Thursday, December 31, 2015

Multiplication is not completely reducible to addition, even for whole numbers

Mathematician Keith Devlin has written several articles about the relationship between multiplication and addition (1, 2, 3, 4, 5). Drawing on a variety of sources, including pure mathematics and empirical research about learning, Devlin argues that since multiplication isn't repeated addition, students shouldn't be taught that it is. I think that's basically right, though in practice it is complicated, as Devlin acknowledges. 

In this sidebar, I would like to mathematize the basic question. "Multiplication is repeated addition" might initially sound like a mathematical claim, something straightforwardly either true or false, but as it stands, this statement is too vague to admit of either proof or disproof. Some precise definitions are necessary before we can consider the question from the standpoint of mathematics.

Everybody agrees that repeated addition is useless as a way to understand multiplication of real numbers, as in cases like π × e. Where I think I differ from most people is that I do not even grant that multiplication reduces to repeated addition for the case of whole numbers. But that claim is also vague, so let's get started. You will let me know if anything below is incorrect or needs fixing.

Definition 1. A whole-number operation is a function that maps every pair of non-negative integers to some non-negative integer, which is called the result of the operation, or the resulting value. Since we only consider whole numbers, for brevity we simply call a whole-number operation an operation. The result of an operation B on two numbers m and n can be denoted mBn.


1) Addition (+) and multiplication (×) are operations. 

2) Let F be defined by mFn = 7 × n. Then F is an operation, in particular a noncommutative operation. For example, 4F6 = 42, but 6F4 = 28. 

3) Let G be defined by mGn = |m − n|. Then G is an operation, in particular a nonassociative operation. For example, 1G(1G2) = 0, but (1G1)G2 = 2. 

Definition 2. An addition expression is a string of symbols consisting of k ≥ 2 whole numbers separated from one another by k − 1 addition symbols. 


4) "2 + 3" is an addition expression.  

5) "2 + 3 + 9" is an addition expression. 

6) " + 1" is not an addition expression. 

7) "1" is not an addition expression. 

8) "6 × 7" is not an addition expression. 

Remark 1. "1" and "6 × 7" are algebraic expressions, but not addition expressions. 

Remark 2. An addition expression can be said to have a value in the obvious way, e.g., the value of "2 + 3 + 9" is 14. 

Definition 3. An operation C is completely reducible to addition if, for all whole numbers m and n, there exists an addition expression that equals mCn and contains no numbers besides m or n


9) The operation mFn = 7 × n is completely reducible to addition. Here is a particular instance: the addition expression 4 + 4 + 4 + 4 + 4 + 4 + 4 equals 5F4 and contains no numbers besides 5 or 4. 

Remark 3. One might have thought that only commutative operations are completely reducible to addition; Example 9 shows that this is not the case.

10) The operation mGn = |m − n| is not completely reducible to addition. To see this, consider that 1G1 = 0, whereas it follows immediately from the definition of an addition expression that an addition expression containing only 1s has value at least 2.

Theorem. Multiplication is not completely reducible to addition. 

Proof. As noted in Example 10, an addition expression containing only 1s has value at least 2. So an addition expression containing only 1s cannot equal 1 × 1 in value. ∎

The theorem states the obvious: 1 × 1 isn't equal in value to any well formed expression consisting of 1s and addition symbols. According to the sense of Definition 3, this means that multiplication is not completely reducible to addition. 

This being obvious, it must be that when people talk of reducing whole-number multiplication to addition, they have something else in mind besides Definition 3.

Wikipedia provides the following quasi-definition of multiplication as repeated addition:

m × n = n + n + ... + n   (m addends). 

I'm not sure the Wikipedia authors have thought about what happens to this prescription when m = n = 1. In this case, it says

1 × 1 = 1   (1 addend). 

The equation is true, of course, but to our point today I can't help noticing that there's no addition in it! Addition is a binary operation. If you are truly adding, then there's never just 1 addend. Whatever the Wikipedia formula reduces multiplication to, it isn't addition. 

Maybe the Wikipedia definition ought to say something like this instead?

m × n = 0 + n + n + ... + n   (m instances of n)

For m = n = 1, this becomes

1 × 1 = 0 + 1   (1 instance of 1)

which is a true equation, and, well, if it contains a number other than 1, then at least it is an addition expression. But now what happens when we take m = 0? Since, according to the prescription, we mustn't write any instances of 5, the result is an ill-formed expression:

0 × 5 = 0 + 

Maybe we could patch things up by agreeing that the phrase "no instances of 5" means "insert the number 0." But wait—isn't that just multiplying?

I'll leave off trying to patch up Wikipedia's formula (feel free to take up that project here) and just list some representative cases that prevent us from interpreting multiplication as repeated addition:

1 × 1
1/5 × 3/4
−1/5 × 3/4
π × e

Hmm. Kind of a lot. It makes me wonder what the value is of defending the idea. 

In my own work, when I have needed to describe the relationship between multiplication and addition, I have pointed first to the distributive property: 

a × (b + c) = a × b + a × c.

Observe that both operations appear on both sides of this identity—it is not a prescription for eliminating one operation in favor of another. Nothing in the axioms is inviting us to undertake that project. 

Indeed the distributive property is the only field axiom that refers to both operations, which means that it captures the entirety of the relationship between the two operations. For that reason, to describe the relationship between multiplication and addition using formulations other than the distributive property itself is to invite error. But is there any true and valuable statement that juxtaposes multiplication with repeated addition? I'll try one:  
Repeated addition is a special-case algorithm for calculating whole-number products, m × n, when m > 1 or n > 1. 
Proving that the algorithm works involves the distributive property, of course:

Proof: By assumption, either m > 1 or n > 1. First suppose n > 1. Then, and only then, can n be written as a sum of 1s: = 1 + 1 + ... + 1 (n addends). So by the distributive property, m × = m × (1 + 1 + ... + 1) = m × 1 + m × 1 + ... + m × 1. Since 1 is the multiplicative identity, we have m × n = m + m + ... + m (n addends). Likewise, if (and only if) m > 1, m can be written as a sum of 1s: m = 1 + 1 + ... + 1 (m addends). So m × n = (1 + 1 + ... + 1) × n = 1 × n + 1 × n + ... + 1 × n = n + n + ... + n (m addends). Thus, if (and only if) n > 1, the product m × n may be calculated by adding n addends m, and if (and only if) m > 1, then the product m × n may be calculated by adding m addends n. ∎

Repeated addition is an algorithm, not an operation. This is one reason among many why multiplication and repeated addition are best thought of as two different things, even for whole numbers. 

Friday, December 25, 2015

Answers to Holiday Challenge

Using all lowercase letters, and using Times New Roman ten-point font with no special formatting...
  • What is the widest seven-letter word you can type?
  • What is the narrowest seven-letter word you can type?
  • Find a four-letter word that is wider than some seven-letter word.

Thanks for sending in so many great answers! As for myself, I sat out the challenge this year—though I did program a computer to try it. Most of the computer's attempts agreed with readers' entries, with some notable exceptions.

I measured candidate words by typing them into Adobe Illustrator and then viewing the type objects at maximum magnification. So without further ado...


The semitransparent blue bars are positioned in such a way that they bookend mugwump. So mommuck just barely pokes into the blue region, and hammaum falls just barely short of it.

These words were familiar to me with the exception of mommuck, which doesn't appear in any of my dictionaries. In Okracoke Island dialect, it means to harass: see this Baltimore Sun article. The widest standard word, mugwump, means a fence-sitter in politics, or a self-important leader. This word also has an interesting history. Hammaum is an archaic alternate spelling of hammam (Turkish bath).

As a reader observed via email, it would have been nice thematically if maximum or mammoth had turned out to be widest!

A couple of tongue-in-cheek responses that I received were mammony, an invented word that means wealthy (its width lies between that of mommuck and that of mugwump) and memmove, a Unix command (between hammaum and whammed).


(Some of these words might be equally wide...I couldn't always distinguish them using the methods I was using.)

Give the computer credit for titlist—I'm a little surprised that no reader submitted this word. The reader favorite instead was illicit. (Maybe the golf ball brand TITLEIST has elbowed titlist out of our collective brains?)

Some of these words are really jargon: my dictionary tells me that illitic is the adjectival form of illite, a kind of mineral clay, and tillite is a stone made of consolidated till. And littlie is a slang term or colloquialism for a small child (chiefly British?).

Interestingly, illitic and illicit are anagrams, yet they differ in width. It seems illicit pays a price for ending with t, because the letterform for t has a little tail sticking out. Evidently, the problem of extremizing word width doesn't reduce to the consideration of individual letter widths. This must have something to do with the way lines of type are assembled. Thinking that it might to be so, I wrote my computer program holistically at the word level: the computer first renders a given word as a vector graphic, then rasterizes the graphic, next discards white pixels, and finally computes the difference between the maximum and minimum column indices among the remaining pixels. This difference is the approximate width of the word.

Four-letter words wider than some seven-letter words

Readers supplied a number of answers to this question. One interesting case is work: it is wider than titlist, but by such a small margin that it might be a candidate for the narrowest four-letter word that is wider than some seven-letter word.

One could ask additional questions along these lines. Is there a three-letter word that is wider than some seven-letter word? (My kids think so...when I put the question to them, they said, 'How about mmm?' As in, 'yummy!' We didn't find mmm in any of our dictionaries, but maybe it's an interjection?) Is there a four-letter word that is wider than some eight-letter word? How about a six-letter word that is wider than some 11-letter word?

Last but not least!

This year's prize, a copy of Word Games 4, goes to reader Melissa. Enjoy the puzzles in this year's volume! And a big thanks to everybody who tried the Holiday Challenge. My best wishes to you for a happy and healthy 2016.

Wednesday, December 16, 2015

Holiday Challenge

Last December, my readers and I played a memorable game of hangman. Here's the 2015 holiday challenge:

Using all lowercase letters, and using Times New Roman ten-point font with no special formatting...
  • What is the widest seven-letter word you can type?
  • What is the narrowest seven-letter word you can type?
  • Find a four-letter word that is wider than some seven-letter word.

Email your answers to

Anybody who tries the challenge this year will be entered into a drawing to win a copy of my latest puzzle book, Word Games 4I'll announce the winner after Christmas!

N.B., the words I like to work with are the “standard” word game variety: this means we avoid proper nouns, foreign words, slang, contractions, hyphenated words, or acronyms. Also, the width of a word is defined as the horizontal separation between its leftmost and rightmost extremes.

Tuesday, December 15, 2015

In Case You Need a Cheap Stocking-Stuffer This Season

A few weeks ago I made this funny toy for my kids...they are still messing around with it!

To make this toy, cut up a cardboard toilet-paper tube into four rings...snip each ring...then tape each ring back together leaving a bit of a gap remaining (tape from the top and bottom so that both faces are slippery). The rings can now turn on the outside of a second toilet-paper tube.

My goal was for the resulting sentences to be funny and/or nonsensical, but I didn't plan carefully what I would write. If you have a great "content model" for this kind of toy, feel free to put it in the comments!

Tuesday, December 8, 2015

The Eagle Have Landed

This image shows an article that was published the other day in The New Republic. I noticed a grammatical error (circled in red) in the sentence just underneath the article title. Errors like this usually get fixed before too long, but this prominent blurb has been up for a while now.

Since I have a few posts up about grammar, I should clarify that whenever I comment about grammar, I'm referring to written language, not speech. That's because speech is usually less formal than writing. Also, speaking is a very personal and revealing sort of performance. For both of these reasons, and probably others, correcting an adult's grammar during a conversation would be an extremely rude thing to do, almost without exception. So when it comes to grammar, my subject is writing—and in particular, professional writing. Like, say, if you worked at a magazine.

Sometimes words mean more than what they say—other times, they mean less. Sometimes they mean something else entirely. Words are used not only to communicate, but to intimidate. Rather than a thesis, a string of words might convey an emotion, an identity, a desire, or some other unspeakable message.

I use language that way too, but mostly in my writing I try to use words denotatively. I have to, because I generally consider writing to be the art of producing propositions that are true of what they describe. So you can imagine how my mind jams when I feed it a sentence like 'The search have led to a new theory' in which the number of searches is simultaneously one and more than one.

Over twenty years ago, Steven Pinker wrote a good article in The New Republic about 'the fallacies of the language mavens.' There, Pinker explained that some of the mavens' favorite grammar rules are really just historical artifacts of recent vintage. Pinker also pointed out some weaknesses in the arguments of famous language writers, like William Safire. Pinker is especially convincing when it comes to certain specific grammar rules, such as the one about split infinitives. But I couldn't help thinking that Pinker's attack on grammar rules might have been more powerful if in his article he had actually broken some of those rules. Well, maybe he tried. Maybe The New Republic had more copy editors in 1994 than it has today.

Pinker, who is a member of the Usage Panel for my dictionary of choice, recently wrote a book about writing called The Sense of Style: A Thinking Person's Guide to Writing in the 21st Century. I haven't read the book, but I'll bet it's a worthwhile read. Here are reviews of it from the New York Times and the New Yorker. Here too is a list of grammar "rules" that Pinker invites us to do away with. I note that subject-verb number agreement isn't among them.

Cutting a Triangle in Fourths, Cont'd

Consider an equilateral triangle in the xy-plane, centered at the origin.

1) By suitably rotating the triangle about its center, is it possible for each quadrant to contain one fourth of the triangle's area?

2) By suitably rotating the triangle about its center, is it possible for each quadrant to contain one fourth of the triangle's perimeter?

(See my previous post for animations.)

In short, the answer to problem (1) is "no." This can be proved from scratch or deduced from the information here.

As for problem (2), again the answer is no. This is not hard to see, if we begin with the premise that the only perimeter-bisecting lines passing through the center of the triangle are the three obvious ones that connect some vertex to the midpoint of its opposite side. More about this premise in a moment, but for now this fact is enough to solve the problem at hand. For suppose the triangle could be rotated so that each quadrant contains one fourth of its perimeter. In that case, the y-axis would bisect the perimeter (since the quarter in x > 0, y > 0 and the quarter in x > 0, y < 0 would add to half), and likewise the x-axis would also bisect the perimeter. There would exist perpendicular perimeter bisectors passing through the center. But none of the lines connecting vertices to midpoints of opposite sides are perpendicular to one another. In other words, there do not exist perpendicular perimeter bisectors passing through the center, and it follows that the triangle cannot be rotated in such a way that each quadrant contains one fourth of the triangle's perimeter.

Now as to that premise: I wrote up a couple of pages of discussion here about the fact that the only perimeter bisectors of an equilateral triangle are the obvious ones. (For amusement's sake, I approach the problem calculus-style.)

I was pleased to learn that a colleague of mine had pursued these problems in the margin of his in-flight magazine! All the mathematical travelers out there will be thanking the Wellendorff ring company for designing an ad with plenty of white space.

By the way, after a conversation with my colleague, I can describe the thought process reflected in these marginalia: what is being shown is that at any time when one quadrant has a quarter of the area, the quadrant immediately "behind" it will demonstrably not have a quarter of the area. Nice approach!

If you liked these problems, some related problems come to mind:

3) If possible, place an equilateral triangle in the xy-plane in such a way that each quadrant has one fourth of the triangle's area.

4) If possible, place an equilateral triangle in the xy-plane in such a way that each quadrant has one fourth of the triangle's perimeter.

5) If possible, place an equilateral triangle in the xy-plane in such a way that each quadrant has one fourth of the triangle's area and one fourth of the triangle's perimeter.

For further reading on these subjects, try "Lines Simultaneously Bisecting the Perimeter and Area of a Triangle," by Paul Yiu. Professor Yiu is the editor of Forum Geometricorum, the journal that published my Pythagoras-free derivation of the identity sin2 θ + cos2 θ = 1.

Thursday, December 3, 2015

Cutting A Triangle In Fourths

Consider an equilateral triangle in the xy-plane, centered at the origin.

1) By suitably rotating the triangle about its center, is it possible for each quadrant to contain one fourth of the triangle's area?

2) By suitably rotating the triangle about its center, is it possible for each quadrant to contain one fourth of the triangle's perimeter?

The answer to the area problem can be deduced from the information here. Just for fun, I also made an animation. If it doesn't show as embedded here, you can watch it on YouTube.

Endless loop GIF version:

Tuesday, December 1, 2015

Guess My Google

In each case below, guess the single word that I typed into Google image search to yield the results shown.

In some cases, the images shown are top results; in other cases, the images were chosen from further down the page.

You can check your answers by mousing over an image and determining the file name of the image file.











Thursday, November 26, 2015

My Year's Best List 2015

The best things that I read, watched, listened to, and otherwise ingested in 2015!

(Previous lists: 2014, 2013.)

Best Books—Fiction

John Banville, The Sea (2005)

Grieving the death of his wife, a man moves to the seaside town where he spent summers as a child. It was there, too, as a child that he met a little girl and fell in love with her. That sets the psychological stage for The Sea, a melancholy and uncanny novel with prose so magnificent that I often gasped or even blurted out "WOW" while reading it.

Every review of the book that I saw online had spoilers, so I won't link to one. Not that it's a novel of suspense or anything—I just tend to think that reviewers tell too much.

Buy it online: The Sea

Elena Ferrante, My Brilliant Friend (2012)

Characters and setting are wonderfully realized in this coming-of-age novel about female friendship set in 1950s Naples. Ferrante has an excellent sense for plot, and meanwhile the prose is good caliber and does what it needs to do. If I could say so without in any way suggesting that the book is a pastiche, I would describe My Brilliant Friend as great Italian cinema committed to paper.

This is the first book in a series of four "Neapolitan novels"—you can read about the series and its famously reclusive author at the New York Review of Books (contains spoilers).

Or, you can just jump right in and buy it online: My Brilliant Friend.

Karl Ove Knausgaard, My Struggle, Book 4 (English edition 2015)

My Struggle is an international literary sensation and a milestone in artistic realism. Knausgaard's hybrid novel-memoir plays back seemingly every event, idea, and urge of the author's first forty-three years of life. That ought to be boring, but if you go into a bookstore and begin reading this book on any randomly chosen page, fifteen minutes later I think you'll still be standing there, pulled along by the ceaseless current of prose.

Four volumes of My Struggle have been translated into English so far. I began the series with Book 4, which was a good starting point; Jeffrey Eugenides calls Book 4 "the fleetest, funniest and, in keeping with its adolescent protagonist, most sophomoric of the volumes translated into English thus far."

Buy it online: My Struggle, Book 4.

Best Book—Nonfiction

Marie Kondo, The Life-Changing Magic of Tidying Up (2014). Japanese lifestyle expert Marie Kondo shares profound insights about clutter and categorization, and she teaches a practical plan for radically changing the complexion of your household. Kondo's pronouncements are amusingly extreme, and she can sound like a cult leader; but if it's a cult, then count me in. Buy it online: The Life-Changing Magic of Tidying up.

Best Short Stories

Joe Dunthorne, "The Line." Winter 2014 Paris Review.  Buy the issue online.

James Lasdun, "Feathered Glory." Spring 2015 Paris Review. Read a teaser excerpt here, and buy the issue online.

Best Poem

Nick Twemlow, "Attributed to the Harrow Painter." Summer 2015 Paris Review. Buy the issue online.

Best Essay

Edward Hoagland, "Walking the Dead Diamond River" (1973). Ted Hoagland is a master of the essay form, and this work displays all of his hallmarks: carefully controlled tone, often wryly elegiac; finely crafted sentences and noteworthy diction; subtle development from paragraph to paragraph, following a logic that is associative and peripatetic yet never meandering; and a wealth of reportage about colorful characters and lore plus loads of physical observation and detail. The essay is reprinted in the 2014 compendium On Nature. Borrow it from a library or, if you have a large library of your own at home, buy it online: On Nature.

Best Long-Form Journalism 

Michael Finkel, "The Strange & Curious Tale of the Last True Hermit."
For nearly thirty years, a phantom haunted the woods of Central Maine. Unseen and unknown, he lived in secret, creeping into homes in the dead of night and surviving on what he could steal. To the spooked locals, he became a legend—or maybe a myth. They wondered how he could possibly be real. Until one day last year, the hermit came out of the forest.
This story appeared on Conor Friedersdorf's list in 2015. Read it online.

Best Play

Fool for Love at the Samuel J. Friedman Theater

Sam Shepard's Fool for Love is an emotionally charged one-act play about two tortured lovers, set in a cheap motel room on the edge of the Mojave desert. This year's Friedman Theater production was Fool for Love's first outing on Broadway (Shepard wrote the play in 1983).

If you read the play, you'll notice that the stage directions at the beginning are peculiarly specific; the producers clearly took that cue and invested enormously in designing the excellent set, lighting, and sound. It would be easy for the actors in such a violent play to overdo their lines and movements, but Sam Rockwell as Eddie was controlled without sacrificing physical potency, and Nina Ariande gave a staggering performance as May. There comes a moment in the play when May collapses to the floor, crying raggedly, unable to speak, suffering a pain so exposed and so pure that I wept along with her. Here is the NY Times review. UPDATE: And here is a New Yorker review of the play combined with an essay on Shepard's life and work.

Best Movies

(Links point to reviews.)

The Conversation – Coppola at his peak...'70s art-house thriller still looks & sounds great.
Fury Road – IMAX auteur-cinema, where have you been all these years??
It Follows – Joins Halloween in the hall of Midwestern hormonal horror.
John Wick – Stylish shoot-em-up with elements of fantasy
Once Upon a Time in the West – Sergio Leone's operatic apotheosis rendered additional Westerns forever unnecessary.

Deserving of special mention is Satyajit Ray's heartbreakingly beautiful 1955 masterpiece, Pather Panchali, in a newly restored print that I was able to see this year at New York's Film Forum.

Biggest disappointment: American Sniper. The film had some excellent aspects, but in key places it suffered from poor artistic taste, and I couldn't escape the conclusion that it was made to serve social purposes as much as artistic ones. (Here is an article analyzing people's reactions to the film.)

Snootiest Cocktail

"Last Caress," Hotel Surrey. Made with, get this, chartreuse snow, foraged immature berries of juniperus virginiana, and house-label champagne. Louis the XVI's favorite slushie flavor, basically. 

Best Meal in an Airport

Saison Bistro, Newark Airport, Terminal C. Over the years I've seen restaurants come and go in this location, including a steakhouse and a Soul Food place. Currently it's a French bistro, and I had a very good piece of salmon there in early 2015. (In fact, it appears to be an Alain Ducasse project.) They have iPads for ordering and you know that never works, but the waitstaff quickly remedied the situation and I made it to my gate in plenty of time.

Best Solo Drive

In July I was doing some work in Boise, so for a change of pace I extended my visit and drove around Idaho, a trip that took me to the impressive Snake River Plain, the eerie landscapes of Craters of the Moon National Monument, and the dramatic Sawtooth Mountains.

Best Diner

Louisa's Place, San Luis Obispo, California. Downtown SLO is a lot more posh than it used to be, with clothing boutiques and single-source coffee purveyors everywhere you look. But walk into Louisa's and you step into a more comfortable past. My brothers, my sisters, and I all grew up in our parents' diner, and being in Louisa's felt to me like being at home. I stopped at Louisa's for breakfast during another great solo drive, one that started in Big Sur and ended in Orange County.

Best Art

On Kawara, "Silence," at the Guggenheim Museum. Kawara (1933–2014) was a conceptual artist, and although his date paintings get most of the attention, his obsessive array of codebooks, postcards, maps, and catalogs are just as important. His decades-long body of work coheres as a whole; Kawara appears to have made his own life into a single staggering artwork. Here is the Guardian UK review of the Guggenheim show, and here is the NY Times review.

Marc Yankus, "The Space Between" (photography, 2014; combination chemical-digital process).  Arresting, gorgeous, and lonely, these images of buildings are simultaneously dreamlike and concrete. (The artist cites de Chirico as an influence.) Online: The Space Between.

Best of the Year—Period. 

Dianne Reeves, Valentine's Day Concert, Jazz at Lincoln Center

Jazz vocalist Dianne Reeves has a godlike musical talent and a generous musical soul. When I wasn't grinning like an idiot at this concert, I was wiping away tears.

Reeves has been tilting toward pop forms in recent years, and on Valentine's Day she sang fabulous versions of "Waiting in Vain" by Bob Marley and "Dreams" by Fleetwood Mac. Another song in a world-music vein transcended the genre, hopping mysteriously from continent to continent before coming in for a smooth landing in South America.

The diva still unleashes power on standards! Her "Don't Explain" was especially memorable— in fact, overwhelming. She sang the song without a microphone (she has no trouble being heard without one), and she slowly paced the stage as a woman stunned, vulnerable, tragically naked. It is, after all, a brutal song.

Here is the NY Times review of the show.

You can check out her latest album on iTunes.

I'll bring this year's list to a close with some vintage Reeves. Here she is singing (the verb seems insufficient) the jazz standard "You Go to My Head." Enjoy!

Friday, November 20, 2015

Sharing Some Math Problems

I thought it might be nice to share some of the problems I've written for Saturday School, just in case other kids might have fun with them too.

The problems are at The grade level of the topics is roughly from Kindergarten to grade 2.

Happy mathing! If you see any errors, please let me know in the comments. (For the sake of context, I should note that I made each of these pages quickly on a Saturday morning.)

Update 11/16/2016: URL corrected

Wednesday, November 18, 2015

Lewis Carrol's Jest

Earlier I wrote about Lewis Carroll's book of "pillow problems." The other night I was reading problem #72 and did a double-take:
A bag contains 2 counters, as to which nothing is known except that each is either black or white. Ascertain their colours without taking them out of the bag.
I thought this must be a trick question. My answer was along the lines of "Ask someone else to take the counters out of the bag for you." But turning to the solution, I found this calculation:

"Whaaaat? That's absurd," I thought. "Both of the original counters might have been white, or both might have been black—a calculation can't retroactively change the counters' colors!"

I thought this must be a jest on Carroll's part. But then again, Carroll took seriously his role as a public communicator of mathematics, and so he would never leave a misleading calculation on the page for people to absorb as if it were true. At the same time, Carroll himself couldn't be mistaken about such a simple problem, especially since some other problems in the book show Carroll creating probability paradoxes and resolving them correctly.

Then I noticed the words "THE END." I thought, It would be just like him to end with a laugh. But still I didn't quite believe that he would commit a fallacy to paper, even for a joke. On the other hand, fallacious proofs like 0 = 1 are a commonplace in mathematical recreations....

I returned to the introduction, which I hadn't read in several months, and there I found this paragraph:
If any of my readers should feel inclined to reproach me with having worked too uniformly in the region of Common-place, and with never having ventured to wander out of the beaten tracks, I can proudly point to my one Problem in 'Transcendental Probabilities'—a subject in which, I believe, very little has yet been done by even the most enterprising of mathematical explorers. To the casual reader it may seem abnormal, and even paradoxical; but I would have such a reader ask himself, candidly, the question "Is not Life itself a Paradox?"
OK, there we go—found the wink-wink! With problem number 72, Carroll was taking tongue in cheek and showing the reader a tempting piece of sophistry.

Next morning, I searched online for discussions of the problem. One is here, but the content is not available for free and I didn't read it. The problem was also mentioned by the great Martin Gardner (1914–2010) in his book The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Puzzles, Games, and Word Plays. (I don't own this book, but in case anybody reading this post is responsible for putting together a Christmas list....)
The book's last problem, number 72, has been the subject of much controversy. ... The proof is so obviously false that it is hard to comprehend how several top mathematicians could have taken it seriously and cited it as an example of how little Carroll understood probability theory!
Gardner goes on to explain how in the book's Introduction Carroll "gives the hoax away." Gardner however doesn't seem to spend any time exploring the paradox. If I come up with any additional thoughts about it, I'll update this post with them. In the meantime, I leave you in the capable hands of another admirer of Lewis Carroll.

Monday, November 16, 2015

Fishing Vacation (A Word Puzzle)

"I don't know if I've ever been more content (or more sunburned) than in East Cape, Baja Peninsula. That was the best of our vacations. The boat weightless on the waves, a sixpack on ice. With all we were catching in the afternoons, even a medium-size roosterfish or bonefish wasn't worth reeling in."

Circle the number words ZERO through TEN in this story.

My kids enjoyed doing this one. A PDF suitable for printing is at

Sunday, November 15, 2015

Why Won't Your Calculator Tell You What 3 Divided By 0 Equals?


Suppose somebody types 3 ÷ 0 into a calculator, hits the = button, and gets an error message. If the person asks you what is going on, what do you say?

I think in these situations we mostly tend to say, "You can't divide by zero." That isn't wrong, but I suddenly find it odd, because it sounds more like physics than mathematics. You can't travel faster than light, you can't build a perpetual motion machine, you can't unscramble an egg, and you can't divide by zero. One of these things is not like the others, yet the formulation doesn't reflect that difference. A scientist could doubt whether it is really impossible to travel faster than light—all scientific knowledge is provisional—but we can say dogmatically that nobody will ever build a calculator that gives a numerical answer for 3 ÷ 0.

Instead of saying "You can't," in a case like 3 ÷ 0, I think it would be better to say to the student,
"There's no such number."
It isn't so much that "you can't divide 3 by 0" as that there's no such number as 3 ÷ 0.

Why is there no such number as 3 ÷ 0? Because remember what 3 ÷ 0 actually means: it means the number that, when multiplied by 0, gives 3.

Think about it! There is no such number!

Any number multiplied by 0 always gives 0, never 3.

"You can't do that" also sounds like an arbitrary rule to remember...just another of the endless rules adults are always hitting kids with. "There's no such number," on the other hand, is a statement of mathematical fact.

Saying "You can't" invites the reply, "Why can't I?" A good answer to which would be: "There's no such number." One might as well skip a step.

It certainly seems absurd to share 3 cookies among no people, or to ask how many servings of size zero are in 3 cups of ice cream...but what is really absurd about the first situation for example is not the vaguely puzzling idea of sharing among no people; it's the concretely outrageous idea that the result of sharing 3 cookies among no people would be shares of half a cookie! Or seven cookies, or any other number of cookies. Likewise, there could not be eleven 0-cup servings in 3 cups of ice cream. No number could be the value of 3 ÷ 0.

So tell me again, why does 3 ÷ 0 give Err on the calculator? Could it be that the school bought cheap calculators to save money? Would a more expensive calculator have told us the number? No. Even at Chichi Academy they get an error message. There is no such number, so the calculator has to display something else.

"Undefined" is the word that careful math educators tend to use for division by zero. And that word is accurate...but saying that "Division by zero is undefined" strikes me as little better or even no better than saying that "You can't divide by zero." After all, what is the force of "undefined," in a discourse where precious little ever gets defined in the first place? I think it means nothing to a student and it gets internally stored in student memory as: "You can't." Anyway, in a case like 3 ÷ 0, "undefined" is shorthand for "there is no such number"! Instead of using shorthand, one could make a statement that has content.


"There's no such number" describes a case like 3 ÷ 0 where the dividend is nonzero. What about the doubly bizarre case of 0 ÷ 0? This, recall, would have to mean the number that, when multiplied by 0, equals 0. But every number, when multiplied by 0, equals 0. So there is a case to be made that any number could be the value of 0 ÷ 0. In a sense, this is the opposite situation to 3 ÷ 0, where there was no such number.

Could there be fifty 0-cup servings of ice cream in 0 cups of ice cream? Sure, why not! Heck, let it be a thousand 0-cup servings. You can feed any number of people with no ice cream, as long as everybody would be satisfied with getting none.

If 0 cookies are shared equally among 0 people, could each person get a dozen cookies? It wouldn't violate any laws of physics!


My own calculator, Mathematica, is pretty chichi, and unsurprisingly it gives two different results for 3 ÷ 0 and 0 ÷ 0, along with some error messages:

Mathematica uses the non-numeric quantity "ComplexInfinity" for various purposes in the analysis of functions. If you ask Mathematica for the value of 0 times ComplexInfinity, you won't get a numeric answer—you'll get "Indeterminate," another non-numeric quantity. So in Mathematica, 0 × (3 ÷ 0) ≠ 3.


I'll close this post with two other cases where "There's no such number" seems like a better answer than "You can't do that":

  • Instead of saying "You can't take the logarithm of a negative number," you could say, "There's no such number as Log(−8)." Why? Because remember what Log(−8) means: it means the number that, when you raise 10 to that power, you get −8. But try a few cases and you'll see that there is no such number...10 raised to any real power is always a positive number, never −8. There's no such number as Log(−8).
  • Instead of saying "You can't take the square root of a negative number," you could say, "There's no such number as Sqrt(−1)." Why? Because remember what Sqrt(x) means: it means the nonnegative number that, when you square it, you get x. But try a few cases and you'll see that there is no nonnegative number that squares to give you −1. There's no such number as Sqrt(−1).

In these two examples, "number" means "real number." High school students learn that there are two complex numbers whose square is −1, and college students in math or physics learn that there are infinitely many complex numbers that, when you raise 10 to that power, you get −8.

Can complex numbers also help us divide by zero? No. Just like the real numbers, the complex numbers are algebraically a field, which implies that even in the complex numbers, there's no such number as 3 ÷ 0.

Wednesday, November 4, 2015

Slot Machine Word Puzzle

Form a long word by choosing one short word from each wheel and then stringing together your three choices.

(You might imagine that the three wheels are the guts of a slot machine—you'll hit the jackpot if the three short words that end up in the display window join up to form a long word.)

Ten words may be formed in all.

Monday, November 2, 2015

How Do You Put 29 Marbles Into 4 Vases? Not With Division.

Elmer has 29 marbles and 4 vases. He puts the same number of marbles in each vase. How many marbles does he put in each vase? How many marbles are left over?

I saw a problem like this in a math textbook, and the intended answer was, "7 marbles in each vase, with 1 left over."

I wondered why the answer couldn't be "1 marble in each vase, with 25 marbles left over." After all, \(4\times 1 + 25\) equals \(29\).

Or "2 marbles in each vase, with 21 marbles left over."   \(4\times 2 + 21\) equals \(29\), too.

If the intent of the problem was to have a unique answer, then it ought to have said something like this:

Elmer had 29 marbles and 4 vases. He put the same number of marbles in each vase. What is the largest number of marbles each vase could have received? In that case, how many marbles would be left over?

Such problems are often presented to students as division problems; students are taught to record the answer as \(29 \div 4 = 7 R 1\), or perhaps \(29 \div 4 \rightarrow 7 R 1\). Notice that neither of these jottings is an equation: in the first case because "7 R 1" isn't a number, and in the second case because \(\rightarrow\) isn't an equals sign. Couldn't students be taught to represent a situation like this using an honest-to-goodness true equation, namely \(29 = 4\times 7 + 1\)?

The equation \(29 = 4\times 7 + 1\) shows clearly what is going on: the 29 marbles can be grouped into 4 equal groups of 7, with 1 left over.

I know it is difficult to break with a tradition like "\(29 \div 4 = 7 R 1\)." Maybe there are even reasons for that tradition that make it inevitable. Perhaps, in that case, students might be taught to represent the problem sometimes using an actual equation like \(29 = 4\times 7 + 1\).

It does bother me, though, this approach of presenting problems as division problems when division doesn't solve them. I mean, the answer to Elmer's problem isn't \(29 \div 4\), so in what sense is this a division problem?

One might wish to say that the reason this is a division problem is that "we are sharing equally." Well, sort of yes, sort of no. We start out with 29 marbles, and in fact we don't share those 29 marbles equally. We share 28 of them equally. And we throw the other one away.

Here's another way to think about it. Take the (true, actual) equation \(29 = 4\times 7 + 1\), divide both sides by 4, and simplify. The result is

\(29\div 4 = 7 + \frac{1}{4}\).

But what is the meaning of "seven and a quarter marbles"? That notion doesn't make sense in the context of the problem.

Division breaks a quantity into equal parts. That's what it does. If the stated number of objects in your problem can't be broken into the stated number of equal parts without the situation turning silly, then that could be a hint that your problem isn't really a division problem.


I thought I'd keep this space for additional thoughts that might accumulate over time.

If Elmer's problem isn't a division problem per se, then we could still see it as a problem with division in it. Along those lines, here is another way to represent the problem and the answer: \((29 - 1)\div 4 = 7\). This equation is another nice account of what's going on: it's an (actual, true) equation that correctly uses division, and moreover it uses subtraction to show the "throwing away" that I alluded to above. 29 marbles, throw away 1, share the rest equally among 4 vases, yields 7 marbles in each vase.

UPDATE 11/14/15 - A one-page PDF of this post, formatted nicely for printing, is available at

Tuesday, October 27, 2015

Are Murder Rates Spiking? Part 3

First, the New York Times erroneously reports that murder rates are spiking all across America. Likely, then, FBI Director James Comey reads the story, or anyway some of the follow-on coverage of it. And so now Comey gets up and gives a speech about the possible causes of this generally non-existent spike—a speech which the New York Times then dutifully covers as front-page news.

I think this is called "epistemic closure."

Actually, it's not so bad. Based on the front-page NYT story as well as this report from NPR, I would say that Comey is improving the quality of this discussion, at least from a statistical point of view, because although he is shopping an unsupported and controversial theory, he is at least honest that he has no data to back it up, and also because whenever he discusses the spiking murder rate itself, he seems careful to add the qualifier "in some cities," which means that he is at least devoting his time to talking about something that exists. The reporters who cover him would do well to catch up.

Sunday, October 25, 2015

ASCII Art Word Search

A couple of years ago I saw "the dog puzzle," which is a one-word version of a word-search puzzle. I wished I'd thought of it myself! Later, I saw some cool examples of ASCII art. Yesterday, it occurred to me that I could combine these neat ideas! So today I made the word search puzzle shown in a thumbnail image below—click here to print it out.

Clown Word Find

Sunday, September 13, 2015

Are Murder Rates Spiking? Cont'd

I've finally seen some pushback against the New York Times article "Murder Rates Rising Sharply in Many U.S. Cities." (Previous post on this here.)

Based on a larger sample of cities, FiveThirtyEight concludes that "Scare Headlines Exaggerated the U.S. Crime Wave."

(However, they compiled the same kind of one-year data the Times showed....)

Wonkblog also pushed back on the Times's overall thesis. And to Wonkblog's credit, when they dug deeper and argued that murder rates are indeed spiking in Baltimore, St. Louis, and Milwaukee, they didn't base their reasoning on one-year data; instead they did the necessary legwork to compile a historical trend:

At a glance, the bumps for these three cities appear significant.

The Wonkblog article also includes other interesting plots along with discussion and interviews with experts.

Friday, September 4, 2015

Animals Inside Animals

In each animal below, find another animal and circle it. The first one has been done for you.


BANDICOOT (answer)
CASSOWARY [two animals inside this one]
CRAYFISH [fish is not the intended answer]
TAPEWORM [worm is not the intended answer]

ALPACA (answer)
IGUANA (answer)
KANGAROO (answer)

All of the following animals have ANT in them. Which animals are they?

Thursday, September 3, 2015

Are Murder Rates Spiking?

Recently the New York Times wrote on the subject of crime statistics. The article, "Murder Rates Rising Sharply in Many U.S. Cities," was backed up by this table of numbers: 

I've seen a lot of pundits adding their two cents to the Times story, but nobody seems to be criticizing the awful data analysis in the article.

First problem: The title of the Times's graph says "Rising Murder Rates," but none of the numbers in the graph are rates (per capita). Of course, if the cities' populations were fairly constant over the past year, then the bar graph would still be fairly accurate. Nevertheless, you shouldn't say "Murder Rates" and then not chart murder rates.  

The much larger problem is that the data shown here in no way demonstrate the article's thesis. For one thing, what is Philadelphia doing in this list? A four percent increase is just noise. Contrary to the article's thrust, the data for Philadelphia show convincingly that there is no spike in the murder rate, at least for that large city.

As one commenter on the Times's website said,
Math. The NYT should learn how to apply it. Crime rates have been historically low the last few years. They have been declining for the better part of a quarter century. Could this be the start of a new trend? Or is it a blip from previous multi generation lows? No one knows for sure but headlines like this are misleading clickbait.
To justify a story like the one the Times wrote, what you want to see is a line plot showing the murder rate in each city over the past five, ten, or twenty years. I made such a line plot for New Orleans, using data from the City of New Orleans Police Department(The numerical values are listed further below in an Appendix; on the plot itself, I have omitted the vertical scale in order to focus your attention on the pattern of fluctuations.)

Seen in context, the blip for 2015 is more noise than signal. So, while the Times article might well contain some interesting sociological analysis, what the article doesn't do is explain the data for New Orleans—because there is nothing in the New Orleans data to explain. 

Maybe if you repeated this exercise for Milwaukee instead of New Orleans, the results would be strikingly different. Even if that were so, it wouldn't change the fact that New Orleans doesn't belong in the list (to say nothing of Philadelphia). In any case, a historical line plot—not the graph the Times presents—is the kind of picture you should be looking at in order to grasp a situation like this. Why didn't the Times do that analysis to begin with? 

Appendix: Data for the line plot

I did this quickly, let me know if there are errors.

Murders by Year (sources: 1, 2, 3, 4, 5)

2010     175
2011     199
2012     193
2013     156
2014     150
2015*   189

*The 2015 value is the Times value, 120, multiplied by 365/232. The end-date range noted in the Times graphic is 8/11–8/31, the average of which, 8/21, is 232 days into the calendar year. 

Population by Year (source)

2010     1,189,866
2011     1,214,235
2012     1,228,375
2013     1,241,949
2014     1,251,849
2015*   1,270,759

*The linked source gives populations for 2010–2014. The population for 2015 is the value obtained from a linear fit to the preceding five years (r^2 = 0.97).

Murder Rate by Year (by dividing)

2010     14.7
2011     16.4
2012     15.7
2013     12.6
2014     12.0
2015     14.9

Monday, August 31, 2015

Three Puzzles Adapted from Lewis Carroll, Cont'd

1. In a certain junkyard, 60% of the cars have lost their wheels, 70% of the cars have lost their doors, and 80% of the cars have lost their seats. What is the least possible fraction of the cars that has lost all three?

2. A bag contains one counter, known to be either white or black. A white counter is put in, the bag shaken, and a counter drawn out, which proves to be white. What is now the chance of drawing a white counter?

3. Consider that:

  • 3 divides evenly into 9. 
  • 37 divides evenly into 999.
  • 7 divides evenly into 999999. 

Show that any odd number not divisible by 5 divides evenly into 99...9, if you include enough 9's.

My answers to the puzzles are here.

You can find the original versions of these puzzles in Carroll's book Pillow Problems—a beautiful edition of which I discovered recently in a used bookstore in Berkeley, California. Here is the eye-catching cover design:

Wednesday, August 12, 2015

Three Puzzles Adapted from Lewis Carroll

1. In a certain junkyard, 60% of the cars have lost their wheels, 70% of the cars have lost their doors, and 80% of the cars have lost their seats. What is the least possible fraction of the cars that has lost all three?

2. A bag contains one counter, known to be either white or black. A white counter is put in, the bag shaken, and a counter drawn out, which proves to be white. What is now the chance of drawing a white counter?

3. Consider that:

  • 3 divides evenly into 9. 
  • 37 divides evenly into 999.
  • 7 divides evenly into 999999. 

Show that any odd number not divisible by 5 divides evenly into 99...9, if you include enough 9's.

All three puzzles can be solved using K–12 mathematics. I have provided some hints here. Answers are here.

Thursday, August 6, 2015

Minimizing Length to a Segment, Cont'd

Place a segment of length 1/2 in such a way as to minimize the sum of its distances to three vertices of a unit square.

In the example shown, the three lengths to be added are shown as dashed lines. (The example shown is not optimal.)

I didn't know how to solve this puzzle when I posted it, but since then I have thought about it some, and I think I have the answer.

Here goes; if I have it wrong, someone will let me know.

To begin with, replace the dashed lines in the figure with solid lines. The result is a network that connects the three vertices of the square. Obviously, this network cannot be shorter than the shortest possible network connecting the three vertices. But what is the shortest possible network connecting three points?

As I learned from this historical study, Pierre de Fermat first posed a similar problem in 1643:
datis tribus punctis, quartum reperire, a quo si ducantur tres rectae ad data puncta, summa trium harum rectarum sit minima quantitatis.

Apparently this is Latin for
given three given points, a fourth is to be found, from which if three straight lines are drawn to the given points, the sum of the three lengths is minimum.
Fermat himself could probably solve this problem, but the first known solution is due to Torricelli (1608–1647). Then, it seems, the problem was forgotten until it was posed again in 1810 by the French mathematician Joseph Gergonne. He posed a number of extensions of the problem, including some that resemble our segment problem. 

Gergenne was the first to pose what has become known in modern times as the Steiner tree problem: Given a collection of points in the plane, find the shortest network that connects them. This is the problem we are interested in today. The Steiner problem differs from Fermat's problem because Gergenne isn't assuming from the outset that the network will meet in a single point (though that turns out to be the case for three boundary points).

Gergenne solved the problem in some specific cases and made impressive progress toward general conclusions. Later, Gauss toyed with the problem, and a mathematician/schoolteacher named Bopp did further work on it, including deriving the first results for a collection of boundary points in three-dimensional space. 

In the 1930's, a couple of Czech mathematicians gave the first modern treatment of the Steiner problem, and in the 1960's interest in the problem exploded. The reason seems to have been twofold. First, problem was presented to a wide readership in the popular 1941 book What Is Mathematics? (still popular!); and second, the problem proved to be directly related to emerging industrial problems such as optimizing integrated circuits and communication networks.

Today, it is known that length-minimizing networks in the Euclidean plane always consist of straight segments that meet in threes at angles of 120 degrees. Like other optimal forms, these minimal networks are often handsome to look at.

(By the way, I first learned about all this when I was an undergraduate and part of the group that proved the planar double-bubble conjecture.)

So much for the history. How does this knowledge help us in the segment problem? Recall that we were interested in the shortest possible network connecting three vertices of the unit square. Based on the summary above, you can probably devise this network yourself; it looks like this:

This length-minimizing network provides a way to solve our segment problem: namely, we lay down our segment along one of the edges of the minimal network:

If we denote the length of the minimal network by L, then we see that the dashed segments in this solution have total length L - 1/2. There is no way to give the dashed lines a smaller total length. For if you could somehow connect the three vertices to the segment using total length D < L - 1/2, then there would exist a network of length D + 1/2 <  L. Since no network exists with length less than L, no better solution to the segment problem exists either.

By the way, our solution has a total dashed length of about 1.43. By comparison, laying the segment vertically down the left-hand edge of the square (touching the upper-left vertex) gives a total dashed length equal to 1.5.