This book is the second in the “Math Girls” series to be translated into English; the series is originally written in Japanese by the author of many books on computer programming. In my review of the “first *Math Girls*” I said that one truly amazing thing about it is, it tells how a small group of teen-agers do math together *only *because they want to, without adult prompting in any form — be it Math Club, Math Circles, Olympiads, etc. These kids meet every day after school in the library, and sometimes in a local coffee shop, specifically to discuss math.

The more I think about it, the more fabulous and radical it seems. These books expose young math enthusiasts to the possibilities of (A) *not *being “math loners”, (B) identifying as mathematicians and giving themselves permission to make math the main, or even only, thing in their lives, and (C) discovering that there are many ways to be “good at math”, as well as many ways to *do *math. *Math Girls*^{2} is even stronger with respect to these properties, and others, than *Math Girls*. More will be said about that later.

First, a quick run-down of the first *Math Girls*. Our narrator, just starting high school when we meet him, expects to have a boring social life, to find no one else to do math with. He assumes, wonderfully, that he will continue to do math, as much and as often as he can, even if it means doing it alone. One can see, from the first page, that he does have a social side, a desire for company. Our opening scene is, I’m thinking now, rather dreamlike (more about that later); suddenly a girl named Miruka, beautiful and perhaps beautifully strange, appears in the midst of cherry blossoms spouting math (Fibonacci series) in riddle form. There’s instant communication between these two.

After a few chapters the twosome develops into a threesome when another “math girl” named Tetra joins them. Tetra’s interest and ability in math take a different form; she’s not a crackerjack like Miruka, or even like our narrator, but her ideas often lead to solutions and Miruka grows to like and respect her. (Like me, perhaps, Tetra is more of a theory creator than a problem solver.) It often seems that both girls have a crush on our narrator (which, I’m now thinking, make the story even more dream-like, or at any rate wish-like). And vice versa; our narrator, every once in a while, gets carried away with Miruka’s “ long lashes. The soft curve of her mouth. Her chest rising and falling as she breathed” (p. 125) or with Tetra’s “chestnut hair” (p. 6).

*Math Girls*^{2} acquires a new “math girl”, our narrator’s cousin Yuri, three grades younger and personifying yet another type of “math-buff”, one who has felt no particular interest in math until she becomes curious to know what her cousin and lifelong friend sees in math. And naturally, *Math Girls*^{2} acquires new math — not only its subtitle, “Fermat’s Last Theorem”, but Pythagorean triplets, rational points on the unit circle, gcds and lcms, proof by contradiction, Gaussian integers and “broken primes”, modular arithmetic, infinite descent, Euler’s Identity, even a little group theory. Most of this is stuff leading up to Fermat’s Last Theorem (and an outline of the proof — but the outline is more detailed than in most “FLT books” for the layperson).

I think of *Math Girls*^{2} as being more “mathy” than its prequel. As our narrator and the “math girls” have matured, so has, not only the math itself, but our characters’ ways of living with it. First, even Tetra understands explanations and proofs more easily, while still acknowledging herself and being acknowledged as the one who needs to take things more slowly. Second, Miruka seems even more able than before to takes things *quickly* and to be already doing her own, if not actual research, at least study of more advanced math. It’s she who teaches (from her hospital bed, recuperating from a non-serious accident) the others about groups, and she does a lot of teaching in general. As a teacher, in fact, Miruka comes across as rather interesting. She can be abrupt and aggressive, but she seems to know how to do this and when to stop. P. 127: “What you really need to do is think of it [the group operation] as a variable standing for some operation over G, the same way that a and b stand for elements in G”. And p. 250: “Yuri held her hands up in surrender. ‘I think I’m gonna head home. I could barely even follow the history lesson.’ Miruka fixed her stare on Yuri. “‘We’ll start with a problem for you.’”

As for our narrator, there are more, and more detailed, accounts of his process of solving math problems. Pages 195 to 211 are devoted to an almost verbatim description of his discovery, with help from a “simple” remark made by Yuri, of infinite descent, in order to prove that (p. 212) “there exists no right triangle with integer length sides and square area.” P. 196:

As much as I’d been pressing Tetra to start using more definitions, to be honest, introducing new variables always put me on edge, too. I could never shake the feeling I’d end up drowning in a sea of letters.

But as always, I decided to trust the math, to let it fly free and see where it would go… Setting aside my insecurities and letting equations do their thing had led me to answers more times than I could count… “If I could get from this to a contradiction, I’d know m + n and m – n were indeed relatively prime, and I’d be able to keep my weapon…”

P. 199:

… At first I had thought of this problem as setting out on an expedition into the great unknown, but now it was starting to feel more like shrinking farther and farther down into the realm of some microcosm. Investigating the molecules A, B, C, D, I had discovered the atoms m, n. Now I was preparing to split those atoms into the elementary particles e,f, s, t.

*If this keeps up, I’m going to end up looking for quarks…” *

P. 210: *“Wait… infinity? No, not infinity! It can’t be!” *and “A single thought repeated itself in my mind like a mantra. *Please let C > C*_{1} … Please let C > C_{1} …”

Such detailed, and resonating, description of the process of doing math research is, in my reading experience, rare. It has, I think, many advantages and uses. First, poetry. Second, honesty. Third, education. When students see how difficult and time-consuming — and human — mathematical thinking can be, they realize that it’s okay to feel awed, frustrated, elated, and so on. And students, even if they can’t or don’t themselves indulge in such mathematical adventure and pleasure, can see the possibilities.

Another difference between the first and second *Math Girls*: these kids are even more on their own. In both books appear index cards with math problems, from a supportive teacher named Mr. Muraki. These questions naturally lead the kids on new mathematical trails, and they’re always excited to report to one another, “I have a new card from Mr. Muraki!” Sometimes two kids get two different cards and the questions turn out to be related, often to the point of being essentially the same question. At any rate, such cards appear less frequently in *Math Girls*^{2}. The kids need Mr. Muraki’s input less and less.

I also get a sense that the kids are getting more and more committed to math. There seems to be more occasion to describe the mathematical life. P. 252:

Perhaps I’ve said this before but I *love *equations. I love their specificity, their consistency. I love how decoding them lets me see the structure of things, and I love the discoveries I make when I rewrite them in new forms. When I can write something as an equation, I know I understand it. When I can’t, I know I don’t.

*But this stuff… *

The proof of Fermat’s last theorem we’d seen at the seminar was way beyond me. In all honesty the equations the lecturer had shown us filled me with dread. It was as though I was seeing advanced mathematics for the first time.

Reading that passage, I recalled the first time I saw a “real” math journal. Throughout most of high school I believed that there was no math problem I couldn’t solve. Indeed, I hadn’t seen a math problem I couldn’t solve. I used to subscribe to math newsletters for high school students and they often sported lists of other things to send away for. One was “101 Proofs of the Pythagorean Theorem” — none of them hard to understand — but another was the latest issue of *Scripta Mathematica*. Well, when it arrived in our mailbox, I was, to put it mildly, surprised. I don’t remember feeling upset, but I had learned that there was math, lots of it, that I could not understand. So yes, the above-quoted passage from Math Girls 2 lets readers know that there is always higher and higher math.

This book contains many other insights into the mathematical life. As with the first Math Girls, I felt that many of the insights are no news to veteran mathematicians, but for some budding mathematicians they’re just right. P. 165:

When all the pieces come together in math, there’s an inherent permanence to it. But an unfinished equation or half-written proof is ephemeral, insubstantial. Open a textbook and you’re presented with the finished product, like a building from which all the construction scaffolding has been removed. All you see is order and perfection, with no hint of the mess made in the process…

I want to say more about the kids’ commitment to math. First, except for intermittent comments, they rarely talk about anything else. I suppose this might be typical of teenagers, and of people in general when they have a specific thing in common and purpose for getting together*. *Still, I was mildly surprised when, for example, the friends, after knowing one another for over a year, had not previously talked about things like siblings and parents. Second, very near the end of the book there’s a party. And, perhaps amusingly, most of what they do at the party (besides eat) is math. P. 279.

“Oh, Miruka, I’d been meaning to tell you,” Tetra said. “I think I finally understand your solution to the primitive Pythagorean triples problem!”

Yuri perked up. “Primitive Pythagorean triples? What’re those?”

What can I say? We knew how to party.

Yes, these kids are true math friends and true math enthusiasts. I have to confess that I felt slightly guilty, even sad, that I probably wouldn’t have wanted, as a teenager and now, to spend quite that much time and energy on math.

Now it’s time to deal with the question, how realistic is the narrative (which consists mostly of math in quotations marks, usually the narrator’s or Miruka’s)? Could math geeks (and math non-geeks) come together like that? Could our narrator have been so lucky as to acquire three “math girls”? And could teenagers talk, for example, like this:

P. 250: “’He stumped the world when he announced a proof in 1993, but unfortunately somebody found an error. He went right back to work, though, and in 1995 published a correction along with Taylor, one of his former students.’” While reading the book I often thought, “Gee, not even geeks talk like that.” In fact, could a teenaged narrator write about the mathematical life with such elegance?

But I have two thoughts regarding all that — besides the “proof in the pudding”, mentioned in my review of the first *Math Girls*; that is, “Math Girls clubs” have, throughout Japan, been forming, though I wonder how many of them are actually formed by the kids themselves. First, the book has “poetic license” (and “mathematical license”). Second, although I didn’t think about this after reading the first Math Girls, writing this review of *Math Girls*^{2} made me think “Hey, he’s got these *three *math girls, all with crushes (of sorts) on him. It’s a math nerd’s dream!” And then I realized, “Perhaps it is a dream.” Or at least wishful thinking. (Only perhaps.) Like a movie, a book doesn’t *have *to be realistic. It could describe a wish, a hope, or a dream. When I view the Math Girls books in this light, much of my skepticism disappears.

I do, however, still have the same questions regarding gender as when reading the first *Math Girls*. Why is the title *Math Girls* (although I do like the sound of it, I must admit)? What if Miruka or Tetra or Yuri were the narrator? What if, in general, the narrator had been female and had acquired math *boys *(as, in high school, I would have loved to, and as many adult female mathematicians have acquired “math *men* “)?

Another gender issue, and perhaps also age issue: the matter of our narrator’s mom. She didn’t appear in the first *Math Girls*. (I don’t know why, other than that the author thought of the idea after writing the first *Math Girls*.). In *Math Girls*^{2} she’s kind of a 50s mom; when our narrator is at home working on math with a math girl or two, she often appears, or calls from downstairs, with offers of snacks and sometimes whole meals. Nothing wrong with this *in real life, *but in a book, in particular a book of this ilk, I believe there is something wrong with it: it contributes negatively to the image of women, in particular because this is the *only *aspect of his mother’s life that readers get to know about.

In Japan more mothers are like 50s mothers than in the U.S. In middle class or affluent families, the fathers are very often businessmen, and in Japan businessmen often go out drinking with their co-workers ’til late in the evening and thus are not around as much as many fathers in the U.S. Still, I wondered about our narrator’s father? Does he *have *a father? Or is his mother single? Does she have a job? Part of me felt a little sorry for her; she doesn’t, as far as readers see, seem to have meaning in life other than her son, who seems to be an only child. She gets extremely excited at the prospect of, and during, the party at the end — and has only pizza to contribute to the party activities.

Indeed, I see another feminist issue here: for some people in our society it’s okay, or more than okay, for *young *females to do math (or do anything) but not necessarily *older *females. Math *Girls *are acceptable, but perhaps not Math *Women. *(My home email is [email protected] ! Perhaps I accomplished something politically as well as personally by creating that address.) It reminds me of the year I adjuncted at an Ivy League university. Although there seemed to be plenty of young women, in their twenties or early thirties, in the Math Department, I was one of only a handful of older women.

The book ends on a philosophical note. The party is over and our narrator, perusing and pondering the stars, walks the math girls to the train station. But I’m admittedly opinionated about what I think *should *be the ending: At the party our narrator’s mother should notice the fun the kids are having discussing math. ”Hm,” she should muse aloud. “What’s this about primitive Pythagorean triplets?” And the kids should fully answer her question. Then they should *all *walk to the train station, as the mother continues to catch the math bug and is initiated into this math friendship, becoming the fourth Math Girl. Perhaps, when the next three *Math Girls* books are translated into English, we’ll see that happen.

Whether or not it does, I’m looking forward to reading those translations.

Marion Cohen teaches math and writing at Arcadia University in Glenside, PA, where she developed a course called Truth and Beauty: Mathematics in Literature. She is the author of *Crossing the Equal Sign, *a book of poetry about the experience of math.