Book Review

Math Girls

Reviewed by Mari Abe and Mei Kobayashi

Math Girls mathematics, the Hiroshi Yuki (translated by Tony Gonzalez) user-friendliness of Bento Books, 2011 the book, and its US$24.99, 304 pages cheerful and play- ISBN-13: 978-0983951315 ful prose make Math Girls an enjoyable When we were asked to review Math Girls, trans- read for a person lated from the Japanese novel 数学ガール , we with an inquisitive were unsure how to respond. A cursory glance at mind and math- the cover of the book might lead naïve prospec- ematical inclination. tive readers to assume it is a pro-math book to We do caution inspire young women to pursue a career in the that Math Girls is mathematical sciences. A quick search on the not for everyone, Internet retrieved a comic book and some particularly those free downloadable short stories. We wondered who have little in- whether it might be an adaptation of a strange terest in or affinity manga (Japanese comic book). Our curiosity was for mathematics. The works of ancient Greek further piqued on finding that many of our female philosophers, scientists, and mathematicians form colleagues had received copies of the book as a gift a leitmotif in the book, and some readers might from their husbands (who appear to be greater fans find it fun to ferret out the clues and allusions of the book than are their wives). So we decided to these works; those readers who are interested to dive in. only in mathematics or are unfamiliar with Greek We hope our review will answer some ques- tions that came to our minds before reading philosophy can remain oblivious to the references the book. Will this tremendously popular Japa- and still find much to enjoy. To help determine the nese book enjoy the same success and appeal suitability of the book as a gift, we provide a brief with the American (or English-speaking) reader- background sketch of the author, profiles of the ship? Or will that special something, that je ne characters, and remarks on the mathematical pre- sais quoi, be lost in translation? Does the book sentation. We do not summarize the mathematical 1 successfully bridge cultural differences (American- content, as it is available in the table of contents. Japanese, male-female)? Can a male writer por- Background of the Author tray young females engaging in mathematical dialog in a positive, exciting, and realistic light? Hiroshi Yuki was an established author of books on Is the title appropriate? Or was it selected to be computer science until a series of vignettes posted intentionally misleading? By the conclusion of on a whim on his website were serendipitously the book, most of these questions seemed less discovered by interested readers. In response to relevant. The journey to the magical world of requests from fans, he compiled the stories into a book, Math Girls. The overwhelming positive re- sponse to the book was highly unusual for a book Mari Abe and Mei Kobayashi are researchers at IBM Re- search in Tokyo. Their email addresses are respectively [email protected] and [email protected]. 1 The table of contents is available for browsing online at DOI: http://dx.doi.org/10.1090/noti886 http://www.amazon.com.

956 NOTICES OF THE AMS VOLUME 59, NUMBER 7 about mathematics. According to the publisher,2 we read on, we find four main characters in the Math Girls is “currently in its eighteenth printing book, each of whom embodies characteristics that in Japan, and has been one of the top 5 general ancient Greeks associated with the elements. The interest mathematics books on amazon.com.jp name Tetra also references Empedocles’ book Tet- since its original release in 2007.” rasomia, or Doctrine of the Four Elements, in which A new Math Girls novel is being published al- the ancient Sicilian (Greek) philosopher, scientist, most annually (in Japanese), with each subsequent and healer proposed, “All matter is comprised of novel covering more advanced material. Major four ‘roots’ or elements of earth, air, fire and water topics covered in the second volume are number … the interaction of (which) is influenced by the theory, abstract algebra (groups, rings, and fields), relationship between the two great life energies methods of proof, Euler’s identity, and Fermat’s of Love and Strife.”3 The two characters who com- last theorem. The third volume introduces the plete the world of four elements are Mr. Muraki, Peano axioms, sets and logic, limits, trigonom- the mathematics teacher, and Mrs. Mizutani, the etry, and Gödel’s incompleteness theorems; and librarian. the fourth, some topics from probability and The young and pure Tetra, who inspires the nar- mathematics in computer science, leading up to rator in his math work, is the embodiment of air, a discussion of randomized algorithms. Japanese which the Greeks regarded as a universal power comic books based on Math Girls have also been and pure substance. Miruka is the embodiment of published, each featuring the imaginative talent of fire, an element said to provide clarity and com- a different artist. The publisher hopes to release monly associated with energy, assertiveness, and an English translation of Math Girls manga and the passion. Mrs. Mizutani is the embodiment of water, second novel in the series within the next twelve an element associated with binding and separation. months. We review here the first Math Girls book Tetra and/or Miruka and the narrator often meet published in 2007. in the library, whose caretaker, Mrs. Mizutani, dismisses (and separates) them at closing time. Overview of the Plot and Profile of Mr. Kimura is the embodiment of earth, an element Characters associated with seasons, crops, and fertility. He Math Girls can be viewed as a collection of conver- assigns new math problems that serve as catalysts sations between high school students about how that drive meetings between the narrator and the to solve mathematics problems that require deep Math Girls (Miruka and Tetra). A Japanese reader thought. Each chapter focuses on an historically would immediately realize Mr. Muraki’s association significant problem which (in some cases) can be with earth from his name, which means village tree, solved via more than one approach. In to and Mrs. Mizutani’s association with water, since tutoring readers on how to solve math problems, her name means valley of water. the conversations between the students provide Three minor characters contribute to the boy- examples of how to think about mathematics, ap- girl/girl-boy subplots. AyAy is a talented pianist proaches for tackling new problems, and ways to who is the president of Fortissimo, the school understand difficulties associated with learning. piano club. She is introduced to us when she gives The protagonist (and narrator) is an awkward a duet recital with Miruka. The Japanese word ay male teenager who loves mathematics and has (more commonly spelled ai) has several different limited experience conversing with girls. He is meanings depending on the Japanese character. simultaneously flattered and bewildered when a It can mean love, meet, companion, or play pal young, confident female math whiz named Miruka (when combined with the character for play) or lavishes attention on him in the form of math- the indigo (blue) color or dye.4 A second minor ematical puzzles and quizzes. In the naming of this character, the author has invoked a typical play 3T. Marks (1998), Elemental: The four elements—from on words by the Japanese. Although the Japanese Ancient Greek Science and Philosophy to Poetry, http:// written characters differ, the pronunciations for www.webwinds.com/thalassa/elemental.htm. Ac- the name Miruka and the phrase “Will you take a cessed 16 Dec 2011. look at this?!” are identical. 4In the original Japanese book, AyAy was named EiEi Separately and equally unexpectedly, the pro- (エイエイ). Its pronunciation would be puzzling for Ameri- tagonist is approached by an awkward younger cans and would lead some to recall lyrics from Old girl named Tetra, who is interested in mathematics MacDonald’s farm (http://en.wikipedia.org/wiki/ but suffers from a lack of confidence. Her intro- Old_McDonald_Had_a_Farm). In Japanese there are sev- duction and the protagonist’s follow-up question, eral characters that are pronounced Ei. For example, in “As in mono-, di-, tri-?” hint at the four elements Japanese, eiei is an onomatopoeic word thought to evoke the sound of cheering (rooting) for someone or something. from ancient Greece: earth, air, fire, and water. As Alternatively, it can mean English, the first letter of the al- phabet A, or to reflect or project (and is the first character 2Bento Books website: http://bentobooks.com/ for the compound word used for movie), or it can serve as 2011/09/math-girls-publication-announcement/. the first character of the two-character word for nutrition. Accessed 6 Dec 2011.

AUGUST 2012 NOTICES OF THE AMS 957 character, Kaito,5 a friend and classmate, provides For reasons we may only venture to guess, the a pair of tickets to a planetarium, which the narra- name of the narrator remains a mystery through tor uses to take Tetra for what was intended to be the entire book, perhaps suggesting that he can be a date (she falls asleep in the dark). In Japanese, any adolescent male. Omission of a name and clear kaito can have several meanings depending on identity helps readers travel to the mathematical the characters used in written form. Two common world of the narrator and empathize with his spellings consist of two characters, the first being challenges. sea (ocean) and the second, the constellation Ursa Some readers may become weary of the girl-boy Major or the character to soar (fly). Several common subplots and the simplistic text, replete with sym- words share the same pronunciation. Some nouns bolism from ancient Greek myths and philosophy. are answer (reply, response), answer (solution, e.g., If they accept Math Girls as a mathematics text- of a math problem), the president, a small ferry; book in disguise and the girl-boy encounters as or the verb to dissolve (e.g., a party); or the verbs (sometimes trite and sometimes more successful) thaw or defrost (e.g., frozen foods) or decompress attempts to introduce approaches to thinking as (a compressed or encoded file) in computer sci- a professional mathematician, the book will read ence. A third character, a flute, appears in the swimmingly. Remarks by the author in the After- background when the narrator and Tetra meet in word section may go over better with the Japanese the library. In ancient Greece the music produced male readership than with females and Americans by flutes was associated with the voice of love.6 due to cultural differences: The characters’ relationships, which are steeped This fascination one can have with in platonic love and conflict, are rather contrived mathematics is similar to the fascina- and puerile. Nevertheless, they serve several pur- tion a boy can have with a girl. You try poses, the most obvious being to enliven the text to solve a difficult problem, but you and tie together what might otherwise be dreary can’t seem to find an answer—you don’t and dry expository dialogs on mathematics. An- even know where to start. But there’s other may be to inspire readers to look into the something about it that draws you in history of mathematics, particularly of ancient and won’t let go. You know that some- Greece. Unfortunately, many hints, jokes, and thing wonderful waits, if you can only double entendres that work in Japanese are lost in coax it into sharing its secrets. That’s translation and would be difficult to replicate in the kind of feeling I hope this book has English, since the Japanese language (which uses brought to you. (p. 271) three different types of characters) can be used more subtly to poke fun. For example, Miruka Mathematics in Math Girls could be translated as CeeCee or C.C. (similar to “J.R.”, as in “Who shot J.R.?”), AyAy as Venus, and Mathematics is clearly the superstar of this novel, Mr. Muraki as Mr. Arbre. However, a second mean- with beautiful derivations and proofs. Equally ing that could be associated with Muraki = Mu + Aki important are the one-on-one discussions that = never grow bored cannot be adequately captured illustrate the need for patient and thoughtful nur- in translation. turing of young people interested in mathematics. We were told by readers that they found the math problems much more interesting than textbook- 5The translator changed the name of this character from style assignments they had encountered in school. Tsunomiya (東宮) in the Japanese text due to its length They also enjoyed the emphasis on thinking—and and the difficulty in pronunciation for Western readers different ways of thinking—about a problem. lacking familiarity with Japanese morphology. In addi- Explanations of different approaches to solving tion, Tsunomiya would have been the only character to be addressed by his last name by other students, which a problem were “exciting”, with each approach would be very odd in the United States. Tsunomiya means boasting its own unique type of mathematical Eastern residence of the Emperor, referring to Tokyo elegance and finesse or high practical value (e.g., (東京), which means Eastern Capital. speed in computation). 6In Greek myths the birth of the flute is attributed to Pan, a The first chapter eases us into the playful world creature who is half human and half goat. One day when of mathematics. The discussions center on the Pan is scampering about in a forest, he encounters Syrinx discovery of patterns in numerical and and immediately falls in love. However, she panics from puzzles based on Fibonacci sequences, primes, unwanted advances by the odd-looking creature and flees and powers of integers. To solve for the nth to the water’s edge. In response to Pan’s pleas for help, , a computer science major ac- Zeus turns Syrinx into a reed, and the heartbroken Pan cuts and bundles the reed to make panpipes. He uses the customed to coding solutions would most likely panpipes to “give voice to his pangs of love.” C. Puscouiu, code a recursive method for finding a solution for B. White, History of the Panflute, http://pan-flute. an input natural number n. The idea of and the com/history/. Accessed 22 Dec 2011. process involved in deriving a closed-form formula

958 NOTICES OF THE AMS VOLUME 59, NUMBER 7 seemed exciting and new to CS majors with whom backtrack. That’s what the journey is we talked. all about.” (p. 252) The final puzzle in Chapter 1 stumped the pro- Math Girls is not a novel. To dub it a “romance” tagonist (as well as us). Upon hearing the answer, or “love story” would be grossly misleading. It is he is unembarrassed, unapologetic, and even a bit a math book that clearly reads like one written defiant: by an author who truly loves mathematics. Some What I didn’t like about this problem view the intended message of the book to be about was … [i]t relies on memorization. the awkwardness of teenagers, the freedom of Math isn’t about dredging up half re- exploring interests outside of prescribed courses membered formulas. It’s about making of study, and the delights of self-directed learning, new discoveries. Sure there are some as can be gleaned from conversations in which the things that require rote memorization narrator urges Tetra to play with math until she … but math is not like that. With a math is satisfied with explanations, to see the beauty in problem you have a set of rules. You elegant proofs and approaches, and not to focus on have tools and materials… math is not solving math problems as a means to some other about memorizing, it’s about thinking. ends (grades, a job, etc.). Or at least that’s what it is to me. (p. 7) Though not intentionally written for a male au- dience, our guess is that Math Girls will resonate This is the first of a series of impromptu philo- more highly with males than with females, due (in sophical discussions that magically pop up in the part) to the depth with which the male narrator midst of mathematical dialogs throughout the is portrayed compared to the female characters. book. Questions such as, “If [a] goes However, we enjoyed the read and look forward on forever, at what point can you figure out the to immersing ourselves in the magical world of rest?” are answered in a delightfully casual, ado- mathematics in future volumes. lescent manner. “There’s always a chance that the In closing, we note that only five years pattern will suddenly change somewhere beyond have passed since former Harvard president Law- what you’ve seen … But that’s the way the world rence H. Summers resigned before facing an all works. You never know what’s going to come next. but certain vote of no confidence following sev- Predictions fail. Check this out … [followed by an eral major and divisive actions, including a public abrupt change in topic]” (p. 9). These moments of speech in which he proposed that women might thoughtful inquiry sprout and dissipate at a mo- lack an intrinsic aptitude for math and science.7 ment’s whim, typifying adolescent dialog. More Last year Princeton celebrated the fiftieth anniver- often than not, they work. sary of its first female degree candidate. The uni- The next several chapters happily zip along in a versity organized an event, She Roars, “to celebrate similar tone, with increasingly more sophisticated and to reflect on the evolving role of women … to and interesting mathematical concepts and deriva- mark the contributions that women have made tions ( (continuous and discrete), har- … as well as the challenges that still confront monic numbers, , the , (them).”8 partitions, …). At the same time, the book helps Challenges and myths still abound. Interest- readers understand the difficulties and hurdles as- ingly, Math Girls was first published in 2007 in sociated with human learning. Among the general Japan, a country considered more backward than populace, for some inexplicable and unjustified the United States in gender equality attitudes and reason, mathematics has come to symbolize the policies. It is refreshing to have a popular book horrors of required schoolwork. When the narra- with females actively participating in mathemati- tor falls into a mathematical rut, Miruka consoles cal discussions and tackling historically significant him with wisdom that all math majors gratefully math problems. We applaud the author for put- received at some point in their careers from their ting politics aside and focusing on the beauty of teachers: mathematics and the development of friendships through collaboration and for highlighting the “I couldn’t solve it,” I said, “I got lost.” importance of thinking and understanding and … “You’re hardly the first mathemati- appreciating different perspectives in approaching cian to go down the wrong path. But you know what? If you don’t go look- 7A. Finer, P. D. Healy, K. Zernike (2006), “President of ing, you’ll never know what you might Harvard resigns, ending stormy 5-year tenure”, NY have found. If you don’t try new things, Times online: http://www.nytimes.com/2006/02/22/ you’ll never know what you’re capable education/22harvard.html. Accessed 5 Dec 2011. of. We’re on a journey here. Sometimes 8C. Ciatt (2011), “She Roars” conference celebrates we get tired, and sometimes we get women at Princeton: http://www.princeton. lost. But that’s no reason to give up. edu/main/news/archive/S30/34/23I46/index. … If you get tired, rest. If you get lost, xml?section=topstories. Accessed 5 Dec 2011.

AUGUST 2012 NOTICES OF THE AMS 959 problems (perhaps not just mathematical). Seen in we found them to be a bit unrepresentative of this context, the book serves as a lively and fun the book; more specifically, we found them to be public ambassador from the world of mathematics among the least interesting. Readers who experi- to the general public and helps free readers from ence difficulty following the math in these samples damaging and pernicious stereotypes. are unlikely to enjoy the remainder of the text. Unfortunately, a large portion of the recommended Addendum supplementary readings were written in Japanese Curious readers may sample the first two chap- and have not been translated. ters of Math Girls as free downloads from the publisher’s website.9 Since the content of the Acknowledgments book advances quickly in mathematical difficulty, We would like to thank the Notices and Bento Books for the opportunity and privilege to read 9Math Girls homepage, sample section: http:// and review Math Girls, and Tony Gonzalez for his bentobooks.com/wp-content/uploads/2011/10/ friendly and courteous help in providing us back- Math-Girls-Sample.pdf. ground information.

Book Review

Magical Mathematics

Reviewed by Donovan H. Van Osdol

Magical Mathematics: The Mathematical Ideas this test of my abil- that Animate Great Magic Tricks ity to forecast the Persi Diaconis and Ron Graham future, I want you Princeton University Press, 2011 to pick any word US$29.95, 258 pages (of which there are ISBN-13: 978-0691151649 thirty-six) in the first sentence of the pre- When I agreed to review this book for the Notices, ceding paragraph. I did not know that the perfect review had already Notice that I have been written—by Martin Gardner—and that it ap- no control over your peared as the preface to the book itself! In the end, choice: there’s noth- I decided to write up my two cents’ worth in hopes ing up my sleeves! of encouraging you to buy Magical Mathematics for Let’s call this word yourself. Of course, if you want to read the perfect a “key” word. review without buying the book or reading this Now please count the number of alphabetic review, you are likely to be able to read the preface characters in your key word and move forward in by having a “Look Inside” the book at Amazon.com. the paragraph that many words to arrive at a new To give a vignette of what you can expect to find key word. For example, if the initial “to” in that flowing from the pens of Diaconis and Graham, first sentence were your chosen key word, then let me describe a magic trick for you. I’m going to your next key word would be “this”. Got it? Okay, write a word on a piece of paper, fold it into fourths now repeat this process (so that in our example so as to hide the word, and put the paper beneath your next key word would be “Notices”), over and this brick here on the table before us (we need over, until you can’t go any further in the second something really weighty to protect my prediction sentence (which ends with “yourself”: always a from being tampered with!). To do your part in good place to stop!). There are twenty-three words in that sentence, so there’s not a very good chance Donovan H. Van Osdol is professor of mathematics, emeri- that I could know your final key word before you tus, at the University of New Hampshire. His email address started, right? Good, I agree… But wait, there’s is [email protected]. this piece of paper under this brick here where I DOI: http://dx.doi.org/10.1090/noti875 wrote a word… Would you pick up the brick, look

960 NOTICES OF THE AMS VOLUME 59, NUMBER 7