AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 9 Mathematical Gems III
Ross Honsberger MATHEMATICAL GEMS III
By ROSS HONSBERGER THE DOLCIANI MATHEMATICAL EXPOSITIONS
Published by THE MATHEMATICAL ASSOCIATION OF AMERICA
COl1znzittee Oil Publicatiolls ALAN TUCKER, Chairman
SUbC011111zittee 011 Dolciani Mathenzatical Expositions JOSEPH MALKEVITCH, Chairman D. MCCARTHY D. SMALL 10.1090/dol/009
The Do/ciani Mathell1atical Expositions
NUMBER NINE
MATHEMATICAL GEMS III
By ROSS HONSBERGER University o.f Waterloo
Published lind Distributed by THE MATHEMATICAL ASSOCIATION OF AMERICA @1985 by The Mathematical Association ofAmerica (Incorporated) Library of Congress Catalog Card Number 061842
Complete Set ISBN 0-88385-300-0 Vol. 9 ISBN 0-88385-318-3
Printed in the United States ofAmerica
Current printing (last digit): 10 9 8 765 4 3 2 The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Math ematical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor ofMathematics at Hunter College of the City University of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successful expositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. The Association, for its part, was delighted to accept the gracious gesture initiating the revolving fund for this series from one who has served the As sociation with distinction, both as a member of the Committee on Publications and as a member of the Board of Governors. It was with genuine pleasure that the Board chose to name the series in her honor. The books in the series are selected for their lucid expository style and stimulating mathematical content. Typically, they contain an ample supply of exercises, many with accompanying solutions. They are intended to be suffi ciently elementary for the undergraduate and even the mathematically inclined high-school student to understand and enjoy, but also to be interesting and sOluetilnes challenging to the more advanced mathematician.
DOLCIANI MATHEMATICAL EXPOSITIONS
1. Mathematical Gems, Ross Honsberger 2. Mathematical Gems II, Ross Honsberger 3. Mathematical Morsels, Ross Honsberger 4. Mathematical Plums, Ross Honsberger (ed.) 5. Great Moments in Mathematics (Before 1650), Howard Eves 6. Maxima and Minima without Calculus, Ivan Niven 7. Great Moments in Mathematics (After 1650), Howard Eves 8. Map Coloring, Polyhedra, and the Four-Color Problem, David Barnette 9. Mathematical Gems III, Ross Honsberger 10. More Mathematical Morsels, Ross Honsberger
PREFACE
The technical background required for the enjoyment of the essays in this collection seldonl goes beyond the level of the college freshman. It is renlarkable ho",- nluch exciting nlathenlatics exists at this ele mentarv level. While \ye can't help learning something from each new problenl, these topics are presented solely for your el{;oy111enf, and al though a certain anlount of drive and concentration is a price that cannot be avoided, I hope you \yill find that these little genls are well worth the effoli necessary for their appreciation. A glossary of technical ternlS and ideas is given at the end of the es says; the ""Yords that are explained there are nlarked in the text with an asterisk (*). I '" ould like to take this opportunity to thank the menlbers of the Subconlnlittee on the Dolciani Expositions-Joe Malkevitch, Ken neth Rebnlan, Alan Tucker. and Donald Snlall-for a careful review of the manuscript; their constructive criticisnl led to nlany inl provements.
Ross HONSBERGER
CONTENTS
CllAfYfER PAGE 1. Gleanings from Combinatorics 1 2. Gleanings from Geometry 18 3. Two Problems in Combinatorial Geometry 36 4. Sheep Fleecing with Walter Funkenbusch 48 5. Two Problems in Graph Theory 56 6. Two Applications of Generating Functions 64 7. Some Problems from the Olympiads 76 8. A Second Look at the Fibonacci and Lucas Numbers 102 9. Some Problems in Combinatorics 139 10. Four Clever Schemes in Cryptography 151 11. Gleanings from Number Theory 174 12. Schur's Theorem: An Application of Ramsey's Theorem 183 13. Two Applications of ReIly's Theorem 186 14. An Introduction to Ramanujan's Highly Composite Numbers 193 15. On Sets of Points in the Plane 201 16. Two Surprises from Algebra 208 17. A Problem of Paul Erdos 215 18. Cai Mao-Cheng's Solution to Katona's Problem on Families of Separating Subsets 224 Solutions to Selected Exercises 241 Glossary 247 Index 249
SOLUTIONS TO SELECTED EXERCISES
Two Problems on Generating Functions 1. The generating function for the partitions in question are
(i) no even part repeated: IIk~l (1 - x 2k - 1)-1(1 + x 2k ); k 2k (ii) no part occurs more than 3 times: rrk~ 1 (1 + x + x + x 3k );
(iii) no part is divisible by 4: IIk~ 1 (1 - x 4k)/(1 - x k ). Easy manipulations show that (i) and (ii) are the same as (iii).
2. The generating function for the number of partitions in which no part occurs more than d times is
1 - x(d+ 1)k dk j(x) = IT (1 + x k + x 2k + ... + x ) = IT 1 _ x k~l k~l k
The generating function for the number of partitions in which no term is a multiple of d + 1 is clearly the same as this latter expression forj(x).
3. The generating function for the number of partitions in which every part occurs either 2, 3, or 5 times is
II (1 + x 2k + x 3k + x Sk ) = IT (1 + x 2k )(1 + x 3k ) k~l k~l
1 - x 4k 1 - x 6k = IT -- k~l 1 - x 2k 1 - x 3k •
241 242 SOLUTIONS
In the denominator, the factor 1 - x 2k provides a term 1 - xi for i = 2, 4, 6, 8, 10, 12, 14, ... , and 1 - x 3k provides a term 1 - xi for i = 3, 6, 9, 12, 15, .... The 1 - x 4k in the numerator cancels terms 1 xi for i = 4, 8, 12, 16, ... , and the 1 - x 6k cancels 1 - xi for i = 6, 12, 18, .... Therefore the ternlS remaining in the denominator are 1 - xi for i = 2, 6, 10, 14, ... , and i = 3, 9, 15, ... , that is, for i = 2,3,6,9, 10 (nlod 12). Hence the conclusion.
4. The generating function for the partitions in \vhich no part oc curs exactly once is
II (1 + x 2k + x 3k + ...) = II [1 + x 2k (1 + x k + x 2k + ... )] k~1 k~1
= II [1 + x 2k (1 - x k )-I] = II [(1 - x k )-1 (1 - x k + x 2k)] k~1 k~1
1 + x 3k 1 + x 3k 1 + x 6k = II (1 - x k )-1 • = II = IT ------k~l 1 + x k k~l 1 - x 2k k~l (1 - x 2k)(1 - x 3k)• In the denonlinator, the factor 1 - x 2k provides ternlS 1 - xi for i = 2,4,6,8, ... , and 1 - x 3k provides terms 1 - xi for i = 3,6,9, . The 1 - x 6k in the nUlnerator cancels 1 - xi for i = 6, 12, 18, , leaving ternlS for i = 2,3, 4, 6, 8, 9, 10, 12, ... , that is, those for i = 0, 2, 3, 4 (mod 6). Hence the conclusion.
A Second Loob~ at the Fibonacci and Lucas Numbers Posed in section on divisibility. For odd111 we have (17,111) = (34,111) = (19,111) = 1(9,11) =Ib13, or19, that is, 1, 2, or 34. Since (17,111) must be odd, then (17, ill) = 1, and 171'111.
Posed in the set of exercises at the end. 2. Since!" = 1,,+2 - !"+b we have
S = E I" = E 111+2 - 111+1 11=1 111+1111+2 ,,=1 1,,+11,,+2 SOLUTIONS 243
= E[_1 1_J (in which both terms have 1l=1 1,,+1 111+2. thesamejornl) = ()2 - )J + ()3 - )J + ()4 - )J + ..., having partial sums SIl = (1/12) - (11In+2). Since 11111+2 ~ 0 as ~ n 00, we have S = limll-+ oo SIl = 1112 = 1.
3. We know that L m 1111 if and only if 111 divides into 1l an even number of times. SinceL4 = 7, we have 7111l if and only if 4 divides into n an even number of times, which requires 1l to be even. If n is odd, we never have 71/".
s. We have
hll(L~1I - 1) = (a: =:211) [(a211 + ~211)2 - 1]
1 = (a2n - {32n)(a 4n + 2a2n {32n + (3411 - 1) a-{3 1 = (a2n - {32n)(a 4n + 1 + (34n) (since cx{3 -1) a-{3 =
1 - ---(a6n + a.2n + {32n - a. 2n - (32n - (3611) a-{3 (since a{3 = -1, then a. 2n {34n = (32n, for example) 1 - --{3- (a61l - (36n) = 16n· a- This givesj211 + 1611 = 12nL~n, and we have IlII + 81211(1211 + 1611) = Iln + 812n ·/2nL 'tz = Il11 + 81~IIL~1l = Iln + 81ln (since/2n L 2n = 1411) = 9/111 = (3/411)2, a perfect square. 244 SOLUTIONS
6. (ii). We have
0·_0 (an - {3n)llll L == linl {fn == linl no~oo n--'OO a - {3 . a[l - ({3/a)l1.Jllll == hnl . n·-"oo (a - (3) 1//1
Now, a == (1 + -)"5)/2 == 1.6 approximately, and {3 == (1 - -JS)/2 == -.6 approximately, and we have 1{3lal < 1, implying
-' linl 11 00 ({31 CI.)l1 == O. Also a - (3 == -J"S > 1, implying
-. linlll oo (-JS) 1//1 == 1. Thus L == CI..
7. We know that,l2n+1 == ,l~+1 + .l~; also L n == .l~+1 + .1',,-1 by definition. Therefore we have fn+l L n+2 - j~+2Ln == .l~+tC/~+J + ,In+l) - .fn+2(l'n+1 + ,I'll-I)
== ,f~+ I + j~ +l,ln+J - .l~ +2.t~ -+ I- ,fn+2.lll-1 == .l~+1 + .In-+I(j~+J - .In+2) - Cln+1 +j~)(ln+1 - .l~)
2 0 -,- /,2n+l +. /..n+12 - (/,2,n+1 -./.. n2) -.- 1.. n+l + ,nf2 -,- 1' 2n+t· Consequently we have
S == E ,l2n+1 E .t~+1 L n+2 - .t~+2Ln n==l LnLn+1L n+2 n=l LnLn+lLn+2
where each term has the saIne fornl
= (/3- 3\) + (3 ~ 4- /7) + (/7- 7:11) + having partial sums Sn == 1/3 -
8. Since 11. is a given constant, and a = {I + .JS)/2 is also a con stant, we have
11 1 11 1 S = E (11) a 3k- 211 = - E (11) (a3)k = - (1 + a 3)11. k===O k a 211 k=O k a 211 Now a 2 = a + 1, nlaking 1 + a 3 = 1. + a{a + 1) = 1 + a 2 + a
= (1 + a) + a 2 = 2a2•
211 2 211 11 Therefore, S = {1/a ){2a )11 = (1/a ) 2" a 211 = 2 •
9. Adding and subtracting the ternlS corresponding to k = 0 and k = r, we get
,.-1E {-l)k(r) ofk = Er {-1)k (r)jOk - {-1)0 (r)10 - (-1)" (r) .f~. k=1 k k=-=O k 0 r
= l.kEo (-l)k(~) a~ =~k-l + fr (sincer is odd)
1 a - f3 [(1 - a)r - (1 - (3)rl + fr
(as argued in part (a» 1 --[{(3)" - {a)"] + fr {since a + (3 = 1) a-(3 a r - (3r a - f3 + fr
= -lr + f,. = o.
10. Since {311 a" (3" I 1 fll=a-f3 a - (3' and Ia - f3 < 2' then III is the integer nearest a"/ {a - (3). 246 SOLUTIONS
Now a == 1..6180339887 ... , andlogloa == 0.20899 ... , IOgl05 == 0.69897. Therefore a lOO 1 IOglO -JS = 100(.20899 .· .) - 2(·69897 )
== 20.899 ... - .349 ... == 20.55 .
Hence .lloo is the integer nearest 10 20 .55 ... , making it a 21-digit inte ger. The .55 in the mantissa places the first digit at 3.
11. Since (n 2 - nln - nl. 2)2 == 1, we have n2 - m·n - m 2 == ±1, n 2 - Inn - (m. 2 ± 1) == 0, and n == (m ± ..JSm 2-f-4)/2, which implies that 5m2 ± 4 nlust be a perfect square (y2). By our result in the text, then, for some positive integer k, it must be that
In == .lk and y == L k == fk-l + fk+l·
Since L k > .lk for all k > 1, it follows that the positive integer n is given by
making nz and 11. two consecutive Fibonacci numbers. For m, n in the 2 2 2 2 range [1,1981], then, the greatest m + 11. is 987 + 1591 • GLOSSARY
Cauchy Inequality. For all real numbers ai' hi' i == 1, 2, ... , 1l, (aT + a~ + ... + a~) (hI + h~ + ... + b?t) ~ (a]h] + a2h2 + ... + all bll )2,
with equality if and only if £Ii and bi are proportional for all i. Reference: G. P6lya and G. Szego, Problenls and Theorems in Analysis I, Springer-Verlag, 1974.
Centroid. The point of concurrency of the medians of a triangle (a me dian joins a vertex to the midpoint of the opposite side). It is the center of gravity of a systenl of equal nlasses suspended at the vertices.
Circumcircle. The circle which passes through all the vertices of a polygon; its center and radius are called the circumcenter and cir cunlradius.
Convex Set. A set of points is convex if, for every two of its points A and B, the entire segment AB belongs to the set.
Convex Hull. The convex hull H of a set of points S is the "smallest" convex set which contains S; H is to be contained in all convex sets that contain S, and is, therefore, defined formally to be the intersec tion of all convex sets that contain S. In the case of finite sets, one can think of H as being given by an elastic band that is allowed to contract around S.
Dilatation. A geometric transformation, sometimes denoted by O(r), which is determined by a center 0 and a ratio r; O(r) carries a point P to a position in line with OP such that its new distance OP I from 0 is r
247 248 GLOSSARY
tinles its old distance (OP' lOP = r). If r is negative, OP' is layed off along OP on the other side of 0 (that is, opposite P).
Eulcl- Line. The centroid, circunlcenter, and orthocenter of a triangle always lie on a straight line called the Euler line of the triangle.
Incircle. The circle which is tangent to all the sides of a polygon. Its center and radius are called the illcellfer and illrlldius.
L-Tronlillo. 'T'he L-shape that results when one square is renloved fronl a set of 4 equal squares that have been put together to fornl a square.
011hoccnter. "rile point of concurrcncy of the altitudes of a triangle.
Pigeonhole Pl·inciple. (A fundalncntal tool of conlbinatorics) The principle is the following and refincnlcnts thcreon: if nlore than 11 ob jects are distributed into a set of 11 cOlnparhnents, sonle cOlnparttncnt nlust receivc l110re than one of the objects.
Regular Polygon. A polygon having equal sides and equal angles.
Translate. To nl0ve \\,ithout turning. INDEX
Alder, Henry, 68 dianleter 186 Alexanderson, G. L., 32 Dijkstra, E. W., 8, 12, 19 Allaire, Frank 28 Dijkstra-Kluyver, Mrs. B. C., 19 Andre's Problenl. 69 dilatation. 91 Andrews George. 39, 45 68 Dodge, Clayton, 89
Baker, Alan. 207 Edwards, Harold, 194 Balasubrarnanian, 208 Elder's generalization, 8 Beal, David, 37 Equilic quadrilateral, 32 Beatty's Theorenl, 181 Erdos, Paul, 37, 146, 196 Benkoski, Stan, 215 Euclidean algorithnl, 131, 170 Bernoulli Daniel, 109 Euler, L., 64. 109, 143, 216 Bertrand's Postulate 179,180,196 Euler line, 100 Binet's Fornlulas, 108, III Euler's Theorenl, 65, 169 Borwein, D. 220 exponential generating function 70 Brousseau Alfred, 105 Brown's Criterion, 124 Fermat's Theorenl, 169 Bruen, A. 220 Ferrers graph, 41 Fine, N. J., 45 Carnot's Theorenl, 25 Catalan numbers, 146 Gallai, Tibor, 37 Cauchy inequality, 3, 4 Galois Field, 1S3 ceiling function, 225 Garfunkel, Jack, 32 centroid, 100 generating function, 43 64, 77, 96, Cesaro's observations, 109, 110, 111 143 Chebyshev polynomials 208 Goulden, Ian 68 99 circular inversion, 29 Graham, Ron, 89 129 conlplenlentary sequences 12 complete graph 60 Hernlite, Charles, 207 conlplete sequence, 123 Hoggatt, Verner Jr., 175 conlplex plane, 202, 206 honeybee, 102 congruent sets, 202, 206 correspondence, 1-1,45,67 Jackson, David, 68 Crux Mathenzaticorul1z, 1, 76, 174, Johnson, Roger, 24 17S Jordan, J. H., 39 249 250 INDEX
Jung's Theorenl, 186 reflection principle, 146 Just, Erwin, 89 regular polygons, 27 representations, 123 Kaplansky, Irving, 146 Richmond, Bruce, 208 Kay, David, 27 Richter, Bruce, 180 key (cryptographic), 151 rotation, 34, 204 knapsack problenl, 163 RSA system, 166 Koether, I{obb, 27 Russian Olynlpiad, 2, 80 Kiirschak's tile, 30 Ryavec, C., 216
Lagrange, Joseph Louis, 177 Saari, Donald, 51 Liu, C. L., 61 Schoenberg, 1. J., 32 Lucas, Edward, 110 security, 171 Lunl Wan, Judith, 174 series 111ultisection, 210 Seydel, Kenneth, 32 MacMahon, Major Percy, 143 Shank, Herb, 180 Mathenlalieal S'peelruI11, 37, 176 Shapiro, David, 10 matrix Q, 106 Sicrpinski, Waclaw, 196 Michael, Glen, 131 signatures, 167 Millin's series, 134 Singnlaster, David, 77 Morsel #9, 21 Stanley's rrheorenl, 6 Morsel #23, 18 sunl-free sets, 89 Moser, Willianl, IH Swanson, Nornlan, 179 sweep line, 18 Newtnan, J. R., 193 Niven, Ivan, 177 Taylor's Condition, 58 translation, 37, 203 Old Japanese Theorenl, 24 trees, 60 orthocenter, 100 triangulation, 24 tro111inos, 85 packing problenl, 60 Turan, Paul, 37 partial fractions, 44 partial sums, 146 u. S. Olympiad, 28 partitions, 6, 10,39, 64, 140, 176 pathological set, 205 perfect partitions, 141 Vanstone, Scott, 173 permutations, 69 pigeonhole principle, 3, 4, 78, 82, Walch, Ray, 39 89, 133, 140 weakly conlplete sequence, 129 Pi Mu Epsilon Journal, 27, 32, 89 weighted paths, 57 Post, K. A., 8, 21 Wisner, R. J., 39 prime nunlbers, 133, 196 public-key systenls, 162 Yzeren, I. van, 21 Putnam problenl, 1 Zaks, S., 61 Ramachandra, 208 Zeckendorf's Theorem, 129 reflection, 203 Zuckerman, Herbert, 177 AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS
Mathematical Ingenuity
Ross Honsberger was born in Toronto, Canada in 1929 and attend- ed the University of Toronto. After more than a decade of teaching mathematics in Toronto, he took advantage of a sabbatical leave to continue his studies at the University of Waterloo, Canada. He joined the faculty in 1964 (Department of Combinatorics and Optimization) and has been there ever since. He is married, the father of three, and grandfather of three. He has published seven bestselling books with the Mathematical Association of America.
Here is a selection of reviews of Ross Honsberger’s books. “The book is designed to appeal to high school teachers and undergraduates par- ticularly, but should fi nd a much wider audience. The clarity of exposition and the care taken with all aspects of the explanations, diagrams, and notations is of a very high standard.” (about Mathematical Gems II) —K. E. Hirst in Mathematical Reviews “Honsberger’s purpose is simply to convey the excitement and delight of mathe- matical discovery at a fairly elementary level. The fact that elementary mathematics is alive and well, and even fl ourishing, is a salutary reminder for professional math- ematicians…the results described in this book gave considerable pleasure to the discoverers and that can be shared by the readers of the book, thanks to the lucidity and enthusiasm of Honsberger’s style.” (about Mathematical Gems III) —N. L. Biggs in the Bulletin of the London Mathematical Society “ All (i.e., the articles in Mathematical Gems III) are written in the very clear style that characterizes the two previous volumes, and there is bound to be something here that will appeal to anyone, both student and teacher alike. For instructors, Mathematical Gems III is useful as a source of thematic ideas around which to build classroom lectures…Mathematical Gems III is to be warmly recommended, and we look forward to the appearance of a fourth volume in the series. —Joseph B. Dence, Mathematics and Computer Education “Although the mathematics involved is quite elementary and requires little back- ground, the book contains much fascinating material…Written in the author’s usual clear and lively style, this book can be enthusiastically recommended to all lovers of mathematics, whether they be teachers or professional mathematicians.” (About Mathematical Gems III) R. J. Clarke, in The Australian Mathematics Teacher
Cover designed by Lisa M. Brzezniak for Barbieri and Green