One Hundred Problems Involving the Number 11 00 00 1 0 0 A Collection of Problems to Celebrate NCTM's First Century G. Patrick Vennebush Copyright © 2020 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM. One Hundred Problems Involving the Number 100 A Collection of Problems to Celebrate NCTM’s First Century G. PATRICK VENNEBUSH Copyright © 2020 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM. Copyright © 2020 by The National Council of Teachers of Mathematics, Inc. 1906 Association Drive, Reston, VA 20191-1502 (703) 620-9840; (800) 235-7566; www.nctm.org Library of Congress Cataloging-in-Publication Data Names: Vennebush, G. Patrick, author. Title: One-hundred problems involving the number 100 : celebrate NCTM’s first century / G. Patrick Vennebush. Description: Reston, VA : National Council of Teachers of Mathematics, Inc., [2020] | Summary: “Math educators always seek great problems and tasks for the classroom, and this collection contains many that could be used in various grades. By using this book, the reader will understand ways that great problems can be used to encourage student participation and to promote powerful mathematical ideas. In addition, suggestions for how problems can be presented in the classroom will provide professional development to teachers in the form of effective routines for promoting problem solving. This book would be both a fun read for NCTM’s membership”—Provided by publisher. Identifiers: LCCN 2020022989 (print) | LCCN 2020022990 (ebook) | ISBN 9781680540659 (paperback) | ISBN 9781680540666 (pdf) Subjects: LCSH: Problem solving. | Number theory. | Arithmetic functions. Classification: LCC QA63 .V46 2020 (print) | LCC QA63 (ebook) | DDC 510.71/2—dc23 LC record available at https://lccn.loc.gov/2020022989 LC ebook record available at https://lccn.loc.gov/2020022990 The National Council of Teachers of Mathematics advocates for high-quality mathematics teaching and learning for each and every student. When forms, problems, and sample documents are included or are made available on NCTM’s website, their use is authorized for educational purposes by educators and noncommercial or nonprofit entities that have purchased this book. Except for that use, permission to photocopy or use material electronically from One Hundred Problems Involving the Number 100: A Collection of Problems to Celebrate NCTM’s First Century must be obtained from www.copyright.com, or contact Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. Permission does not automatically extend to any items identified as reprinted by permission of other publishers and copyright holders. Such items must be excluded unless separate permissions are obtained. It will be the responsibility of the user to identify such materials and obtain the permissions. The publications of the National Council of Teachers of Mathematics present a variety of viewpoints. The views expressed or implied in this publication, unless otherwise noted, should not be interpreted as official positions of the Council. Printed in the United States of America Copyright © 2020 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM. Table of Contents Acknowledgments .........................................................................................................v Part 1: Why This Book? ................................................................................................1 Why 100? ........................................................................................................................................1 What Makes a Great Problem? .......................................................................................................2 How to Use This Book ...................................................................................................................6 How This Book Is Organized .........................................................................................................7 Problem Solving in the Classroom .................................................................................................9 Classroom Practices That Promote a Problem-Solving Culture ................................................11 Actions That Inhibit a Problem-Solving Culture .......................................................................16 Part 2: The Problems .................................................................................................19 Part 3: Solutions and Suggestions ...........................................................................41 Problems 1–9 ................................................................................................................................48 Problems 10–19 ............................................................................................................................56 Problems 20–29 ............................................................................................................................65 Problems 30–39 ............................................................................................................................77 Problems 40–49 ............................................................................................................................86 Problems 50–59 ............................................................................................................................96 Problems 60–69 ..........................................................................................................................107 Problems 70–79 ..........................................................................................................................123 Problems 80–89 ..........................................................................................................................137 Problems 90–101 ........................................................................................................................152 References ................................................................................................................171 iiiiii Copyright © 2020 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM. Acknowledgments Over the years, I’ve had the privilege of working with myriad wonderful educators, solved many problems from math competitions and conference sessions, and read many amazing books by my peers, so every problem in this book represents an amalgamation of ideas. To say that I’ve been fortunate in my career would be like saying that Euclid was adequate at geometry. I spent several summers at the Johns Hopkins University Center for Talented Youth with a lot of brilliant educators, and countless amazing problems made the rounds among staff. I worked with presidential award winners and lifetime achievement award recipients at the Public Broadcasting Service, Education Testing Service, National Council of Teachers of Mathematics, and Discovery Education. I’ve attended conferences, professional development workshops, webinars, and math circles where great problems were shared on a regular basis. I was the liaison to the Question Writing Committee (QWC) when I worked for the MathCounts Foundation, and later I served as a member of the QWC and eventually as the chair. Which is to say, these problems come from the collective wisdom of hundreds of educators and thousands of resources. Where possible, I give attribution in the text. But to pinpoint the exact source of each problem would be like attempting to identify which butterfly flapped its wings in Madagascar and caused a typhoon in Japan. Instead, I’ll offer gratitude to the math educators who have freely shared their ideas and great problems with me. Through writing, solving, and discussing, my out-of-school education was greatly enhanced by David Barnes, Art Benjamin, Carey Bolster, Judy Ann Brown, Tom Butts, Edward Early, Skip Fennell, Carol Findell, Peg Hartwig, Marjan Hong, Liz Marquez, Harold Reiter, Nicole Rigelman, Jim Rubillo, Marian Small, Dave Sundin, Kris Warloe, and Joshua Zucker. Their influence can be seen throughout the pages of this book. There are hundreds of others who should probably be listed, but the margins are too narrow to contain all their names. I offer a collective thank you to them all. Finally, the completion of this book happened in no small part thanks to the efforts of my family. My wife, Nadine Block, allowed me the time and space to write when I should have been helping with chores or giving her the time and attention she deserves. I am eternally grateful that a wonderful woman was able to find the coping mechanisms necessary to tolerate my shortcomings and idiosyncrasies. My twin sons, Alex and Eli, served as the preliminary editors of this volume, reading every problem multiple times, correcting errors, and suggesting improvements. They are my favorite students, my greatest joy, and the best audience for my terrible jokes, and the quality of this book is a direct result of their tremendous help. I give thanks daily for a family that I love dearly but would never claim to deserve. —G. Patrick Vennebush vv Copyright © 2020 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM. Part 1 Why This Book? The century of problems in this book range from elementary to advanced. They cover topics from number theory, probability and statistics, geometry, algebra, and operations. Some will inspire you; others will challenge you; but hopefully all of them will intrigue you. Some of them are classics that you’ve seen before; others are brand new, and they are being published for the first time in this volume. All of them have brought joy to me and the students with whom I’ve worked, and I hope that you and your students have a similar experience. In the interest of complete transparency, know that the cover contains a lie. There are actually 101 problems in this book. The extra 1 percent has been provided absolutely free of charge. WHY 100? Any number that “turns the dial” from a string of 9s to a 1 followed by some 0s is important, but it’s possible that 100 may be the most important power of 10. For elementary teachers and students, the 100th day of school is a big deal, partially because it indicates that the year is more than half over, but also because the number 100 represents a significant milestone in regard to place value. The number 100 is used as the basis for comparisons with percentages. A temperature of 100° C will cause water to boil, there are 100 members of the U.S. Senate, there are 100 years in a century, and there are 100 letter tiles in a Scrabble® game. The 100 most common words in the English language account for more than half of the words used in speaking and writing (Stuart et al. 2003; McNally and Murray 1964). The number 100 is a square number, and it’s the sum of the first nine prime numbers, the sum of the first ten odd numbers, the sum of two square numbers, and the sum of the first four cube numbers: n 100 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 n 100 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 n 100 = 64 + 36 n 100 = 64 + 27 + 8 + 1 Why This Book? 1 Copyright © 2020 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM. The number 100 is also the sum of several pairs of prime numbers (see problem 44, Prime Time, on page 89). When asked why 100 is special, my 12-year-old son said, “It’s big enough, but it’s not too big” (E. Vennebush, personal communication, April 12, 2020). For instance, counting from 1 to 10 can be done rather quickly, but counting to 100 requires some effort, and getting there feels like an accomplishment. But then continuing to 1,000 or a higher power of 10 would take too long, and just doesn’t seem worth it. That same idea is true beyond counting, too. For patterns and problem solving, the number 100 serves as a good guidepost. Asking students to find the sum of the first 10 positive integers doesn’t feel like much of a challenge. Asking students to find the sum of the first 1,000 positive integers feels like too much of a challenge. But asking students to find the sum of the first 100 positive integers—as Carl Friedrich Gauss was supposedly asked to do by his teacher (see problem 21, Gauss and Check, page 64)—gives the impression that the sum could be calculated by hand or found with a calculator if necessary, but even using a calculator would be tedious enough to suggest that, just maybe, considering a simpler problem or looking for a pattern could be beneficial. The number 100 is just big enough to exhilarate, but not so big as to intimidate. Every problem in this book involves the number 100. Sometimes, a sequence or series will contain 100 terms; other times, the sum of an expression will be 100. Some problems use the number 100 as an exponent, a product, an area, or a perimeter; other problems use 100 as a constant in an equation, as the number of objects in a pile, or as the difference in heights between two dogs. Some problems include a ladder with 100 rungs, a pile with 100 coins, a deck with 100 cards, or a jug that holds 100 ounces. There is something magical about the number 100, and these problems attempt to capture some of that magic. But no matter how 100 is used, each problem is meant to spark curiosity and motivate students (and their teachers) to want to solve it. WHAT MAKES A GREAT PROBLEM? Consider the following three problems: Problem 1. Let d(n) denote the number of positive divisors of the integer n. Prove that d(n) is odd if and only if n is a square. Problem 2. Which positive integers have an odd number of factors. Justify your answer. Problem 3. Imagine n lockers, all closed, and n men. Suppose the first man goes along and opens every locker. Then the second man goes along and closes every other locker beginning with #2. The third man goes along and changes the state of every third locker beginning with #3 (i.e., if it’s open, he closes it, and vice versa). If this procedure is continued until all n men have passed by all the lockers, which lockers are then open? (Butts 1980, p. 257) 2 ONE HUNDRED PROBLEMS INVOLVING THE NUMBER 100 Copyright © 2020 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM. This book contains 100 problems, and if you’re a math educator, then you’re likely far more interested in those problems than in any information contained in this introduction. I suspect that prior to reading this section, you’ve at least perused the rest of the book, or quite possibly, you’ve attempted every problem contained herein. Consequently, it’s extremely likely that the third problem above looks familiar; it’s very similar to problem 29, The Locker Problem, on page TK. The Locker Problem is a classic that’s been around for quite a while. I first learned of the problem from colleagues in the mid-90s. The above presentation by Butts originally appeared in NCTM’s Forty-Second Yearbook, Problem Solving in School Mathematics, in 1980. I suspect that versions of the problem existed long before that, too. What is it about this problem that has allowed it to survive for more than four decades? Why is this problem and the details of its solution retained by students, while thousands of other traditional textbook exercises are completed and then quickly forgotten? The three problems above are, in fact, the same problem presented in three different ways. That is, their solutions rely on the same underlying mathematics, but the first version states a fact and asks for proof, the second asks a question, while the third offers a novel presentation. Butts (1980) contends, “The phrasing of the third version would significantly motivate the potential solver to tackle the problem” (p. 257). While the format in which the problem is presented is certainly part of its allure, I believe that there are two deeper, more fundamental reasons why the problem appeals to students. First, the mathematics of the Locker Problem is beautifully disguised. Whereas the first two versions use the words divisors and factors, the third version contains very little mathematical language. It isn’t presented as a mathematical problem; it’s just a puzzle. Second, the solution doesn’t rely on the application of previously learned material, but instead it requires students to discover something new about the underlying structure of mathematics. In the vernacular of Common Core, that’s mathematical practice 7. In common parlance, that’s just fun and exciting. The Locker Problem isn’t just an exercise; it’s an actual problem without an obvious solution strategy. The K−8 Publishers’ Criteria (CCSSO 2012) draws a distinction between problems and exercises: “In solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery” (p. 17). Because students may learn some new mathematics while solving the Locker Problem, that is exactly why it should be used in the classroom. Unfortunately, not everyone holds that belief. I shared the Locker Problem during a conference session recently, and afterwards a middle school teacher told me that she thought it would be a great problem to share with her students after they learned about factors. A similar sentiment was expressed by a student online, claiming that the problem is not appropriate for sixth-grade homework because “there are other ways to teach kids common factorization” (Lollos 2014). Both of those comments unfortunately miss the point and fail to recognize the benefit of great problems. They should not be reserved merely to reinforce Why This Book? 3 Copyright © 2020 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM. mathematical skills; they should instead be used to promote the development of conceptual understanding. In the book Switch, Heath and Heath (2010) state, “Script the critical moves. Don’t think big picture, think in terms of specific behaviors” (p. 259). They contend that change happens not when people are asked to make many changes, but when they instead are asked to make just one important change. Using problems at the beginning of lessons is a simple but critical change that could greatly affect mathematics education. Because many great problems already appear in the curriculum, implementing this change would only require a change as to when problems are presented. Rather than assigning them as homework or introducing them after direct instruction, use them instead to open lessons and form the basis for instruction. This shift to a problems-first approach would emphasize problem solving, focus on conceptual development, allow for the natural introduction of vocabulary, and increase learning and retention. Seeley (2014) refers to this as “upside-down teaching” and describes it as having three parts: 1. S tudents “mess around with a task for a while, ideally engaging in some thinking, trying things out, and generally wrestling with or constructively struggling with mathematics arising from the problem” (p. 90). 2. T hen, possible solution strategies are presented in a whole-class discussion, but students will explain, clarify, and continue to struggle with the mathematics. 3. Finally, the teacher connects the students’ work to big ideas, ensuring that students understand how the problem connects to the important mathematics that they are expected to learn. Note that only in the third part does the teacher have the role of providing any explanation or direct instruction—after students have struggled and presented their work. The Locker Problem, when solved prior to learning about factors, provides an opportunity to introduce terms such as factor, divisor, prime, composite, and square number when they arise organically as part of a class discussion; moreover, it has the potential to unveil deep insights about the structure of our number system and the nature of mathematics: Most integers have an even number of factors because the factors occur in pairs; square numbers have an odd number of factors because the square root of the number occurs as a repeated factor; and the integer 1 is the only number with a single factor. Indeed, when students attempt problems for which they don’t immediately have a solution strategy, they learn more. Brown, Roediger, and McDaniel (2014) note that “Trying to solve a problem before being taught the solution leads to better learning, even when errors are made in the attempt” (p. 4). It is my sincere hope that all of the problems in this book—like the Locker Problem—serve to initiate discovery, exploration, and learning. They are intended to stimulate interest while providing an opportunity for students to better understand some aspect of mathematics. I have always 4 ONE HUNDRED PROBLEMS INVOLVING THE NUMBER 100 Copyright © 2020 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM. contended that great math problems are like good jokes: Both make you think, and the punch line is unexpected. Those are two important elements of a rich mathematical task; the list below provides a more extensive set of criteria. Criteria of a Great Math Problem n It’s exploratory. That is, the problem requires students to pose conjectures and test them. In short, students will need to get “messy.” n Students want to get messy with the problem because it’s posed in an interesting way. Often, they contain real-life contexts—but remember, math is a context! n The solution strategy isn’t obvious, and there are multiple solution strategies. n It feels more like a puzzle than a problem, more like play than work. n Something—and, hopefully, something mathematical—will be learned by solving the problem. n It is based on important mathematics. n The problem is low-floor, high-ceiling, meaning that there are entry points for every student but also challenges for high-achieving students. n At least one solution is understandable to every student. n Although the problem can be solved independently, students will benefit from collaboration and discussion. n The problem provides opportunities for extension. The Locker Problem meets all of those criteria, including the possibility of an extension: One day, some students are out sick. Regardless, those present repeat the process and just skip the students who are absent—for instance, if student 3 was absent, then no one would change the state of every third locker. When they finish, only locker #1 is open, and the other lockers are all closed. How many students were absent? (Math Jokes 4 Mathy Folks 2011). Well, would you look at that? Only TK pages into this book, and already an Easter egg! I hope you had fun solving that extension. The criteria on the list above were generated several years ago, following a problem-solving workshop that I offered to middle school teachers. It was a four-hour session, and the first activity required attendees to put some of their favorite problems onto pieces of chart paper. Those problems were to serve as motivation for the work we’d do for the rest of the day. The instructions specifically Why This Book? 5 Copyright © 2020 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.