Table Of ContentMathematical
Recreations from the
Tournament of the
Towns
Mathematical Recreations from the Tournament of the Towns contains the
complete list of problems and solutions to the International Mathematics Tour-
nament of the Towns from Fall 2007 to Spring 2021. The primary audience for
this book is the army of recreational mathematicians united under the banner of
Martin Gardner. It should also have great value to students preparing for math-
ematics competitions and trainers of such students. This book also provides an
entry point for students in upper elementary schools.
Features
• Huge recreational value to mathematics enthusiasts
• Accessible to upper-level high school students
• Problems classified by topics such as two-player games, weighing problems,
mathematical tasks etc.
AK Peters/CRC Recreational Mathematics Series
Series Editors
Robert Fathauer
Snezana Lawrence
Jun Mitani
Colm Mulcahy
Peter Winkler
Carolyn Yackel
Six Simple Twists
The Pleat Pattern Approach to Origami Tessellation Design, Second Edition
Benjamin DiLeonardo-Parker
Tessellations
Mathematics, Art, and Recreation
Robert Fathauer
Mathematics of Casino Carnival Games
Mark Bollman
Mathematical Puzzles
Peter Winkler
X Marks the Spot
The Lost Inheritance of Mathematics
Richard Garfinkle, David Garfinkle
Luck, Logic, and White Lies
The Mathematics of Games, Second Edition
Jörg Bewersdorff
Mathematics of The Big Four Casino Table Games
Blackjack, Baccarat, Craps, & Roulette
Mark Bollman
Star Origami
The Starrygami™ Galaxy of Modular Origami Stars, Rings and Wreaths
Tung Ken Lam
Mathematical Recreations from the Tournament of the Towns
Andy Liu, Peter Taylor
For more information about this series please visit: https://www.routledge.com/AK-PetersCRC-Recreational-
Mathematics-Series/book-series/RECMATH?pd=published,forthcoming&pg=2&pp=12&so=pub&view=list
Mathematical
Recreations from the
Tournament of the
Towns
Andy Liu
Peter Taylor
First edition published 2023
by CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Liu, A. C. F. (Andrew Chiang-Fung), author. | Taylor, P. J. (Peter
James), 1947- author. | Konstantinov, N. N. (Nikolaĭ Nikolaevich),
1932-2021, honoree.
Title: Mathematical recreations from the Tournament of the Towns / Andy
Liu, Peter Taylor.
Description: First edition. | Boca Raton : AK Peters/CRC Press, 2023. |
Series: AK Peters/CRC recreational mathematics series | “Dedicated to
the memory of Nikolay N. Konstantinov”-- title page verso. | Includes
bibliographical references and index.
Identifiers: LCCN 2022023876 (print) | LCCN 2022023877 (ebook) | ISBN
9781032352923 (hardback) | ISBN 9781032352909 (paperback) | ISBN
9781003326212 (ebook)
Subjects: LCSH: International Mathematics Tournament of the Towns
(Competition) | Mathematical recreations.
Classification: LCC QA95 .L765 2023 (print) | LCC QA95 (ebook) | DDC
793.74--dc23/eng20221007
LC record available at https://lccn.loc.gov/2022023876
LC ebook record available at https://lccn.loc.gov/2022023877
ISBN: 978-1-032-35292-3 (hbk)
ISBN: 978-1-032-35290-9 (pbk)
ISBN: 978-1-003-32621-2 (ebk)
DOI: 10.1201/ 9781003326212
Typeset in Latin Roman
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Dedicated to the Memory of
NIKOLAY N.
KONSTANTINOV
Table of Contents
A Brief History of Tournament of the Towns viii
Preface xi
Acknowledgement xiii
Authors Bio xiv
Part I: Mathematical Recreations 1
Problems 4
Answers 31
Part II: Mathematics Education 33
Arithmetical Recreations 34
Geometrical Recreations 38
Combinatorial Recreations 42
Solutions 46
Part III: International Mathematics
Tournament of the Towns 63
Tournament 29 65
Solutions 74
Tournament 30 96
Solutions 104
Tournament 31 124
Solutions 132
Tournament 32 154
Solutions 162
Tournament 33 183
Solutions 191
Tournament 34 212
Solutions 220
Tournament 35 240
Solutions 248
Tournament 36 267
Solutions 275
Tournament 37 291
Solutions 299
Tournament 38 318
Solutions 325
Tournament 39 346
Solutions 354
Tournament 40 374
Solutions 381
Tournament 41 403
Solutions 410
Tournament 42 432
Solutions 440
vii
A Brief History of the
Tournament of Towns
The International Mathematics Tournament of the Towns is a mathematical
problem-solvingcompetitionwhichstartedintheformerSovietUnion(USSR)
in 1980.
In the years immediately preceding 1980, there was considerable concern
about the opportunities available to students from the larger cities, such as
Moscow,LeningradandKiev,toparticipateintheUSSRAll-UnionOlympiad,
under the format of this event. The All-Union Olympiad had, for many years
been organised by A. N. Kolmogorov and his jury. Due to the difficulty en-
countered by Kolmogorov and his colleagues in changing the format to allow
more students from the larger cities to participate, a new competition was
organised.
In the first year (1980) this competition was known as the “Olympiad of
ThreeTowns”(itwasheldjustinMoscow,LeningradandRiga).Othertowns
wereallowedtoparticipateafterthefirstyearandthecompetition’snamewas
changed to “Tournament of the Towns”. In the early years the competition
hadnoofficialstatus,butin1984thecompetitionbecameofficiallyrecognised
andtheorganisingcommitteebecameasubcommitteeoftheUSSRAcademy
of Sciences.
Ataboutthistimetownsfromothercountrieswereinvitedtoparticipate.
Towns from Bulgaria first took part in the 6th Tournament (1984 to 1985).
The infrastructure for supporting mathematics competitions was so strong in
Bulgaria that most major towns participated and a national committee was
established to coordinate Bulgarian entries. Bulgarian towns were frequently
among the most successful participants in the Tournament. During this time
townsfromothercountriesinthe(then)EasternBlocalsoparticipated.These
countries included Romania and Poland.
In1988IattendedICME-6inBudapestandreceivedaninvitationviatwo
Bulgarian mathematicians, Jordan Tabov and Petar Kenderov, to enter Aus-
tralian teams in the Tournament. At the time Australia had become under
viii
Tournament of the Towns ix
increasing notice for its successful Australian Mathematics Competition and
hostingthe1988InternationalMathematicalOlympiad,bothunderthedirec-
tion of my colleague at the University of Canberra, Peter O’Halloran. Under
Peter’sinfluencetheWorldFederationofNationalMathematicsCompetitions
(WFNMC) had also been founded in 1984. So I think Jordan and Petar were
acting on behalf of the organisers wanting to spread the Tournament to the
West, and they felt that Australia might have the infrastructure to start.
I looked at the problems from the previous Tournament. They looked
very creative, not purely technical as in most Olympiads, but requiring good
structural thinking, as well as technical knowledge. The problems were often
presented in real-life settings which could be explained to people who were
notmathematiciansandhavingunexpectedresults.Itlookedveryinteresting,
enough to form a mathematics circle in Canberra which could continue that
which had been run for some time by colleagues Mike Newman and Laci
Kovacs. So we reassembled their circle with the Tournament problems as
material.
So then at short notice students from Canberra did participate in the
10th Tournament (1988 to 1989), becoming the first Western students and
first English-speaking students to participate in the Tournament. The 11th
Tournament (1989 to 1990) saw further growth in the Tournament. More
Australian towns participated, and a national committee was established to
coordinateAustralianparticipation.EntrieswerealsoreceivedfromHamburg
(then West Germany) and colourado Springs (USA).
In May 1989, for the first time I met Andy Liu, whom we had invited to
Canberra to help set the 1990 papers of the Australian Mathematics Com-
petition. I acquainted him with the Tournament, and he was obviously just
as impressed by the quality of the problems as I had been. He and I became
close working colleagues on the Tournament in the years to come.
In1990IvisitedMoscowfordiscussions,mainlywithKonstantinov,prob-
lems committee chairman Nikolay Vasiliev and others. I explained to Kon-
stantinov that we were enthusiastic, and had started organising the Tourna-
ment,butinordertohavecompleteintegrityinEnglishweneededalltheback
papersinEnglish.Hewenttoafilingcabinetandgavemeacompleteset,but
ofcourseallinRussian.IundertooktotranslateallthequestionsintoEnglish
and to publish them with solutions via the Australian Mathematics Trust.
Many of my colleagues contributed the solutions, and we published the
best, but Andy Liu was now working closely with me, and he produced the
greatest number of solutions, and some of really outstanding quality.
The 12th Tournament saw even further growth. Towns from Canada,
Spain, Czechoslovakia, Iran, Yugoslavia, New Zealand, the United Kingdom,
Zimbabwe, French Polynesia, India, Indonesia, Singapore, Israel, Colombia,
HongKongandthePhilippineseitherenteredorweregearingupforentering
in the near future.
