Sunday, November 19, 2017

Overrated, Underrated, and Correctly Rated - Another Round


The equation ei π + 1 = 0. It’s on T-shirts, coffee mugs, and book covers. "The most beautiful equation," everybody says. I have to confess: I don’t get it. I mean, it’s cool—all the celebrity numbers are there, with no wasted space. Was that something we were particularly trying to accomplish in math? Euler's identity must have been a blast at parties in 1750, but as soon as you know that eix = cos x + i sin x, how interesting is the special case x = π ?

Curry. People crave curry like a drug, but of all spicy foods curry is the least flavorful. Chilis, hot peppers, red pepper flakes, even aromatic peppercorns—all are more flavorful than curry. Next time you're making curry at home, try replacing half of the curry powder with garam masala, and see if the food doesn’t taste better.

Buffalo wings and their variations. Chicken wings are less food than food-scraps, something you might find while excavating a midden. Buffalo sauce doesn’t taste very good, and when I do have a taste for it, I prefer a Buffalo chicken wrap. The wrap format gives you more meat with less mess, and none of those weird hairs you sometimes see on chicken wings. Even chicken nuggets are a better choice than wings, and they bring you to the end-state of self-hating nausea even faster.

Ai Weiwei. Overrated as an artist. (Sorry!)

Street fairs. Street fairs block traffic, make you feel guilty for not buying anything, and tempt you into eating food that will only give you diarrhea later. Street fairs are too small (or else the wrong shape) to create a carnival atmosphere, so that you just feel that everybody has gone to too much trouble. Street fairs aren’t just overrated; they’re depressing.


Instant coffee. Years ago, a friend and I were traveling in Chile, and we spent the night at high altitude in an Aymara village. In the morning, the woman who’d given us a bed in her house offered to make us café con leche. "This is going to be good," I thought. The woman brought us steamed milk and two packets of Nescafé freeze-dried instant coffee; it was delicious. I certainly enjoy artisanal coffee roasted and prepared by obsessives, but at home I usually drink this instant coffee, and when traveling I often carry these. I reconstitute my instant coffee with milk—just like they do high in the Andes.

TSA Pre-Check. I hesitated to include it in this list, because if everybody gets pre-check then the benefits disappear. But when you fly often, you'd be surprised what a relief it is to be able to keep your shoes on. Pro-tip for security screening: if you have a choice of which X-ray line to get into, choose the line that puts you behind the gray-faced traveler with this carry-on. His slipstream is the one you want to be in.


Maple syrup. Maple syrup is the only substance produced in the Northern Hemisphere that begins to approach the elusive, paradisaical qualities of tropical vanilla. The world at large must be rating maple syrup correctly, because it's expensive: around a dollar for a quarter-cup. At home we only buy Vermont maple syrup, because as we learned when we resided there, Vermont agricultural regulations require that syrup sold as "Vermont maple syrup" must be 100% pure and produced entirely within the state.

Cats. I love cats and currently have two of them, but one thing we can infer from the eternal battle between cat-people and dog-people is that the cat is a mixed bag. Everything his supporters say is true, and so is everything his detractors say. Regal in bearing, beautiful in form, he nevertheless shits in a box. Self-sufficient and untamed by nature, he flashes teeth and claws at his ostensible master. Graceful and fluid in motion, he eludes your clumsy attempts to get him to the vet. Half pet, half living-room decoration, the cat is an Old God who fulfills the masochistic desire to worship and receive nothing in return.

Saturday, November 4, 2017

A Curious Equation

Earlier this year, a colleague showed me the following equation, which he sometimes uses in conversation with teachers:

x  +  1/x  =  1/x

What do you make of it?


One might add a term −1/x to both sides. That gives the result

x = 0.

Easy! We got the solution in one step. Or did we?

We did not. There is actually no number that satisfies the equation x1/x = 1/x.

No number, when added to its reciprocal, equals its reciprocal.

(Naturally in order for the sum of a number and its reciprocal to equal its reciprocal, the number would have to have a reciprocal! This rules out 0 as a possible solution.)

But surely we didn't do anything wrong in adding the term −1/x to both sides? After all, "Equals added to equals are equal," right?

Yes, equals added to equals are equal; we didn't do anything wrong by adding the term −1/x to both sides. Our technique was impeccable!

Still, was it surprising that a legal move resulted in a value that doesn't solve the equation? It almost seems like a glitch in the Matrix. Shouldn't legal moves always produce valid output?

One might wish it were so. But let's remember that "equals added to equals are equal" is hypothetical:

If a = b, then a + c = b + c

By this principle, we can say that if x1/x = 1/x holds, then x must equal zero. Fine…but since x + 1/x = 1/x never actually holds, this conclusion isn't worth much.


The equation x + 1/x = 1/x probably wouldn't be worth using with students, but it has led to some interesting conversations with fellow math educators. For example, I showed the equation to a colleague, and she suggested viewing it as a statement of equality between function values, f(x) = g(x), where f(x) = x1/x and g(x) = 1/x. That way, as she noted, we could graph the functions f and g on the same set of axes and see if the two curves intersect. It's a good technique to know. Here's what the graph looks like:

The functions f and g are defined for nonzero real inputs. Our question is whether f(x) ever equals g(x) for values of x in the common domain of the two functions.

As the values of x get smaller and smaller, the two curves approach closer and closer to one another; it's as if the curves are "trying" to touch! However, the blue curve is always displaced from the red curve by some amount. (The amount is x.)

Although f(x) = g(x) never occurs, Calculus students might want to show that

limx→0 (f(x) − g(x)) = 0.

This statement makes precise the intuition that the curves are "trying to touch." It's a valid statement because calculating the limit as x → 0 never actually requires us to evaluate f(x), g(x), or f(x) − g(x) at x = 0.

By the way, I don't want to leave the impression that extraneous roots are always good approximations to f(x) = g(x). For example, consider the equation Sqrt[x] = −1, which lends itself to finding an extraneous root x = 1. In this case, there are no values of x for which the left-hand side approaches the right-hand side arbitrarily closely. 


I showed the equation to a second colleague, and she asked for a real-world situation leading to the equation—something with a context, not just an English translation "A number added to its reciprocal equals its reciprocal." Her question was really more general: how could it happen that a real-world situation leads to an equation with no solutions? So I made up this word problem, which relates in some way to the equation:

Find the dimensions of a rectangular 100-square-foot kitchen so that the perimeter is twice one of the sides.

(This situation could be represented by 2x + 200/x = 200/x which is our equation multiplied by factor of 2, or it could be represented by 200/x + 2x = 2x, another equation with no solutions.)

A nice thing about having a context is that there's a contextual reason why x can't be zero. If a kitchen has area 100 square feet, then no side of the kitchen could have zero length! (Whereas without the context, the grounds for excluding x = 0 are abstract.) The context also gives concreteness to the fundamental absurdity of the problem: since the perimeter is twice a pair of sides, it can't also be twice one of the sides, again because that would require a side of zero length.

The kitchen problem asks for something impossible, and that's why the equation that represents the question has no solution. Such absurdities aren't always apparent at first. I think of the many research findings in physics, economics, or other fields, arguing that "such-and-such a thing cannot happen." Often the argument is that the thing of interest can only happen if a certain equation has solutions; yet the equation in question turns out to have no solutions. (More often the mathematics centers on an inequality, but the logic is the same in either case.)

Even more interesting are the reverse cases, in which a scientist takes seriously an "extraneous root" and thereby discovers new phenomena. A famous case of this in physics was when Dirac took seriously the negative-energy solutions that arose from his relativistic version of the Schrodinger equation (what we now refer to as the Dirac equation). After a few years of striving for understanding, Dirac proposed the existence of a new particle, the anti-electron, also known as the positron. It turned out to exist! Dirac persisted because he believed that in a beautiful cosmos, the mathematics in question was too beautiful to ignore.

James Clerk Maxwell had done something similar but even more important. When he looked hard at the equations of electromagnetism as they were known in 1865, he deduced that the equations allowed for traveling wave solutions with a speed consistent with the known value of the speed of light. Thus was born the revolutionary idea of light as an electromagnetic phenomenon. About twenty years later, Heinrich Hertz verified the existence of electromagnetic waves in the laboratory. We are lucky enough to be witnessing a similar confirming moment, thanks to the discovery in 2015 of gravitational waves, which were predicted to exist a hundred years ago by Albert Einstein on the basis of his gravitational field equations.


Getting back to our more humble equation, I was shocked to discover that the computer program Mathematica gives the wrong answer to it! Mathematica thinks the solution is x = 0. Not only is this mathematically wrong, it's internally inconsistent, because substituting x = 0 into the equation doesn't yield the result True. No value or token in Mathematica's language can be substituted into the equation yielding True. Yet when asked to analyze the equation, Mathematica fails to return False.

Computers are a powerful tool for doing math, but you have to stay on your toes.


Let's look at another equation with no solutions.

(x + 3)/(2x + 6) = 1. 

My steps in solving this equation are going to sound like this inside my head:

[Well, if it's true that x satisfies]

(x + 3)/(2x + 6) = 1

[then x must also satisfy]

x + 3 = 2x + 6.

[And if that's true, then x must satisfy]

x = −3.

Conclusion: if there's a solution of (x + 3)/(2x + 6) = 1, it can only be x = − 3.

But does x = −3 satisfy the original equation? No. So the equation has no solutions.

When solving equations, we typically record only symbols; but whether we know it or not, our mathematics also includes the logical "connecting tissue" shown above in italics.

The intermediate equation x + 3 = 2x + 6 has more solutions than the equation we started with, (x + 3)/(2x + 6) = 1. Ideally, each "step" in your solution process would be a new equation with exactly the same solutions as the previous equation. But that isn't always feasible when solving complicated equations—which means that you might have to cull extraneous solutions and/or restore dropped solutions in order to solve the equation correctly. That sounds complicated, even to me as I write it, but if solving could easily be made algorithmic, then Mathematica would probably be getting it right by now.

Some years ago, I asked a group of math educators to write down an equation with the same solution as the equation 3(y − 1) = 8. Many did the natural thing: they first solved for the value of y, namely y = 11/3. (They wanted to know the solution, so they could write down an equation with that same solution.) What some didn't realize, however, was that their very first step in solving for y, whether it was 3y − 3 = 8 or y − 1 = 8/3, was already a valid answer to my question. Each "step" in their solution process was an equation with the same solution as the original. The accumulating "steps" of their work weren't only a sequence of moves; they were also a chain of logically equivalent claims about y.


Analyzing equations isn't always synonymous with solving them. In my physics textbook, for example, there's a problem about stars in which physics insight emerges from analyzing the equation

k(LD) = GM2/D2 

where the variable is D. The goal isn't to produce the solutions to the equation; those roots are a mess! Instead the goal is to understand how the number of solutions depends on the physical parameters k, L, G, and M. (The best method is to sketch non-numerical graphs of the functions f(D) = k(LD) and g(D) = GM2/D2 on the same set of axes, and think about intersections.) These mathematical conclusions in turn tell us about how the star's fate depends on the physical parameters. For example, if the star is too massive, then we find that there are no solutions to the equation, and this suggests a runaway collapse to form a black hole.

A simpler example of a problem in the analyzing category might be,

Show that the equation x3 − 9x2 + 23x − 15 = 0 has no negative solutions.

Here, all you have to do is argue that substituting a negative value for x results in a negative value for the left-hand side. Conclusion: there are no negative solutions.

This problem is also an example of looking for and making use of structure, a valuable practice in mathematics. Studying the equation, we see a pattern: coefficients of odd-power terms are all positive, while coefficients of even-power terms are all negative; this ensures that a negative value for x leads to a negative value for the left-hand side as a whole.

In college-level science and beyond, the equations that arise can be so difficult that solving them is impractical or even impossible. Being able to extract insight from the equations anyway is a skill that science students and practicing scientists depend on.

Because solving can involve a lot of mechanical steps and strategic moves, practicing the moves and gaining strategic competence are a big part of "getting good at algebra." However, equation-solving can't only be viewed as choosing and executing moves. We could get wrong answers in advanced cases if we fail to understand that equation solving is fundamentally a process of playing out hypotheticals. That understanding is unlikely to emerge if the curriculum consists entirely of problems in which students are given an equation and asked to return a number or several numbers. Students should also occasionally be given an equation and asked to produce, not a number, but rather a conclusion.

Thursday, November 2, 2017

Another Court Injunction Weighs Against Setting Policy via Twitter

In a previous post, I noted that the Ninth Circuit had issued an injunction against a "travel ban" executive order on the basis that (1) the law requires a "finding" of threat to national security, whereas (2) the executive order was mere assertion, thus short of a finding (that is, lacking the rationale implied by the word "finding"). This argument appeared to have been lifted from the earlier decision of a sister court.

A similar situation has now arisen with the recent injunction against Trump's Twitter-ban on transgender military service. (Opinion here.) A number of affected military personnel and aspiring military personnel filed suit against the policy, which had been announced via Tweet in August and elaborated via Presidential Memo in September.

Some key passages in the new transgender ban opinion reminded me of the earlier travel ban opinion:

...the President abruptly announced, via Twitter—without any of the formality or deliberative processes that generally accompany the development and announcement of major policy changes that will gravely affect the lives of many Americans—that all transgender individuals would be precluded from participating in the military in any capacity. 

[In the 1981 case of Rostker v. Goldberg], the Court reviewed the particular facts before it and found that the district court in that case had not sufficiently deferred to the reasoned decision of Congress in the context of a particular military personnel-related decision.

The study and evaluation of evidence that the Rostker Court found warranted judicial deference is completely absent from the current record. ... Accordingly, unlike the district court in Rostker, the Court’s analysis in this Opinion has not been based on an independent evaluation of evidence or faulting of the President for choosing between two alternatives based on competing evidence. 

(Emphases mine.)

All the stress being laid here upon reasoning, evidence, and deliberation reminds me of a trip I took to Tennessee earlier this year. There, in a small-town bar, I had a conversation with a gentleman in his 80s who had voted for Donald Trump. He gave many reasons for his vote; the reason most relevant here was the desire he'd had to see "government run like a business." One hears this opinion from time to time (director Steven Soderburgh once said something not too different). And you can see the temptation…how easy it would all be, if we could just let one person do all the deciding! A real executive, you know?

But there are fatal problems with this model. If you don't like your boss's decisions, you can find another job. But if you don't like the President's decisions, you can't find another country—nor are you obligated to. Because unlike in a business, where there's an official hierarchy, in America every citizen has equal political rights. That means you are entitled to challenge the President's decisions. At your workplace, the boss is giving you a job; that's why it's "his way or the highway." But in America, we're the boss. The government serves at our pleasure.

Every policy decision advantages some people and disadvantages some other people. This gives the government a responsibility to account for its decisions. In the transgender case, why did the government take away the plaintiffs' life option to serve in the military? The plaintiffs have a right to know the reasons. They have a right for the reasons to exist. That Donald Trump wants something isn't reason enough in itself.

Tuesday, October 31, 2017

Minimax Approximation of a Cubic by a Quadratic

Here's a problem from my notebook:

If f(x) = xx3, find the quadratic function g(x) that minimizes the value of max|f(x) − g(x)| on the interval 0 ≤ x ≤ 1.

In other words, what is that "quadratic hump" which best approximates a "cubic hump" over the entire interval [0, 1]?

I thought this was going to be easy, but when it turned out to be harder I did some googling and found that the problem belongs to the interesting subject of minimax approximation, or uniform approximation by polynomials (see Wikipedia).

My problem is a pretty trivial case, because the function I'm approximating is already a polynomial. And the approximating function is of low order (quadratic). Still, I didn't see a worked example online of uniformly approximating a cubic by a quadratic. (This slide show gives the example of uniformly approximating a quadratic function by a linear function.) Nor did I see the answer to the x − x3 problem anywhere online. So for the benefit of future googlers, I'll post my own solution here. I didn't use any fancy machinery, just an intuition about what the optimum situation looks like. (I think this intuition proves to be general…anyway, that might be what the Chebyshev Equioscillation Theorem is about. But I haven't read enough into it to be sure.)

Let the quadratic we seek be written as g(x) = ax2 + (1 + b)x + c. One of the first things I did was to make some guesses and view them graphically, for example like this:

The blue curve is the target cubic. The red curve is the approximating quadratic. The yellow curve at the bottom is the error.

I felt it was suboptimal to have so much error residing in the right-hand lobe…it seemed to me that I should deform the quadratic to trade off some of the right-hand error, at the cost of raising the left-hand error somewhat. Now in the optimum situation no such rebalancing should be possible, so I made a requirement that the two lobes' errors be equal.

Soon I got into trouble, though, because I realized I could make both lobes' errors simultaneously zero!

The trouble was that I hadn't been accounting for the edge errors. With those in mind, again it seemed to me that in the optimum situation, further rebalancing across the various errors would be impossible because all of the errors would be evenly allocated; that is, the following equations would hold simultaneously:

g(0) = D

g(x1) − f(x1) = D

f(x2) − g(x2) = D

g(1) = D

where D is the common value of the difference in function values. Here, x1,2 are the values in (0, 1) for which (f(x) − g(x))' vanishes.

The solution now proceeds in three steps:

1. The system implies −g(0) = g(1), which gives c = −½(a + b + 1).

2. The system also implies g(x1) − f(x1) = f(x2) − g(x2), which gives a = −3/2 or a = ¾(1 − Sqrt[9 + 8b]).

3. The system implies −g(0) = g(x1) − f(x1), and with a = −3/2 this gives b = 9/16 and hence c = −1/32. So g(x) = −(3/2)x2 + (25/16)x − 1/32. (Alternatively, if a = ¾(1 − Sqrt[9 + 8b]), we get g(x) = −6x2 + 10x − 2, which isn't optimal.) So the answer is

g(x) = −(3/2)x2 + (25/16)x − 1/32

for which

max|f(x) − g(x)| = 1/32.

This, I think, is the best one can do. Here's a graph:

The Weierstrass Polynomial Approximation Theorem (1855) shows that good polynomial approximations to continuous functions on a closed interval exist. This theorem is important in mathematical physics, but it's an existence proof; until now, I think I'd never actually calculated any specific uniform polynomial approximations, even in simple cases like the present one.

Note that I solved (f(x) − g(x))' = 0 by hand for x1,2, but I used Mathematica for the subsequent grunt-work. The commands are shown at bottom.


In my googling around, I saw it claimed that a good approximation to the minimax polynomial can often be found by calculating a partial Chebyshev series for the target function. This raises the question of whether my solution −(3/2)x2 + (25/16)x − 1/32 is just the Chebyshev series for the function xx3, up to the second order. In fact, that is the case, as one can easily verify using a table of coefficients. (I used Table 22.3 in my old copy of Abramowitz and Stegun.) It takes a little work, because you first have to shift the Chebyshev polynomials from [−1, 1] to [0, 1]. But it's easier work, and less of it, than my naive approach of equipartitioning the error. Here are the steps:

Use Table 22.3 in A & S to calculate the shifted Chebyshev polynomials T0*(x) = 1, T1*(x) = 2x − 1, T2*(x) = 8x2 − 8x + 1, and T3*(x) = 32x3 − 48x2 + 18x − 1. These polynomials are orthogonal on [0, 1] with weighting function (xx2)−1/2, and they have the same normalization as the unshifted polynomials.

By inspection, x = (1/2)T0* + (1/2)T1*.

By inspection and a bit of scratchwork, x3 = (1/32)(T3* + 6T2* + 15T1* + 10T0*).

Thus xx3 = (3/16)T0* + (1/32)T1* − (3/16)T2* − (1/32)T3*.

Truncate this at the second order:

(3/16)T0 + (1/32)T1 − (3/16)T2 

−(3/2)x2 + (25/16)x − 1/32,

which is the solution found earlier. The method of Chebyshev expansion will generalize pretty easily to a problem like quadratically approximating xx4, whereas that problem would be substantially more difficult using the naive approach I took for xx3.


Sunday, October 29, 2017

Book Review: Transit, by Rachel Cusk

Transit, by Rachel Cusk
Farrar, Straus and Giroux, 2016
259 pages

[Louis's] torn leather jacket and stained jeans made so obvious a contrast with Julian's luxurious navy suit and mauve silk cravat that his appearance seemed, despite his attitude of slouching indifference, premeditated and deliberate.

Why describe such shabbiness as being both premeditated and deliberate? The two words are often considered synonymous. But when they're being used together, what's common about them gets reinforced and what's different gets enhanced: Louis's performative wardrobe was both a decision taken beforehand and a thing painstakingly designed.

By such writing the narrator of Transit trains the reader, so that elsewhere when she writes, "It was further than it looked and the rain unleashed itself with a sudden burst of intensity while we ran," you don't skip past that word unleashed, the way you ordinarily would. You see instead what she wants you to see: a wild something being let off the chain.

There is also this question: premeditated and deliberate on the part of whom? Louis, or Rachel Cusk?


Like other significant novelists of recent decades, Rachel Cusk wonders what the novel is supposed to be now. Shall we keep faith with Charles Dickens and tell tales, as Donna Tartt did in The Goldfinch? Or are we postmoderns stuck with "stories about nothing"—Iowa Workshop meanderings that sidle up to quiet epiphanies? Karl Ove Knausgaard's epic My Struggle arose at least in part from his own frustrations with the novel form, and I don't think it's an accident that Rachel Cusk reviewed Book 2 of the series. In her review, she wrote that Knausgaard

...shows us, by the route of life, that there is no story, and in so doing he finds, at last, authenticity. For that alone, this deserves to be called perhaps the most significant literary enterprise of our times.

Outline, the first book in Cusk's trilogy, was, I thought, "another brilliant solution to the problem of the novel." But of that book too, it could almost be said that "there is no story." The good news for anyone who was bored by Outline is that Transit is more eventful, and funnier. We also learn more about the main character, primarily through the rejoinders she makes and the questions she asks of the people to whom she speaks. These exchanges, I felt, revealed her to be a somewhat clenched person.

And yet we don't learn much. Louis, a character who writes plotless novels, bares his soul during a public reading, but when it comes to the narrator's turn to give a reading from one of her works, we don't even get to hear the passage that she reads. She's a writer, but what is her writing like? But what an odd question, because we are reading her writing. She is the narrator.


Like Outline, Transit is structured by conversation. The first-person narrator retells conversations she's had. Often, the subject of the conversation is some earlier conversation the narrator is hearing about. In these nested conversations there's a lot of career talk. She talks with a contractor, a hairdresser, a photojournalist, several writers; and her interlocutors talk about work. An appearance at a writers' event gives the narrrator a chance to work out some ideas about fiction.

Transit is an astronomical term, and the book opens with a description of an astrologer's spam email. The book's many curious and obvious repetitions make me think of Jung. A baby-faced boy; a discussion of "baby people"; a writer who looks like a baby. A hairdresser talking to the narrator about his autistic nephew, when (as it turns out) an autistic child is sitting in the neighboring chair. A walk with an old boyfriend is ruled by a "tacit agreement," and in another walk later on, "they seemed to be walking towards some agreement." Twice, a man and a woman find themselves standing at the door of a hotel room. Sheds also come up twice.

One detail the narrator focuses on is the roots under the pavement in her backyard, and likewise the roots under the sidewalks in a town with a literary festival. The subtext of the book—and its text, in a gorgeous penultimate sentence—appears to be the subterranean movements of the self, and the effort to impose narrative on them. As the story progresses, it becomes more about fate, choice, control—and how a life story relates to Story. One character is described as "the opposite of fatalistic," seeing life "as a fantastical plot full of contrivances." Another character says, "It felt wrong to be choosing [which country to live in]. ... It felt wrong for the whole of life to be based on a choice." A modern person's dilemma, and a modern novelist's.

Although this second book offers more pleasures than the first (I dog-eared dozens of pages), I do feel that something extremely subtle, a little beyond me, was accomplished in the first installment. Maybe, like the Knausgaard books, somebody might even want to read these out of order. In any case, I'm extremely eager to read the third book in Cusk's trilogy, titled Kudos and due out next year.

Read this review on Amazon

Thursday, October 26, 2017

Solution to Halloween Challenge: Haunted Hayride

The bottom row says B × BOO = BOO. Since BOO is nonzero, this implies B = 1.

Could O equal zero? No, because then the Y in HAY would have been O.

Could O equal 1? No, because 1 is represented by B.

Could O equal 2? No, because then the A and Y in HAY would have been the same letter. By a similar argument, O couldn't equal 3.

O could equal 4. In fact, if we just try O = 4, we will find that the substitution works. And this is the answer to the puzzle: 144 × 144 (576, 576, 144) = 20,736.

But is BOO = 144 the only solution? Traditionally in these kinds of puzzles, the solution is unique; let's be sure.

Could O equal 5? No, because 5 × 5 ends in 5, so the Y in HAY would have been O. By a similar argument, O couldn't equal 6.

Could O equal 7? No, because 7 × 177 has more than three digits, whereas HAY only has three digits. And if 7 × 177 has more than three digits, then the larger products 8 × 188 and 9 × 199 also have more than three digits.

So BOO = 144 is the unique solution to the puzzle.

To the readers who commented on the previous post, sent email, worked on the puzzle with me over a drink in Chicago, and shared the puzzle with their students: thanks to all!

Monday, October 23, 2017

Halloween Challenge: Haunted Hayride

Replace each letter with a digit to make a correct multiplication problem.

As usual in puzzles like this, a given letter stands for the same digit everywhere it appears, and a given digit is always represented by the same letter anywhere it appears.