Anyway, Celeste Ng is also the parent of a future math teacher.

I, for one, warmly welcome our 9-year-old hero to the time-honored profession of foisting equations upon the reluctant.

As a kid, I made similar efforts to stump my dad. Mine were mile-long arithmetic problems: “13406824360 times 78645103465” and the like. Such multiplication is certainly *hard.* But it’s also tedious and shallow (as my dad’s taxed patience can attest).

These equations, in contrast, run deep. Deeper than their inventor could have known.

That’s the way it is with math. When you put a shovel in the dirt, sometimes you get a shovelful of dirt; but sometimes, a 200-foot geyser springs forth. Beneath the innocuous patch of surface you’re exploring, there may lurk terrific geological forces.

This is one of those times. (Beware: spoilers for the problem follow.)

Our system involves four variables: **f**, **a**, **c**, and **e **(which pleasingly spell “face”). Before we solve algebraically, it’s worth investigating the matter geometrically.

Here’s the relationship betweenÂ **a** andÂ **e**:

BetweenÂ **a** andÂ **f**:

BetweenÂ **f** andÂ **c**:

And betweenÂ **a** andÂ **cÂ **(which I derive from the final equation by replacing **e** with **11 – a**, an equivalence established by the first equation):

Since each equation relates just two variables, it can be visualized in two dimensions. The results are two lines, a parabola, and a hyperbola.

But what if we want to combine them all? With four variables in play, we need four dimensions. I don’t want to speak for you, but when my brain tries to visualize four dimensions, it mostly achieves something like this:

In context, each 2-variable relationship becomes a 3-dimensional solid, living in an ambient 4-dimensional space. The solution we seek is a point that belongs to all of the solids simultaneously.

Pretty heavy dinner-table conversation for a 9-year-old.

Thwarted by geometry, we turn to more algebraic methods. What do you do with 4 equations involving 4 unknowns?

Turn it into 3 equations involving 3 unknowns.

(In this case, I used the substitution mentioned above, then factored the left-hand-side.)

And what do you do from there? Why eliminate another variable, of course, boiling it down to 2 equations with 2 unknowns.

(In this case, I added the first and second equations from before.)

And from there, you can guess our next move: dessert.

Then, with our sweet tooth sated: boil our pair of equations down to a single one, in a single unknown.

(In this case, I solved the second equation for c, then substituted into the first equation.)

The result is a cubic. Potentially a nasty piece of work. Consult this table:

In this case, fortunately, our 9-year-old teacher has thrown us a bone. One of the solutions to this cubic can be guessed without too much trouble: **a = 4**.

(Ben Dickman, my arch-rival in being a mathy Ben on the internet, has a nice explanation of why, if you’re curious and also a traitor who is forever dead to me.)

Knowing this solution, we can factor the cubic into a linear and a quadratic:

Then, pick your favorite quadratic-solving method, and voila:

All of these solutions check out, giving us a total of three solutions to the original system of equations:

Back to the geometry: apparently those 3-dimensional surfaces, crisscrossing in 4-dimensional space, meet at exactly three points. To higher-dimensional aliens, this would perhaps be as obvious as noting that two lines cross at a single point. Or that a line and a parabola at a pair of points. It’d be elementary geometry, almost too trivial to speak of. But despite my siblings’ insistence, I am not an alien, and cannot see such things.

(Though if we retreat back to the step where we had two equations in two unknowns, Desmos can show us why three solutions emerge: the hyperbola and parabola intersect in three places.)

This is, if I may reiterate, some hefty dinner-table chitchat.Â Even my wife and I (together comprising one math professor, one author of math books, and two incurable nerds) don’t delve this deep on your typical weeknight. Hats off to the Ng household.

Anyway, this all got me thinking about the vast pointlessness of algebra.

I know I’m not supposed to say algebra is pointless. But I can’t help it. The other day, as we moved into factoring quadratics, my Algebra I students raised the timeless chorus: “Why do we have to learn this?”

I had no good reply. I never do.

It’s not for lack of thought. I’ve spent years meditating on this very question, which is exactly why I find the standard answers so empty and unsatisfying.

If algebra class is to be of daily use to the citizenry, why not focus on probability, statistical literacy, and personal finance?

If algebra class to impart valuable professional skills, why not focus on spreadsheets (plus maybe a little Python)?

If algebra class is to teach us “how to think” (which was once my preferred answer), then what do we make of research revealing that lessons in reasoning don’t generalize? That algebra teaches us to think, not about life or logic as a whole, but about equations only?

Finally, if algebra class is simply a platform upon which to compete for college admissions… then please excuse me while I scream into the void.

So. Why *do* I teach this stuff?

Here’s one reason. It’s an idiosyncratic reason, inspired by the exploits of the Ng household, offered with the caveat that your mileage may vary. It goes like this.

Arithmetic raises questions that it cannot answer.

Arithmetic poses riddles that it cannot solve.

Arithmetic uncovers patterns that it cannot explain.

But algebra can.

Arithmetic is the solid ground, the numerical surface of things. You figure it must be dirt all the way down. But if you stick your shovel in the right spot, as the Ng family did, you’ll be greeted with a geyser.

How can you not love those eruptions? How can you not want to know their source? How can you resist the urge to probe the depths?

So, three cheers for algebra, and for all the 9-year-olds at all the dinner tables tossing off questions so profound that, even on a well-caffeinated day, a half an answer is the very best I can muster.

*A NOTE ON THE TITLE OF THIS POST: Apparently the TV show is spelled “Kids Say the Darndest Things,” with only one “e” in “Darndest.” This is an abomination and I will have no part of it.*

EDIT: A cool animation from Adam in the comments, showing what the graphical approach looks like with three equations.

]]>

The question: **What’s this equivalent to, for a typical person like you or me?**

As inevitably happens during brief radio interviews, I got a bit fumbly and mumbly, so to do greater justice to their excellent question, here are three different answers:

1. **As a percentage of wealth:**

Bloomberg spent a little under 1% of his wealth on the campaign. For a typical U.S. household, that amounts to perhaps $800. In this sense, Bloomberg’s campaign was the equivalent of, as Peter put it, buying a new laptop.

2. **As a sustainable daily expenditure:**

Bloomberg spent about $5 million per day on ads. If he cashed out his wealth, he could continue to do this, day after day, for another 35 years.

The proportional daily expenditure, for a typical U.S. household, would be $8 per day. In this sense, Bloomberg’s campaign was the equivalent of a daily sandwich.

3. **As an amount of labor:**

If I spend $800, I need to go earn $800 more, sooner or later, in order to keep financially afloat. That’s what $800 means to me, really: it’s the work I’ll need to do to earn it back.

The same isn’t true of Michael Bloomberg. On an ordinary day, he gains a passive income greater than my lifetime earnings will likely be. (Indeed, he ended his presidential campaign several billion dollars richer than he began it.)

Thus, Bloomberg does not need to go earn $500 million. The $500 million come to him. In this sense, the most closely analogous price in your or my experience is “free.”

*You can hear the interview at Here & Now. Thanks again to Peter and the team for having me on!*

- A mathematical topic arises.
- “You know,” I say, “someone has a great tweet about this… somewhere…”
- In order to find it, I am forced to read all of the tweets, ever.
- I am reminded that “all of the tweets ever” is rather too many tweets.

So about a year ago I started a compendium. Tweets, yes, but also videos, apps, memes… anything stimulating or arresting that I can use to embroider my lessons. For a while, this document lived where all important documents live: as a gmail draft. But now I share it as a blog post, and I intend to continue updating it as new ones cross my ken.

NOTE: I will, where convenient, use screenshots and links, because WordPress’s embedded tweets sometimes take ages to load.

A very strange pricing scheme:

A brilliant anagram from Colin Beveridge:

A gorgeous visualization of prime factors (from this Smithsonian blog post).

The timeless classic Powers of Ten, arguably the best film of 1977 (suck it, Annie Hall):

The mesmerizing interactive “Scale of the Universe” app (which requires you to enable Flash, but just do it).

Also, this black hole:

Four-story slides shaped like parabolas:

An ellipse as the maximum heights of a family of projectiles:

Throwing an object at the same speed but different angles defines an ellipse via its maximum height https://t.co/vQ8NMssCMf

—

〈 Berger | Dillon 〉 (@InertialObservr) July 22, 2019

And again, this time for figuring out the scoring system in Australian Rules Football:

Four place mats, arranged to make a quadratic identity at the dinner table:

Polar coordinates on pizza:

Putting sauce on a pizza. https://t.co/Oe9gsZaSjz

—

Machine Pix (@MachinePix) August 28, 2016

Volumes of a cylinder, a sphere, and a cone:

Volumes of earth, earth’s air, and earth’s water:

Animated visual proof that any polygon can be rearranged into any other polygon of equal area:

(You’ve just got to click here, it’s amazing.)

For your trigonometric Halloween, the blood function:

Defining a radian with a wooden model:

Tragic Tweet Delete! -- I thought I would at least add it back : ) We are interested in sending these to folks, es… twitter.com/i/web/status/1âŚ

—

MathHappens (@MathHappensOrg) October 01, 2019

Simple harmonic motion:

Beautiful shapes created by simple harmonic motion 🧐 https://t.co/ifsFX4nfN9

—

Fermat's Library (@fermatslibrary) January 02, 2020

Riemann sums (comparing upper and lower sums as the grid is refined):

Concepts without words: Integration and Riemann Sums
bit.ly/2E7iNU3
#math #science #iteachmath #mtbos… twitter.com/i/web/status/1âŚ

—

Tungsteno (@74WTungsteno) August 11, 2019

A professor solves an optimization problem (“smallest surface area for a given volume”), writes a company that makes cat food to ask why they don’t use this solution, and receives an incredibly thoughtful and interesting reply:

A real-life butterfly effect:

In office hours, sophomore @JackSillin showed me this real world example of the butterfly effect. An unexpectedly… twitter.com/i/web/status/1âŚ

—

Steven Strogatz (@stevenstrogatz) September 10, 2019

The exquisite sensitivity of the double pendulum:

50 double pendulums, whose initial velocities differ only by 1 part in 1000 https://t.co/3b75BDkwF1

—

〈 Berger | Dillon 〉 (@InertialObservr) September 30, 2019

Independence is a delicate and rare phenomenon:

What do probabilistic words really mean?

I see you Anscombe’s Quartet, and I raise you the Datasaurus:

A delightful game aptly called Guess the Correlation:

The dangers of using r^2 as an effect size estimate:

The normal distribution in action:

Moore’s Law, and the glorious improbability of that exponential growth:

Fascinating: Moore’s Law predictions vs actual growth in transistor count.
by @datagrapha
reddit.com/r/dataisbeautiâŚ https://t.co/ZwN1dBGE1n

—

Lionel Page (@page_eco) September 03, 2019

Quick sort, in an image:

Centrifugal force to restore a whiteboard marker:

I had no idea you could do this to take a dead whiteboard marker and give it life again!
Source:… twitter.com/i/web/status/1âŚ

—

Robert Kaplinsky (@robertkaplinsky) July 06, 2019

Voronoi diagrams (i.e., which national park is closest to you?):

Set theory (specifically, the power set), where each rectangle is one of the possible sets of these 4 elements (ranging from the empty set in the middle, to the set of all four):

Mathematics in nature:

Stumbled into reading about hermit crabs and wut: https://t.co/ERdEs8TwUK

—

Derrick (@geekandahalf) November 29, 2019

Most are questions I am wholly ill-qualified to answer.

Take this doozy from high school student Charlie in Florida:Â **What makes a great teacher and what makes a bad teacher in your eyes?**

I fumbled over how to answer… then realized that I don’t have to. I can ask some great teachers to answer for me! From four hand-picked heroes of mine, I got a variety of insightful replies, to which I have added my feeble illustrations.

by Fawn Nguyen

School is inherently stressful for many. Itâs mandatory. Itâs an institution of compliance: take this class, follow this schedule, sit here, do this, and more often than not, shut up.

A great teacher makes school less stressful.

She understands that her students would rather be elsewhere, but they are here now and sheâll make the best of it for them. She will not waste her studentsâ time. She is pleased and excited that everyone is present because she has crafted the best lesson and together they will explore.

A great teacher does not take his class too seriously, he laughs with them, makes light of his vulnerability. He yearns to hear each studentâs voice and works on inviting that voice to come out in whatever way it needs to.

A great teacher loses her mind because sheâs so happy that one of her students has asked a wonderful question that leads the discussion down some glorious path that she had not intended.

A bad teacher does not care. About anything other than putting in the hours.

Iâm reluctant to use the phrase âBad Teacher.â Faced with hundreds of interactions and decisions every day, we all have good and bad moments. Those moments accumulate over a semester, a year, a career, and in most cases yield a net positive result Iâd say.

But you can tell a lot about a teacher by how they respond when students don’t succeed. Some will say, “What’s wrong with you?” Others will ask, “What’s wrong with me?”

A wise man once told me that a great teachers know three things: they know their subject, they know how to explain their subject, and they know their students.

Subject knowledge is so often misunderstood by teachers. âI have degree in mathematics, of course I can teach itâ. But itâs so much more than that. Itâs knowing how to explain things, in multiple ways, in ways that make sense. Itâs about making complex ideas crystal clear. Itâs bringing topics alive, and that can only be done by teachers with a deep understanding of what theyâre teaching.Â Knowing the history of a topic, the interesting problems, the common misconceptions, the multiple approaches, the links to other topics. And itâs about expert communication of all of that.

Thatâs not to say a great teacher needs to be an entertainer – an expert communicator does so with clarity, not with gimmicks. But passion goes a long way. A great teacher loves their subject, and canât wait to tell their students about it.

No one sets out to be bad at teaching. But there are those that enter the field for reasons that don’t serve them (or anyone).

I believe that these teachers do not have a well-articulated “why.” Why are they teachers? Why their particular school? Why their grade band? They also don’t know the whys of their pedagogical moves, instructional questions, or mindset.

Since they are not adept at self-analysis, they also can not appreciate this in their students. They are unable to understand students (or colleagues) who need to know why. They see their incessant questions as unnecessary interrogation and even insubordination.

And when they grow up to become administrators…

Now, what about greatness? I see it as on par with wisdom. An educator can approach greatness only after much time (perhaps many years) honing their craft. Even then, it is never really achieved, but approached asymptotically.

Great educators have the humility to know that they do not know everything, and are forever seeking to know.

They seek this knowledge from unpopular sources.

They take risks daily, because what other way is there to live?

They know that the world is inequitable, and they seek their own place in creating lasting change.

They co-create educational experiences with their students because they realize that students hold key knowledge of their own that most don’t acknowledge or even recognize.

They do not expect perfection from their students or from themselves, but have unwavering faith in collective knowledge-making.

They hold each other accountable.

They are quick to apologize, quick to praise, yet do not hesitate to correct where needed.

They have a way of being that students respect.

They do not see education as subject-specific. They recognize the interconnectedness of disciplines and do their best to model this for students.

They are honest with themselves and with students.

They seek to be antiracist in belief and action; they value being racially fluent.

They understand that race is a topic that is hard to discuss openly, and take opportunities to do it well.

They understand that ultimately, the answers to the questions that plague us are found through communication across race.

They are courageous not because they don’t know fear, but because they don’t allow fear to keep them from questioning existing structures, challenging the opinions of their colleagues and administrators, or investigating their own participation and complicity in a system they find subpar.

Yet, they do more than just reflect. They push for change, within and without their classroom.

]]>Contrary to popular opinion, U.S. mathematics education has for decades achieved near-universal success in its goals. Virtually all citizens acquire a set of “core” mathematical competencies, which they use to great effect throughout their lives.

All that remains is to articulate, for the public, precisely what those goals and competencies are…

Proposal for the Adoption of the

Truly Common Core Standards, p. 37

**Students will learn to deflect any and all mathematical conversations with self-effacing phrases like âIâm not really a math person” and “Ha, numbers are the worst, right?”****Students will groan amiably when asked to calculate a tip.****Students will internalize a deep and nameless sense of intellectual inferiority.***Â***Students will consider multiple representations of mathematical ideas, and find them all equally baffling.****Students will revere mathematics as a kind of magic. Boring, irrelevant magic.****Students will blithely defer to dubious statistics – except for any statistics that challenge their preconceptions, which they will reject out of hand.****When faced with an unfamiliar problem, students will cross their fingers and combine the numbers at random.****Students will look up one day, decades into adulthood, and realize with a sense of scandal and outrage that they never actually needed any algebra.**

They are arranged in roughly ascending order of challenge, but can be consumed in any order (unlike most four-course meals).

People kept asking Catriona if this one was a “trick question.” It’s not; just a tasty little hors d’oeuvre, which you may wish to nibble at for a while, or to eat in a single bite. (See the original here.)

I love this construction. Nothing but midpoints, connected with the elegance and economy of a calligrapher. “I’m sure this is a classic that’s been done many times,” says Catriona, “but it was new to me.”

(See the original here, including a beautiful animated solution, as well as a lovely solution by dissection.)

Catriona describes this as a riff on her very first puzzle (which I dubbed Garden of Clocks). I’ll take this moment to make the vapid point that, in addition to impeccable geometric taste, Catriona always chooses the perfect colors. (See the original here.)

Some of Catriona’s puzzles yield to a single geometric insight. This one offers a different kind of pleasure. Catriona explains: “I doubt many people could do this one in their head – I couldn’t – but it makes the highlight reel on account of an unreasonably nice answer.” (See the original here.)

]]>All paradoxes are basically the same. You’ve got the barber who cannot shave himself, the set which cannot contain itself, the sentence that cannot describe itself… and, in this case, the glass-breaker that cannot break its own glass.

**Deadly Notational Sins**

The same paradox returns as the punchline here. What’s the sixth way? Is the infuriating failure to list a sixth way, in itself, the sixth way? And if so, doesn’t that mean there reallyÂ *are* six ways shown… and thus, no infuriating failure… and thus, no sixth way?

(Hint: this paradox is the sixth way.)

Several commenters informed me that they didn’t realize e-theism was an option, and now intend to embrace it.

This cartoon was inspired by my readings of C.S. Lewis and G.K. Chesterton, both of whom seem to suggest the best explanation for the trinity is that you can’t understand it, and that’s okay, because divinity. And who am I to argue with double-initialed writers?

**And After All….
You’re My Variable…**

This is true of much pop music, but Wonderwall is a singularly impressive specimen. EveryoneÂ *thinks* that Wonderwall has some kind of story lurking behind it, but then you look at the lyrics, and it’s like… huh?

(“Wonderwall,” by the way, was the name of George Harrison’s Record label. In other words, it’s a nonsensical placeholder Beatles reference. Just like a variable!)

Only semi-relevant, but my friend Adam persuaded me a few years back to use the “wow” reaction on friends’ posts much more often, his arguments being: (a) It works for most news, whether happy and exciting or angering and frustrating, and (b) Nobody else uses it, so whereas your “like” or “love” will go unnoticed among the hordes, your “wow” will stand out, and your friend will know you care.

Anyway, if too many people take up his advice, it ruins the equilibrium, so you didn’t hear it from me.

**Flat Earth Society**

I try to avoid puns.

Which makes it suspicious that I write so many of them.

But trust me: for every mediocre pun I publish, there are twenty even worse ones that languish on my hard drive.

**Two Points Make a Line**

The actual quote (one of my dad’s absolute faves) is: “Make everything as simple as possible, but no simpler.”

It’s a glorious example of following one’s own advice: The modified version “Make everything as simple as possible” would be simpler, but Einstein’s original is the simplest oneÂ *possible* while still being wise counsel.

**My Kind of Xenophobia**

I feel this way about my brother-in-law Farid. He moved here speaking what little English he’d picked up from Dave Chappelle routines and Lakers broadcasts… now, less than a decade later, he’s added English to his existing fluencies (Algerian, classical Arabic, and French, plus some Berber) and reads harder books than I do.

Screw that guy, right?

**The Fruits of a Life Writing for the Internet**

“Impressions” is a word for “number of people who saw your post.” They are a dubious statistic which social networks wave before their users’ faces, a bit like a hypnotist’s swaying watch, to lull them into a dreamy sense of false accomplishment.

I know it’s not a good joke if you have to explain it, but: “nonlinear” has nine letters, and also, it’s effectively a curse in mathematics; hence, a “four-letter word.”

See? Now it’s a great joke, right?

**The Mysteries of the Circle**

This comes up every time I teach about polygons. And not because I’m bringing it up! Students love the leap towards infinity.

**Arithmetic vs. Algebra**

Came up with this one while running a workshop for teachers on how to make math memes. One participant had a far better version: the “Before Algebra” panel showed a student solving a problem by a roundabout guess-and-check method, and the “After Algebra” panel showed the student still solving it exactly the same way.

**The Self-Curving Exam**

It’s silly how often we judge a test’s efficacy by the distribution of scores, rather than by whether it actually assessed students’ mastery.

And by “silly,” I mean “horrifying.”

Several commenters raised the question of whether there are separate coin flips for each student, or a communal set of coin flips for the class. The former will pretty much guarantee a nice binomial distribution; the latter may create a lumpier distribution, if the students’ guesses are highly correlated.

The again, one set of coin flips will make for faster grading. And given a choice between speed and efficacy, we know which one math education tends to pick…

This cartoon is a work of fiction, obviously.

In reality, both kids and adults use exponential to mean “really fast.”

**9 Stories, 9! Readings**

J.D. Salinger’s books, ranked by desirability of reading them in every permutation:

*Franny and Zooey*: 2 stories, hence 2 permutations; well worth it*Nine Stories*: 9 stories, hence 362,800 permutations; maybe a tough slog*Catcher in the Rye*: 1 story, hence 1 permutation; eh, take it or leave it

**The Curse of the Three-Day Weekend**

What’s more fun than millennial burnout, right kids?!

For more thoughts, see my post on this urgent question.

**Lesser-Known Kinds of Circles**

Not depicted: the squircle.

**The Tragic End of a Proof By Contradiction**

Drew this one for a Jim Propp essay over at Mathematical Enchantments. If you ever wanted a more intellectually serious but still playfully accessible version of Math with Bad Drawings, check it out!

This one went viral on Twitter; see further discussion here.

This one has been spotted on a few office doors in math departments. Y’all are very brave and crazy and I wish you the best!

]]>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 usÂ *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.Â **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 âHold That Line,â but I call itâŚ

**Â **

**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 (thatâs 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:Â Â Â Â Â Â Â Â Â

- 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.

Itâs a breezy yet strategic game. If youâre nimble, you may even discover a guaranteed winning strategy. Unfortunately, this rather undermines the fun.

(No spoilers here, but Iâll 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 Sacksonâs original, youâre ready for my preferred variant: Snakes!

(Aliases include âSnake Breederâ and âSnakes in the Coordinate Plane.â)

**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 itâs a tie, then look at each playerâs 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. Thatâs triumph itself.)

The multi-snake variant also spawns a solitaire versionâor, 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 âsnake numbersâ 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?