But let’s begin at the beginning.

I was browsing *FiveThirtyEight* (one of these days, I should really go back and read *One* through *FiveThirtySeven*) when I came across a lovely probabilistic table:

There are four teams left in contention for the Super Bowl this year. The match-ups look like this:

FiveThirtyEight uses a fancy probabilistic model to simulate these games, and then, for the plebeians like us, it reports the resulting probabilities.

From the table, it’s easy to see their odds for the coming pair of games. If the weekend were played 10,000 times:

**Kansas City**would win ~6900 times, and**Tennessee**would win ~3100.**San Francisco**would win ~6400 times, and**Green Bay**would win ~3600.

So much for the AFC and NFC Championships. But what about the Super Bowl?

We can tell that Kansas City is in good shape. If we played this season 10,000 times, they’d make 6900 Super Bowls, and win 4300 of them. Thus, assuming they make the big game, they’ve got about a 62% chance of winning it.

Here’s each team’s chance of victory, conditioned on making the Super Bowl:

**Kansas City**: 62%**Tennessee**: 55%**San Francisco**: 41%**Green Bay**: 39%

This was my first surprise. If San Francisco is a strong favorite to beat Green Bay, then why do they have such similar odds in the Super Bowl?

The answer, for the football fan, is obvious: cheeseheads.

No, sorry, I meant to say: match-ups.

Each team has its own idiosyncratic strengths and weaknesses (and fan headgear). That means that the odds of victory depend on who, exactly, you’re playing. Maybe A has a big advantage when playing B, but they’re equally matched against C.

That raises the question: What are the odds, according to FiveThirtyEight’s model, for each of the four possible Super Bowl match-ups?

First, I created a variable representing the AFC team’s probability of victory in each case:

Then, I converted the FiveThirtyEight numbers into equations. For example, here’s the equation for Kansas City:

And here are the analogous equations for Tennessee, San Francisco, and Green Bay. (The latter two are a little more complicated, because I defined my probabilities in terms of Kansas City and Tennessee victories.)

Beautiful. Four linear equations, four unknown variables. Not so different from the simultaneous equations I’ve been teaching my middle schoolers. And so I did exactly what I tell them to do, solving them step by painstaking step.

Ha ha, gotcha! Of course I didn’t! I plugged them into Wolfram Alpha, because I am a lazy hypocrite, and unashamed of it.

Here came my second surprise. The solution was not unique!

How does this happen? It must be that the fourth equation gives no unique information. It can be deduced directly from the other three.

“Madness!” I cried. “I must have done something wrong.”

And so I thrashed. I thought and I fought and I wrought. I looked for my error. I found it nowhere, which I was sure must constitute another error in its own right. And then I realized the truth. There was no error.

The solution really *wasn’t* unique.

To see why, imagine a simpler scenario. Suppose, hypothetically, that these four teams were perfectly evenly matched. Each has a 50% chance of making the Super Bowl, and a 25% chance of winning it.

From this, can we deduce the odds of each match-up?

Not at all! Here are two equally valid solutions:

**Solution #1:** Every single match-up is an even 50/50 split. Thus, no matter who plays whom in the Super Bowl, it’s as good as a coin toss.

**Solution #2: **KC is guaranteed to beat SF, who is in turn guaranteed to beat Tennessee, who is guaranteed to beat Green Bay, who is guaranteed to beat KC. Thus, once the Super Bowl match-up is set, the winner is a lock.

In fact, there are *infinite* solutions here.

Just pick any probability p between 0% and 100%. Then, KC beats SF, and SF beats Tennessee, and Tennessee beats Green Bay, and Green Bay beats KC, all with the same probability p.

Our four equations tell us *nothing* about p. It could be anything!

The actual data offer a little more information than this worst-case scenario. (Thankfully, lest all my travails be for nothing!) Using the results from Wolfram, I generated this list of possible match-up probabilities in Excel:

A more interesting solution than this silly puzzle deserved, right? Two observations:

**The likelier the match-up, the more information we have about it**. The likeliest match-up is KC vs. SF, and for this, we know that KC’s probability of victory is somewhere between 56% and 71%. The least likely match-up, meanwhile, is GB vs. TN, and for this, we know absolutely nothing! In theory, it could be a guaranteed TN victory, a guaranteed GB victory, or anything in between.

**Stepping outside the problem, some combinations of variables seem more plausible than others. **The ones at the bottom and the top of the table seem incongruous. (Could Tennessee really be a huge favorite against one of the NFC teams, and a huge underdog against the other?) Meanwhile, the ones in the middle seem pretty plausible.

**Now, the $1 billion question: Which solution is the likeliest?**

To make my guess, I’m going to go against the whole premise of this puzzle, and argue that match-ups shouldn’t matter *that* much.

The original table suggests that SF is stronger than GB. So let’s eliminate any solutions where KC’s odds of victory violate this assumption. Goodbye, rows 11 and below.

Also, the original table tells us that KC is stronger than GB. So let’s eliminate any solutions where GB’s odds of victory would suggest the reverse. Farewell, rows 9 and up.

What remains? Only row 10.

That’s my final answer:

Final observation: KC and GB are not very match-up sensitive. Their odds of victory change by only a point or two, depending on the opponent.

But SF and TN are quite sensitive. Their odds of victory change by roughly 10%, depending on the opponent.

Ball’s in your court now, FiveThirtyEight. Did my laborious, low-information approach accurately recreate the probabilities?

EDIT: Looks like Peter Owen on Twitter has the solution:

]]>Apparently she’s going through a semicircles phase, which I’m sure will be remembered with the same fervent enthusiasm as Picasso’s blue period, or the Era of Peak TV, or the year the Beatles got really into acid.

Speaking of which: when will Catriona’s blue period arrive? More urgently, what would tripping acid do for one’s geometric imagination?

Without further ado, six puzzles. Feel free to discuss and solve below.

**1.
The Three Amigos**

See also Catriona’s original tweet (and the ensuing discussion).

**2.
The Broken Purple Moon**

When it comes to this puzzle, Catriona explains:

I spent a week thinking about how to pack two semicircles into a larger one, with very little progress. I only managed to get anywhere when I made a scale drawing. I hoped someone would show me why the solution is obvious; I learned lots from reading the solutions, but it seems it is genuinely tricky.

**Double Decker**

Catriona’s preferred solution involves a hidden insight, but she also gives props to this “more physical” solution. See also her original tweet.

**The Box of Tangents**

I’m very fond of this one. More discussion at the original tweet.

**Sizing the Aquarium**

Check out this tweet for a beautiful animated hint.

**The Trisected Corner**

Original here. Catriona explains:

]]>Most people I showed it to (including my students) managed the correct answer in their first guess but then got into all sorts of a muddle trying to explain why.

I did it with trigonometry, but there are nice ways without – such as this.

The first comes from master game designer Sid Sackson. Encountering it in his writing was like finding a new creature in the underbrush, an unknown reptile, with its own strange form of locomotion.

The game-taxonomist in me delighted. Sackson called it “Hold That Line,” but I call it…

** **

**Players: **Two, although under conditions of extreme boredom or lack of paper, another one or two could join.

**What You Need: **

- A pen
- A four-by-four array of dots (or larger, if you like)
- A healthy fear of snakes (that’s ophidiophobia, for my Ancient Roman readers)

**The Goal: **Force your opponent to complete the snake.

**How to Play:**

- One player begins by connecting any two dots via a vertical, horizontal, or diagonal line. Here are three possible opening moves:

- Now, players take turns growing the snake from either end, using horizontal, vertical, or diagonal lines, like so.

- There are some restrictions. The snake (a) must never cross itself, (b) must never revisit a used dot, and (c) must grow only vertically, horizontally, or at a 45
^{o }angle.

- Eventually, the snake can grow no further. At this moment, it springs to life, and discharges its hateful venom into the most recent hand to touch it. Or, more prosaically: Whoever completes the snake is the loser.

It’s a breezy yet strategic game. If you’re nimble, you may even discover a guaranteed winning strategy. Unfortunately, this rather undermines the fun.

(No spoilers here, but I’ll offer two hints: (1) Try it on a 3-by-3 array; and (2) Try the variant where the person who makes the last move is the winner. The solution to this version be adapted, without much trouble, to the traditional game. For more analysis, see Jim Henle’s discussion, and the pertinent MathOverflow thread.)

In any case, once you have a handle on the flow of Sackson’s original, you’re ready for my preferred variant: Snakes!

(Aliases include “Snake Breeder” and “Snakes in the Coordinate Plane.”)

**Players:** Two, though if you want to add a third, just use a larger board and another pen.

**What You Need:**

- Two pens (different colors)
- A five-by-five array of dots (or larger; any size works)
- A logic-defying love of snakes (parseltongue fluency encouraged)

**The Goal: **Draw as many snakes as possible.

**How to Play:**

- Play proceeds much as in Snake. (First, you begin a snake. Then, on each move, you connect a free end of the snake to an unused dot, via a vertical, horizontal, or 45
^{o}line, all without crossing lines or reusing dots.) But there’s a key difference:**in this game, each player is growing their own personal snake.**

- Moreover, in this version, you want to finish your snake as fast as possible. That’s because, when it can no longer grow, you get to begin a new one.

- Eventually, a player will be ready to begin a new snake, but have no space to do so. In that case, the other player simply finishes their snake, and the game ends.

- The winner is whoever created more snakes. If it’s a tie, then look at each player’s snakes, and count up the number of dots. Whoever used fewer dots is the winner.

In this game, both players made two snakes, but blue used fewer dots (11 vs. 13), and thus is the victor.

Once you get the hang of it, the choices you face are subtle and satisfying. Nothing is sweeter than stealing a dot that your opponent was relying on, thus forcing them to go careening off into the open board.

(Nothing, that is, except completing a late-game snake in just a single move. That’s triumph itself.)

The multi-snake variant also spawns a solitaire version—or, really, a collection of puzzles. With no opponent, just creating snakes on your own, how many snakes can you pack into a board of a given size?

For example, in the 3-by-3 board, the best I can achieve is two snakes:

On the 4-by-4 board, meanwhile, I can manage four snakes:

On a 2-by-n board, you can fit n/2 snakes if n is even, and (n+1)/2 snakes if n is odd.

At this point, I cede the floor. Open questions:

- What other “snake numbers” can you figure out? Can you prove that they are optimal? Is there a formula for the 3-by-n board, or the 4-by-n?
- What strategic gambits can you devise for the two-player version? Is there a learnable winning strategy? Are there useful heuristics?
- Who wins in the two-player version, if both players move optimally? Does it depend on the board size?

All of these books are certified 100% great and you should read them so we can chat about them, like a little two-person online book club.

(Note that my attention span favors books which are (1) written with a distinctive voice, (2) intellectually dense, and (3) short. Your tastes and mileage may vary.)

*I read a lot of books about math, for the purpose of stealing the authors’ ideas and, eventually, their identities drawing inspiration.*

**The Weil Conjectures, by Karen Olsson**. Beautiful, subtle reflections on the elusive nature of modern mathematics. Composed from an outsider’s vantage, yet with an insider’s ear and finesse.

**Humble Pi, by Matt Parker**. Riveting, wide-ranging exploration of mathematical mistakes. Includes clever “mistakes” of its own – e.g., the pages are numbered in reverse.

**Euler’s Gem, by David Richeson. **Dave’s two books – this one on topology, and his new one on impossible problems such as squaring the circle – are superb. His selection and presentation of mathematical ideas is exquisite: an unmatched combination of accessibility and depth.

**Tales of Impossibility, by David Richeson**. My back-cover blurb: “The story of a mathematical treasure hunt, and a treasure chest in its own right.”

**What Is the Name of This Book? by Raymond Smullyan**. A classic collection of great puzzles, including those about knaves who always lie, and knights who never do. I especially loved the pages of anecdotes, jokes, and stray thoughts.

**Infinite Powers, by Steven Strogatz**. In contrast to my book on calculus – a silly, literary, personal affair – Strogatz’s is epic and sweeping. It weaves together historical storytelling and surprising accounts of modern applications.

**Mathematics for Human Flourishing, by Francis Su**. My back-cover blurb: “Francis Su believes that math can make us better humans—and he leads by example. Every page is a work of generosity and compassion. Plus, the puzzles will haunt you for weeks.”

*My pleasure reading. A good one gives a quick, stimulating burst of “whoa.” The four authors below – some of my absolute favorite writers – deliver fireworks.*

**Exhalation, by Ted Chiang**. The most carefully crafted and rigorously imagined sci-fi in the business. The final story, on parallel universes, is worth the price of admission.

**How Long ’til Black Future Month?, by N.K. Jemisin**. A collection of extraordinary diversity, from distant-planet exploration to a prose-poem meditation on NYC to a spy caper set in an alternate, high-tech 1800’s New Orleans.

**Changing Planes, by Ursula Le Guin**. Stimulating, funny thought experiments about imaginary civilizations. Silent people; perpetually migrating people; people with wings…

**Sorry Please Thank You, by Charles Yu**. Witty and wildly imaginative meditations on relationships, meaning, and capitalism.

*I guess sometimes I want to be sad? I read all of these before my daughter was born. Having an infant now, I don’t lack for strong emotions in my diet.*

**Night, by Elie Wiesel**. This Holocaust memoir was required reading for 9th graders at the first school where I taught. Ten years later, I’m finally caught up.

**Alex: The Life of a Child, by Frank Deford**. I love Deford’s sportswriting; this is a memoir about his daughter, who died of cystic fibrosis. Sat on my shelf for years; had to read it before my own daughter came along, lest it wreck me even more than it did.

**The Best We Could Do, by Thi Bui**. Graphic memoir of a family’s journey from Vietnam. Full of hurt and compassion, with colors so beautiful they register as music. One of the best books I read all year.

*Not coincidentally, those are adjectives you might use to describe my 2019 book, Change is the Only Constant.*

**The Lonesome Bodybuilder, by Yukiko Motoya**. Short stories; a Japanese sort of magical realism. One of the strangest books I read this year.

**Romeo And/Or Juliet, by Ryan North**. A choose-your-own-path version of Shakespeare’s classic. North is one of my favorite humorists, and he explores every corner and permutation of his delightful premise.

**Love Dishonor Marry Die Cherish Perish, by David Rakoff**. A short novel written entirely in verse – and what struck me, given the sardonic edge of Rakoff’s essays, entirely in earnest.

**Franny and Zooey, by J.D. Salinger**. Two linked novellas. I still don’t understand why *Catcher in the Rye* gets all the glory; Salinger’s other stories are deeper, funnier, and more virtuosic.

**Tenth of December, by George Saunders**. Incisive, witty short stories, ranging from pedestrian to sci-fi fantastical. Saunders has a keen and devastating eye for the flattering lies that we tell ourselves.

**To Say Nothing of the Dog, by Connie Willis**. A time travel romp through Victorian England. Leisurely yet propulsive, full of fun moments. To say nothing of the dog!

*These books, for me, push the bounds on what books can be and do. All three are full of serious, interesting ideas – and all three are playful in presentation.*

**The Dialogues, by Clifford Johnson**. The author, a physics professor, drew this series of cartoon dialogues about science himself. Clearly a multi-talented fellow.

**How To, by Randall Munroe**. I write in Munroe’s shadow, and it’s a big, beautiful shadow. This book shows off his chops not just as a humorist, but a researcher; he has a nose for the quirky and fascinating.

**Basketball (and other things), by Shea Serrano**. I wound up giving this as a gift to every basketball fan I know. A mixture of meticulous argumentation and delicious pop culture lunacy.

*Usually I read stuff that’s years or decades old, but in 2019 I actually read some modern nonfiction about modern concerns that might be relevant to a modern person! Go me!*

**Because Internet, by Gretchen McCulloch**. Whereas oral speech has always had formal and informal registers, writing had only the former. Until the internet. This bestseller thoughtfully unpacks how we write online.

**Hacking Life, by Joseph M. Reagle, Jr.** An affectionate but unflinching critique of the form of self-help known as “life hacks,” and its obsession with optimization.

**How Eskimos Keep Their Babies Warm, by Mei-Ling Hopgood**. Every chapter, more or less: “Here’s a question about child-rearing. Here’s a society that does it totally differently than the US. And guess what? Both have their ups and downs.” Formulaic but immensely reassuring for the new parent.

**Bringing Up Bebe, by Pamela Druckerman**. Polar opposite of Hopgood’s book; celebrating the Parisian style of child-rearing as lower-effort and superior. Probably right on food; maybe insightful on discipline; dubious elsewhere, but well-written.

*I went on a brief kick of this stuff, and it’s great! Good job, teen girls! I mean, not that you wrote these books, but you created the market demand for them, which is the highest form of virtue in a capitalist society!*

**The Fault in Our Stars, by John Green**. Tear-jerking mega-bestseller. I borrowed it from a friend because I fell in love with Green’s podcast, The Anthropocene Reviewed, which I recommend fanatically.

**Catfishing on Catnet, by Naomi Kritzer**. Charming YA thriller, based on a Hugo-winning short story about an AI that gains sentience… and demands cat pictures.

**Grace and the Fever, by Zan Romanoff**. After years of obsessing over a boy band, what if you got to meet them? Half the joy is sheer wish fulfillment; the other half is the surprisingly delicate character study of our guarded narrator.

“How big is the book publishing market in the U.S.?” my friend John asked me as we strolled through the Harvard Coop.

I remembered reading somewhere that the average American buys 5 books per year. Or maybe I made it up; either way, seems about right. Call that $100 per capita. With 300 million of us, that’s a $30 billion market.

(Actual number: $26 billion.)

“Guess how much blood a 16-week-old fetus pumps per day?” my wife quizzed me.

Well, I know an adult has 8 liters of blood. Say that it circulates once per minute. That’s 1500ish times per day. Hence, 12,000 liters per day. But a fetus is perhaps 1/10th the height, so its blood volume is 1/1000, giving us 12 liters per day. Then again, their heart-rate is twice as fast, so let’s double that to 24 liters per day.

(Actual number: 25 quarts. Basically identical.)

You might know them as “Google interview questions”; those funky estimation problems that you’d never know off the top of your head, but to which you can reason your way.

Whatever you call these exercises, I find them inordinately fun.

Since Fermi questions are so fun and useful, why aren’t they more widely taught? Why isn’t every middle school student doing one of these per week all year long?

I suspect a prosaic reason: they’re hard to grade fairly. Math education is accustomed to cut-and-dry answers. Fermi work is more like an essay, where there are many plausible answers, and reasoning trumps conclusions.

All the more reason to embrace them, I say!

]]>A daily gap whose dimensions are, say, 5 inches by 7 inches?

I know what you’re missing. It’s the new American Mathematical Society page-a-day calendar, the brainchild of witty and trenchant math writer Evelyn Lamb.

I got my hands on a copy last month and found myself reading January and February like a book: a quirky, brainy, immaculately researched book that you slowly tear apart as you read it. (Note: I do this with more books than I should.)

I asked Evelyn to tell this calendar’s story.

**How did this project begin?**

A friend of mine called me out of the blue and asked whether I knew of a math page-a-day calendar and if I didn’t, did it sound like a good idea?

I hadn’t, and it did!

**Why a calendar, rather than a 366-page book of mathematical goodies?**

I’ve thought about writing books. One possibility would be a compilation of posts from my blog Roots of Unity. (Publishers, talk to me if that’s something you’re interested in!)

But I’ve not quite found the right way to organize something that eclectic. With the calendar, I don’t really mind that there’s no big narrative arc. I just want to give people something interesting related to math every day!

**Eclectic it is! We get puzzles, art, Sudoku variants, jokes, recipes, history, music, conjectures, coloring pages, quotations, poetry, theorems… did I miss anything?**

There are hands-on activity pages too, that involve cutting or folding the page. Also, there are quite a few pages about different number systems, like the base twenty system developed by Inupiat students in Alaska to make it easier to do arithmetic using their native language.

**It’s deliberately reusable, with no reference to the year (e.g., no days of the week). Did that increase the level of difficulty?**

AMS books don’t usually have expiration dates; if they misjudged the size of the print run, they’d be stuck with a lot of product they couldn’t sell. Unfortunately, that did make it tricky to include things like Mother’s/Father’s Days, Hanukkah, Easter, and other holidays that don’t fall on the same date every year. But they were committed to the idea, so I grudgingly went with it.

In the end I’m glad we did. The calendar was initially intended for 2019. (Everything takes longer than you think it will!) So I would have had a lot of work to redo when it took an extra year to finish. Also, now that I know how much work this calendar was, I’d like for it to be out there for at least a few years.

**How did you handle holidays that change dates from year to year? And what about Leap Day?**

For some of the holidays that move around from year to year, I put the relevant page on the first possible day. (E.g., there’s a vaguely turkey-looking coloring page on November 22, which is the earliest possible date of U.S. Thanksgiving.) For some, I put it on the day the holiday would occur in 2020. And for some, I just put it vaguely near the right day.

And don’t worry! The February 29 page is actually a flowchart so you can determine whether the February 29 page is needed.

**I’m impressed at the depth of research.**

I joked a bit with my spouse about this being a very inefficient project in some ways. I read two academic books about khipus to make one 5×7-inch calendar page!

**How did you figure out the right date for each morsel? Conspiracy wall full of newsprint and red string?**

It was a bit of a conspiracy wall!

I had a file on my computer called “calendar date thing” (I am great at file names!) where I kept track of where I had coloring pages, different types of puzzles, birthdays, Pi Days (there’s one per month), art, and so on. I wanted to avoid clumping things too much.

There ended up being a lot of shuffling because I’d write a page, put it on some arbitrary date, and then find someone whose birthday was that day, so I’d have to shift things around. I definitely made it harder than necessary, but it was also a fun puzzle.

**How did you decide which people to celebrate?**

I made a lot of rules for myself. There are six biography pages a month, and they are all about dead people (because I thought it would be weird for a living person to read a page about themselves).

For most, the pages are on their birthdays. I thought putting them on an anniversary of their death would be too depressing (though I did make Hausdorff’s depressing; he committed suicide rather than be taken to a concentration camp, and we should be sad about that).

Half of the biography pages are about men, and half are about women. I did my best to keep this balance each month as well. (I might have a couple months with four and two, because people rudely chose not to be born on days that are equally distributed throughout the year!) I worked at having a lot of racial/ethnic diversity, though I didn’t set myself any explicit number goals for that.

I included some mathematicians who are really well known (hi, Gauss!), but I also wanted to include people I had never heard of before, or people who might not be thought of as mathematicians but had some math connections in their lives, like W. E. B. Du Bois, James Garfield, and Anna Julia Cooper.

I thought about including some people who really sucked (and writing about how much they sucked) like Oswald Teichmüller, an actual Nazi, but decided I wanted to focus on more admirable people, like Lee Lorch, who got fired from multiple jobs for his civil rights activism, or Thyrsa Frazier Svager: she and her husband saved about half of their income to create scholarships for black women math majors.

**Anything I didn’t ask about?**

It’s so funny you picked February 6th to share above, because when I first flipped through the final calendar, I looked at that page—which, like all the pages, I had read through maybe a dozen times already—and thought, “This page makes no sense!” This is more a converse to Poincaré’s quote: different names for the same thing. But hopefully people will just think I’m trolling them. (In general, please assume I’m playing an elaborate prank if and when you see errors in this calendar.)

Anyway, thanks for letting me ramble on about my baby! I’m very proud of this project.

For anyone who’s bought one or gets one as a gift, I’ll be at the Joint Math Meetings in Denver in January, and I’d be happy to sign your birthday page or other important date.

And I’d love to chat with you about the calendar as you’re enjoying it next year. You can find me on Twitter: @evelynjlamb.

*You can buy the calendar through the American Mathematical Society.*

You know the guy.

Anyway, he is master of many things, among them the peculiar genre of the “Wanna Feel Old?” joke. Consider this exemplary specimen:

I remember reading this one – I was 24 at the time – and getting blown away. It totally worked on me. It felt so customized, so personal.

But I guess I’m an easy mark. Munroe even jokes about the ubiquity of these jokes:

How do these temporal assaults work?

First, it helps to riff on pop culture. When a new movie comes out, it gets tagged in our minds as “new.” Years later, you can sometimes catch the brain having forgotten to update the tags.

More to the point, we don’t experience time linearly. A year is not a year – at least, not in the janky clockwork of the human mind.

The older we get, the faster time seems to go.

It’s almost as if we consider each year as a *percentage* of our life. For example, the next year will add 50% to my nephew’s lifespan; it will add about 3% to mine. You can guess who will feel time is passing faster.

(By the way, if you want a gorgeous interactive capturing this phenomenon, you’re in luck. Seriously: click it.)

Knowing this mechanism, I find, doesn’t blunt the impact of jokes like Munroe’s. It still gives an electric shock to notice time racing by.

Here is my own humble contribution to the genre:

Huh. Didn’t quite nail it.

Let’s try again:

Okay, maybe this art form is best left to the master.

]]>It is December on the beaches. It is December in the streets. It is very specifically and intensely December outside my bedroom window, where 3-foot death icicles dangle, ready to fall and impale me as I step out into the frigid December of it all.

(Yes, we’re loving Minnesota, thanks for asking!)

Where can we find warmth in this frigid time? Yes, yes, the Southern hemisphere, but more usefully: *in the life-giving geometry problems of Catriona Shearer, Puzzle Magician.*

Here are three of her favorites from November. Come, friend; heat yourself by this fire.

Why do I dub this one “stained glass”? Because it has the delicate and symmetric beauty of a stained glass window, obviously.

But why “sci-fi”? That takes more explanation.

On Twitter, acclaimed sci-fi author Greg Egan chimed in with a crucial point: as stated, the problem is under-defined, allowing for an infinite family of solutions. Thus, Catriona added another condition: *the whole diagram has bilateral symmetry*.

To see why this is necessary, check out the nifty animation that Greg created. I, for one, am eager to watch the sci-fi film where all stained glass windows rotate like this.

(Note to my fellow U.S. readers: please forgive the Britishism “trapezium.” Actually, scratch that: don’t forgive it, embrace it! Trapezium is an adorable word.)

Catriona explained the genesis of this one:

Alison Kiddle, who coordinates my local MathsJam, asked me a question at last month’s meeting: could I draw a trapezium so that when I drew in the diagonals the four areas each had a different area? Could I draw one where 2 areas were equal, or 3, or all 4?

I was pretty confident that I could, but [spoiler] after a fair bit of scribbling and failing to convince Alison (who is very good at asking questions!) I realised I shouldn’t have been quite so hasty.

Naturally I’d brought my felt tips to the pub, so by the time I left I’d made this puzzle from my discoveries.

I found this one the trickiest of the three. Indeed, Catriona says that these three are “in ascending order of difficulty (in my mind, at least).” She also explains, tantalizingly:

Unless I missed it, I don’t think anyone posted the solution I originally came up with. It’s unusual for me to feel like my method is unique!

As always, feel free to post your solutions in the comments (which means you should watch out for spoilers if you don’t want them)!

]]>I say “faceless government headquarters,” you say…

Yes, that’s right: surprisingly strong TripAdvisor reviews!

No joke. The Dallas branch of the Federal Reserve is apparently a real tourist-pleaser. So when I had a free morning in Dallas this February, I gave it a shot. A daunting security system, like the kind of airlock you’d need on an intergenerational space voyage, led into a mostly-empty lobby. Only two other tourists joined me.

And let me tell you, it was the other 7+ billion people on earth’s loss. The building housed a nifty museum on the Fed’s history, as well as the general topic of macroeconomics (which I find about as intuitive and graspable as general relativity).

In particular, I loved this visualization of inflation. Rotate the wheel, and you change the decade, revealing how prices rose (or, in some cases, stagnated):

Inflation is strange enough in itself. But the uneven way that prices rise – coffee and clothing and eggs each following their own unique trajectory – is even stranger. Just one of many “real-world” subtleties that we elide in our teaching.

When I mentioned this exhibit to my father, he pointed me towards an impressive project by a colleague of his: The Billion Prices Project, which creates a day-by-day inflation tracker via crowd-sourcing. A billion prices, it turns out, is about the level of nuance you want!

This strikes me as the raw material for a killer math project. Some brainstorms:

Pick 5 goods. Graph their changing prices over time. Discuss what you notice and what you wonder.**Mild**:: Pick 5 pairs of goods. Graph their changing prices over time, as well as the change in their ratio. Discuss the significance.*Medium*: Design your own inflation metric by picking a basket of goods. Compare your metric to the consumer price index over time. Discuss their relative merits.*Spicy*

Other ideas?

]]>I see why they do it. Working with equations is pretty straightforward: do the same thing to both sides, and they’ll remain equal.

Not so with inequalities. Sure, **5 > -4**, but multiply both sides by -1, and suddenly you’ve got the claim that **-5 > 4**. If you believe that, then I’ve got a bridge I’d like to sell you for $4. (And then buy back for -$5. And then sell to you again.)

The point, as any algebra teacher can tell you: Inequalities reverse when you multiply (or divide) by a negative.

One way around this is the **student sidestep**, like so:

17 – x > 18 – 3x

17 – x = 18 – 3x

17 = 18 – 2x

-1 = -2x

0.5 = x

So far, so good. But now we’ve got to turn our result back into an inequality. We suspect the real answer is either **x > 0.5 **or **x < 0.5**, but we’ve got to check the possibilities back in the original inequality. It’s almost like having to solve the problem twice.

Instead, I prefer a different sidestep: the **ads and subs only** sidestep. Wherever possible, I try to avoid multiplication and division by negatives, like so:

17 – x > 18 – 3x

17 > 18 – 2x

-1 > -2x

(*don’t divide by -2; instead, add 2x to both sides…*)

2x – 1 > 0

(*then, add 1 to both sides…*)

2x > 1

(*now, it’s safe to divide by 2*)

x > 0.5

It can feel a bit like working with one hand tied behind your back. And it doesn’t cover all cases. Still, I strongly prefer it to the turn-it-into-an-equation sidestep, which feels to me like a betrayal of the whole spirit of inequalities!

Inequalities are a special resource to the mathematician, richer and more complex than those garden-variety creatures we call equations.

Take the inequality x > 4. It opens up conversations about the boundary line between solutions and non-solutions. In particular, 4 doesn’t work, but 4.00000000001 does. What’s the smallest solution? Does a “smallest” even exist?

By contrast, the equation x = 4 is a flavorless, gray nothingburger. It says that x is four, and there’s nothing more to add.

Perhaps that’s what Stephen Hawking had in mind when he gave this famous quip:

]]>