, Advanced Calculus, 3rd ed. , McGraw-Hill, New York, NY, 1 978. M. Esser and 0. Shisha, A Modified Differentiation, Amer. Math. Monthly 71 ( 1 964), 904--906. W. Fulks, Advanced Calculus, 3rd ed. , John Wiley, New York, NY, 1 978. G. Peano, Sur Ia definition de Ia derivee, Mathesis 2 ( 1 894), 1 2- 1 4, (Opere scelte, vol. 1, Edizioni Cremonese, Roma, 1 957, pp. 2 1 0-2 1 2). 6. J. Stewart, Calculus, 4th ed. , Brooks/Cole, Pacific Grove, CA, 1 999. 7 . K. R. Stromberg, An Introduction to Classical Real Analysis, Wadsworth, Bel mont, CA, 1 98 1 . 8 . A . E . Taylor and W. R . Mann, Advanced Calculus, 3rd ed., John Wiley, New York, NY, 1 9 8 3 .
1. 2. 3. 4. 5.
70 Yea rs Ago in the M A G AZ I N E (ca l l ed, at that t i me,
Newsletter)
The Ma thematics
From "Improving the Teaching of College Mathematics," by May M. Beenken, Vol. 8, No. 5 , (Feb. , 1 934), 97- 1 03 : I n order to keep mentally alert, the college teacher o f mathematics should himself [sic] be working and learning constantly. We m ay well harken to the words of J. W. Young in his retiring presidential address to the Mathe matical Association of America. He said, "The sin of the mathematician is not that he [sic] doesn't do research, the sin is idleness, when there i s work to be done. If there be sinners in my audience, I would urge them to sin no more. If your interest is in research, do that; if you are of a philosophical temperament, cultivate the gardens of criticism, evaluation, and interpreta tion; if your interest is historical, do your plowing in the field of history ; if you have the insight to see simplicity in apparent complexity, cultivate the field of advanced mathematics from the elementary point of view; if you have the gift of popular exposition, develop your abilities in that direction; if you have executive and organizing ability, place that ability at the dis posal of your organization. Whatever your abilities there is work for you to do,-for the greater glory of mathematics." And m ay I add, "Whatever you do, do it for the greater glory of teaching, which is the chief purpose for which you are employed." The editor hopes that those evaluating the scholarly achievement of faculty today will reward all the various types of endeavor advocated by Young.
PRO B L EMS ELG IN H . J O HNSTON,
Editor
I owa State U n i ve r s i ty
RAZVAN GELCA, Texas Tec h U n i vers i ty; ROBERT GREGORAC, I owa GERA L D HEUER, Concord i a Co l l ege; VAN IA MASCION I , B a l l State U n i PAU L ZEITZ, The U n ivers i ty o f S a n Fra n c i sco
Assista n t Editors: State U n i ve rs i ty; vers i ty;
Proposa l s
To be considered for p ublica tion, solutions should be received by july 1 , 2 004 .
1686.
Proposed by Shahin Amrahov, Ari College, Turkey.
Find all positive integer solutions
2/ 1687.
=
(x , y ) to the equation x 4 + 8x 3 + 8x 2 - 3 2x + 1 5 .
Proposed by Sung Soo Kim, Hanyang University, Ansan Kyunggi, Korea.
A two-player game starts with two sticks, one of length n and one of length n + 1 , where n i s a pos itive i nteger. Pl ayers alternate turns. A turn consi sts of breaking a sti ck into two sticks of pos itive i nteger length s. or re mov i ng k sticks of length k for some positive i n teger k. The player who makes the l ast move wins. W h i ch pl ayer can force a win?
1688.
Proposed by Mihai Manea, Princeton University, Princeton, NJ. Let p be an odd prime, and let P (x) + ap_ 1 x p - I be a a0 + a 1 x + a 2 x 2 + polynomial of degree p - I with integral coefficients . S uppose that p f ( P (b ) - P (a)) whenever a and b are integers s u c h that p f (b - a ) . Prove that p I a p - I · 1689. Proposed by Ali Nabi Duman, student, Bilkent University, Ankara, Turkey. ·
=
·
·
Triangle ABC i s a ri ght tri angl e with right angle at A . Circle C is tangent to AB and BC at K and N, respective ly, and i ntersects AC in points M ( =/= A) and P , with AM < AP. The line perpendicular to BC at N intersects the median from A , the circle C, and AB in points L , F, and E , respectively. Prove that if FLjEF = LNjEN, then a.
K,
L, and M are collinear. = EA/ EK.
b. cos (2 LA B C)
We invite readers to submit problems believed to be new and appealing to students and teachers of advanced undergraduate mathematic s . Proposals must, in general , be accompanied by solutions and by any bibliographical information that will assist the editors and referees. A problem submitted as a Qui ckie should have an unexpected, succinct solution. S olutions should be written in a style appropriate for th i s separate sheet.
MAGAZINE.
Each sol ution should begin on a
S olutions and new proposals should be mailed to Elgin Johnston, Problems Editor, Department of Mathe matics, Iowa State University, Ames IA
500 1 1 ,
or mail ed electronically (ideally as a
Ie.Tpc
file) to ehj ohnst @
iast ate . edu. All communications should include the readers name, full addres s , and an e-mail address and/or
FAX number.
69
70
MAT H EMAT I C S MAGAZI N E
1690. Proposed by Costas Efthimiou, Department of Physics, and Peter Hilton, De partment of Mathematics, University of Central Florida, Orlando, FL. Prove that there exist functions f lR lR that satisfy :
for all x ,
y
---+
f (x - f (y) )
= f (x) +
Y
E JR, and show how such functions can be constructed.
Q u i ck i es
Answers to the Quickies are on page 75.
Q937. Proposed by Bill Chen, Philadelphia, PA, Clark Kimberling, Evansville, IN, and Paul R. Pudaite, Glen Ellyn IL. Let n be a positive integer. Prove that
ln
I: -k + -2 - I: n
k=l
1
n
J
k= l
l
n
�
k+
2
J=
n.
Proposed by William P. Wardlaw, U. S. Naval Academy, Annapolis, MD. Let R be a ring, let G be a finite subset of R that forms a multiplicative group under the multiplication of R, and let s be the sum of the elements of G . Prove that if G has more than one element, then s is either zero or a zero divi sor in R. Give examples in which s is a nonzero divi sor of zero. Q938.
Sol uti o n s A Square Bound
February 2003
Proposed by Erwin Just (Emeritus) and Norman Schaumberger (Emeritus), Bronx Community College of the City of New York, Bronx, NY. Let xb 1 ::=: k ::=: be positive real numbers with L�= x;k - l ::=: n. Prove that l 2 L� = 1 (2k - 1 ) xk :S n . I. Solution by Michael G. Neubauer, California State University, Northridge, CA. Bernoulli ' s Inequality states that if r :::: 1 and x :::: 0, then x' - 1 :::: r (x - 1 ) . Re place x by xk and r by 2k - 1 , then do some rearranging to obtain 1662.
n,
(2k - 1 ) xk
::=:
x;k - l
- 1 + (2k - 1 ) .
It follows that
L (2k - 1 ) xk :S L x;k - l - n + L (2k - 1 ) ::=: n - n + n 2 = n 2 •
II.
n
n
n
k=l
k=l
k=l
Solution by Heinz-Jiirgen Seiffert, Berlin, Germany.
We prove the following generalization:
Let I be a real interval containing 1 and let fk : I --+ JR, 1 ::=: k ::=: n , be dif ferentiable and convex on I . If c is a real number, and xk E I, 1 :S k :S n , with
VOL . 7 7, NO. 1
I
71
F E B R U A RY 2 004
n n I: Jl ( l )xk :::: c + I: u:co - !k o ) ) . k=l k=l The result in the problem statement follows by taking fk (x) = x2k -l , xk E I = (0, oo ) , and c= n . To establish the generalization, first observe that for 1 :::; k :::; n , the function gk defined by gk (x)= j{(l)x - fk (x) satisfies
g� (x) 2: 0 g� (x) :::; 0 so
gk (x) :::; gk (l) for all x
!.
E
X
E
xE
J
and
X
Hence
II
L f{(l)xk :::; c + L ( fj ( I )xk - fk (xd) k=l n
k=l
k=l
k=l
i\ lso solved h v Rem ,1 khlo;;hi, Tsefw ve ,1 ndehrhan. Mi,·lwel A n dreoli, Cor/ Axness (Spu in ), Mich el Ratoille
( Frunce ). Jean R og ue rt ( Rel;;ium ) . Cui Polr Pomona Pm!Jiem Solving G ro up, Minh Can . Murio Catolon i ( /tu l v ) .
C o n /\ m o re Problem G m up (Denmark ) . Kn u t D a l e (Nonvoy). Donieh· Do n ini ( !ta l\' ) . Rohert L. fJoucette. Peter Driunot· ( Canada ) , ,1 amn Du tle. F G C U Prohlem Grmlf J , (J\'idiu Furdui, G . R . A . 2 0 Pm/Jiems G ml lf J ( ltu h i . Julien
y
Grit·'(((/.\ ( 1-iw!Ce ). Enke/ Ih.1neluj (,1ustmliu ). The It h u m College Sol t ·crs, St!Te Ka c�ko\\ ·ski, ,1 ch im Kehrein
z
(Gem! W l \' ) , Murra Y S. Klumkin ( Can ado ) . lc'l i o s LWUJill kis ( G reece ). Kee - Hl1i Lou ( Ch i n o ) .
Nortil lvntern Un i P Ray. Rolf' R ich h('{H (Gcrmon v), Joel Sclz lo sb etH . HarrY Se dil g et: A ch illeas Sincf'a kOfJOulos, Niclw lo s C. SingCI : John W Spellnwnn. Nom Thorn ber, Dave Tm utmon. Chu Wenc/wng ond Mo gli Pic rl uig i (lta l v ) , Michael Vr!\Ve (Stvit�erland), John T ZoHo: l.i Zho11, and the prop o se rs. ve rsit Moth Prohlem So!t·ing G ro up, A lbert D. Polimeni. Roh Pmtt. Phillip
Much Ado About Nothing
February
2003
1663. Proposed by Michel Bataille, Rauen, France. Let m and n be integers such that l
� ( (k + k= 1
l)
sink - !
(
ln m n l
+
:::;
m
0. Similarly, we may assume A > 0. If D > 0 then F has two distinct real roots r" r2 , and on taking x = r1 , x = r2 , we deduce that f also vanishes at the points ri . Thus f (x) = a (x - r1 ) (x - r2 ) and F (x) = A (x - r1 ) (x - r2 ) . The given inequality implies that a :S A. Then d = a 2 (r1 - r2 ) 2 :::; A 2 (r1 - r2 ) 2 = D. If D :S 0, we consider two cases : Case 1. d < 0. On letting x 0 < a :::; A. Since Solution to A4 Put
--+
min x real
( B)=-
F (x) = F -
2A
oo,
D
4A
and
min
x real
f (x) =
f
( - -b ) = - - , d
2a
4a
it follows that
and so 0 :::; -d :::; -D. Case 2. d :::: 0. Since F ± f is :S 0. That is,
F (x) ± .f (x)
(B + b) 2 - 4 (A + a) (C + c )
:::: 0 for all x it fol lows that the discriminant of :::; 0,
(B - b) 2 - 4 (A - a) (C - c) :::;
0.
On adding these two inequalities we find that 2(D + d) :::; 0, so 0 :::; d :::; -D. Solution to AS We exhibit a bij ection between the two sets. Suppose we are given a Dyck n -path with no returns of even length. It begins with U and later returns to the x axis. Now delete the path's first step U and the last step D o f the first such return. The result is a Dyck (n - 1 )-path. This map is the desired bijection. To reverse it, suppose a Dyck (n - 1 )-path P is given; if P has no returns of even length, prepend UD to P , otherwise locate the initial segment of P through the last even-length return and "elevate" this segment, that is, put a U in front and a D after it.
B
Solution to A6 Yes . Let A be the set of nonnegative integers whose binary expansion has an even number of 1 s and let be those with an odd number of I s . Given n and a1 I= a2 E A with sum n, locate the first position in which the binary digits of a1 , a2 differ (starting from the units digit) and interchange these digits. This gives a bij ection from the ordered pairs counted by rA (n) to those counted by r B ( n ) , with inverse given by the same procedure. So A , B form a partition as desired. Solution to B l There do not exist such polynomials . To see this, suppose there are
such polynomials, and write a (x) = a0 + a1x + a 2 x 2 + · · · and b(x) = b0 + b1x + b2 x 2 + · · · . Then, equating coefficients of 1 , x, and x 2 , we would have the system of equations
2 004 1
VOL. 77, NO. 1 , F E B R U A RY
=
=
y
81 aoc (y) + bod (y)
a 1 c(y) + b 1 d (y)
i = azc(y) + bzd (y) . This system has no solution because c (y) , d (y) span at most a 2-dimensional sub space of the space of polynomials in y and { 1 , y, y 2 } would belong to thi s span, but these three polynomials are linearly independent. Solution to B2 More generally, if we start with the sequence inductively that the kth sequence i s
(k
( �( (L (k )
+ I )st sequence is
1 2 =
]
2k
1
we show
I i s trivial. Assume the result for k ; then the i th entry i n the
=
The base case k
a 1 , a2 , . . . an ,
k- I k
2k- 1
_
1
r
k- 1
_
]
r
r =O
)
1
a i + r + 2k- 1
a; +r +
k- I
L r= l
k - (k � 1
(k ) a _
r -
] I
; +r
So, the fi nal number X 1 1 i s
( )U J +r 2 1 1 - 1 r =O ) n- I ( 1
n- I
'""' �
I
= -
1
-
11
=
r
I
2" - 1 L r=O n
11
r
+ I
=
_
)
1
r
)
a i +r + l ]
2k L r=O
=
1 '""' n-I 2n - I � r =O
(n )
I -- (2 "
-
n 2" - 1
1
_
r
I)
k
o L n I p k J . k Thus it suffices to prove that the power of p on the right-hand side i s the same . The power of p in lcm { l , 2, . . . , m } is pi where pi ::=: m < pi + I . Thus if k i s given then the power of p in { I , 2, . . . , Ln / i J } is preci sely p k if n / p k + I < i ::=: n / p k . There are exactly Ln / p k + 1 J - Ln / p k J such i . Hence the power of p in the right-hand side above Solution to B3 For each prime
IS
L k ( l p�+I J - l p
17k
k
Solution to B4 Clearly f ( r 1 +
r2
see that
f ( r i + rz )
is rational . Hence
=
=
(
=
1)
k= l
=
p
·
k=l
) i s a rational number. By the factored form of f we
a r 1 rz r 1 a r 1 rz
n j) f(k - l nk j f l pk j
+ rz
- r3
) ( r 1 + rz
( r 1 + rz ) ( r i + rz
- r4
- r3 - r4
)
) + a r 1 r2 r3 r4 .
82
MAT H EMAT I CS MAGAZI N E
f (r i + r2 ) = r1r2 Cr1 + r2 ) a (r1 + r2 - r3 - r4 ) is rational. If r1 + r2 f=- 0 then we are done. If r2 = -r1 then ar{ + b r � + cr� + dr1 + = 0= ar{ - b r r + cr� - dr1 e
-------
e
- e,
which gives ( b r l + d )r1 = 0. I f r1 = 0 then r1r2 = 0, a rational number, and w e are done. Note that b f=- 0, since b = 0 and r1 + r2= 0 together imply that r 1 + r2= r3 + r4 , contrary to our hypothesis. Thus r1 = ±,J-djb and r2= �,J-djb, so that r1r2 = djb is rational in this case also. This completes the proof. Solution to B5 Representing points in the plane by complex numbers, we may take A = 1 , B = w, C = w2 , where w = (- 1 + i ) j 2 is a cube root of unity. The line segments AP, BP, and CP then have lengths I P - 1 1 , I P - w l , and I P - w2 1 , which form the sides of a triangle if and only if there exist complex numbers z 1 , z 2 , z3 (the tri angle vertices) such that lz 1 - ZJ I , lz 2 - ZJ I , and lz3 - z 2 l are equal to I P - 1 1 , I P - w l , and I P - w 2 1 . Such numbers do exist, defined by Z1 - Z3 = P - 1 , z 2 - z 1 = w ( P - w), Z3 z 2 = w 2 (P - w2 ) . The sum of these three complex numbers is zero, so, when consid ered as vectors, they are the sides of a triangle. Write P = x + i y . The area of the triangle, found by computing the cross product of two of the sides, is (up to sign)equal to
,J3
( (,J3
� ex 4
l)
But x 2
2
,J3) - ( ,J3 )) ,J3
x - �Y + 2 2
y+� y - �x 2 2 2
=
2
(x 2 + l - 1 ) .
+ y 2 1 , since P is inside the circle, and so the area of the triangle is given by ,f3( 1 - r 2 )j4, where r is the distance from P to 0 . Solution to B6 Let P = {x E [0, I ] : f (x) 2: 0 } and N = {x E [0, 1 ] : f (x) 0} . Put P = !1- (P) , N = /1- (N) (the measures of P and N, respectively) . If either P = 0 or N = 0 then the inequality is obvious. Thus we may assume P 0 and N 0. Consider the average values of I f I on P and N:
_2_N 1N f (x) dx .
f (x) by - f (x ) if necessary, we may assume /1- p
2:
fi-n · Clearly
{ l f (x) + f (y) l dx dy = 2P 2 fl-p , !'}NxN { l f (x) + f (y) l dx dy = 2N2 fl-n · !,lPxP
I n addition,
flxN l f (x) + f (y) l dx dy lfl xN (f(x) + f (y)) dx dy l 2:
= INP/1-p - NPfl-n l = NP(/1-p - !1- n) , and the same inequality holds for the integral over N P. Thus, the given left-hand x
side is greater than or equal to
2P 2 !1-p + 2N P ( /1-p - /1-n) + 2 N 2 f1-n = P (/1-p - !1-n) + (2N - 1 ) 2 !1-n + P /1-p + N fl-n 2:
P /1-p + Nfl-n = t l f (x) l dx
Jo
Thanks to B yron Walden for editorial assistance.
.
from the
Mathematical Association of America
Envi ronmenta l
Mathematics i n the Classroom
B.A. Fusaro & P. C. Kenschaft, Editors Series : Classroom Resou rce M ateri a l s E n v i ro n mental
Mathematics
in
the
Classroom seeks t o m a r ry t h e m ost p ress
i n g c h a l l e n ge of our t i m e with t h e m ost powerfu l
t e c h n o l ogy
math e m at i c s .
of
our
t i m e
It does so at a n e l e m e n ta ry
l eve l , d e m o n strat i n g a w i d e va r i ety of s i g n i f i c a n t e n v i ro n m e n ta l
a p p l i c at i o n s
t h at c a n b e e x p l o red w i t h o u t resort i n g to t h e ca l c u l u s . Seve ra l c h a pters a re access i b l e e n o u g h t o be a text i n a ge n e r a l ed u c at i o n cou rse, o r t o e n r i c h a n e l e m e n ta ry a l geb ra cou rse . G ro u n d l eve l ozone, pol l u t i o n and wate r u se, p rese rvat i o n of wh a l es, m a t h e m at i c a l eco n o m i cs, t h e move m e n t of c l o u d s ove r a m o u n ta i n ra n ge, at l east o n e pop u l at i o n mode l , a n d a s m o rgasbord of newspaper m a t h e m a t i c s c a n be stu d i ed at th i s l eve l a n d c a n be t h e bas i s o f a sti m u l at i n g cou rse that p repa res futu re tea c h e rs n ot o n l y t o l e a r n bas i c m a t h e m a t i cs, b u t t o u n de rsta n d h ow they c a n i n teg rate i t i n to oth e r top i c s . A l so, many of the c h a pte rs a re adva n ced e n o u g h to c h a l l e n ge p rospective m a t h e m a t i c s
m aj ors. Environ mental Mathematics in the Classroom c a n be a text fo r a n i n de p e n d e n t m a t h e m a t i c s cou rse . With the experti se of a n ot h e r tea c h e r, i t cou l d be t h e bas i s of a n i n terd i sc i p l i n a ry cou rse re l at i n g to m a t h e m a t i c s a n d sc i e n c e . I t c a n a l so se rve as a n exce l l e n t s u p p l e m e n ta ry read e r fo r tea c h e rs a n d th e i r stu d e n ts, e i t h e r fo r recreat i o n or as t h e bas i s of i n depe n d e n t stu dy. Cata l og Code: E MC/J R
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Mathematical Puzzle Tales M a rti n G a rd n e r Series: Spectrum Ma rtin offers everybody (not just mathematicians) creative refuge for the imagination. The puzzles in th is book are not just puzzles. Very often, they embody deep mathematical principals that deal with mat ters not yet well enough understood to be applied to the practical wo rld. Such "games" are not more trivial than "real " mathematics. They may well be more important and may be the foreshadowing of future mathematics. -Isaac Asimov from the Preface Martin Gardner published his first book in 1 9 3 5 .
Since then he has published more than 60 books, most
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T h e M A A is p r o u d to reissue t h i s collection o f thirty-six stories taken from I s a a c Asimov's Science Fiction Magazine. Brilliant, amusing, these brainteasers will help yo u sharpen yo ur wits and prepare for takeoff into uncharted universes of the future. The challenging problems presented here are based on geomet ry, logarithms, topology, p robability, weird number sequences, logic and virtually every other aspect of mathe matics as well as wordplay. Included are: Lost on Capra the Bagel
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Space Po ol
The Th ree Robots of Professor Tinker Oracle
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The Explosion of Blabb age's
Remarkable Mathemati cians
F r o m E u l e r t o vo n N e u m a n n loan }ames
l o a n J a mes i nt rod uces a n d profi l es s i xty mathemat i c i a n s from the era when mathemat i cs was freed from i ts c l assica l ori g i n s to deve lop i nto i ts modern for m . The s u bj ects, a l l born between
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CONTENTS A RT I C L ES 3
15
E m i l Post a nd H i s A n t i c i pa t i o n of Gi)del a nd Tu r i ng,
by John Stillwell
E nter, Sta ge Center:
The Early
D r a ma of the H yperbo l i c
F u nctions, by janet Heine Barnett
30
Proof Without Word s : C a u chy-Schwa rz I neq u a l i ty,
31
S i m pson Symmetri zed a n d Su rpassed,
by Cla udi A lsina by Da niel ). Velleman
N OT E S 46
Ti l i ng Recta ngles and Defi c i ent Recta n g l es w i t h
55
Proof W i t h o u t Words: Z by Des MacHale
Tro m i noes, by ). Marshall Ash and Solomon W Golomb x
Z I s a Co u n ta b l e Set,
56
Extrem a l C u rves o f a Rota t i ng E l l i pse, by Carl V. L utzer
61
A Q u est i o n of L i m i ts, by Paul H. Schuette
and )ames E. Ma rengo
P RO B L E M S
69
Proposa l s 1 68 6- 1 690
70
Q u i c k i es 9 3 7-9 3 8
75
A n swers 9 3 7-9 3 8
70
So l u t i o n s 1 6 62 - 1 665
R EVI EWS 76
N EWS A N D L ETT E RS
78
6 4 t h A n n u a l Wi l l i a m Lowe l l P u t n a m Mat hemati ca l Compet i t i o n
