Seeds and Stalks

The grails of math activities, for me, are those that involve almost no special knowledge to get into, but have near-infinite depth. (Like this one.) We sometimes describe them as having a short barrier to entry, and no ceiling. (A common suggestion when we work with teachers is to “remove the ceiling,” that is, find ways to change the problem so that the learning doesn’t end when you get the answer…)

Here’s a spanking new puzzle I’ve been playing with, and it feels like a perfect example of a problem with virtually no barrier to entry, and no ceiling either. I call it Seeds and Stalks.

Here’s how it works. We’ll generate sequences (in a kind of Fibonacci-like way) by choosing a number and adding it’s digits to itself to get the next number. For example:

16 goes to 16 + 1 + 6 = 23.
23 goes to 23 + 2 + 3 = 28.
28 goes to 28 + 2 + 8 = 38. And so on.

So we have a sequence that goes 16, 23, 28, 38, … I call this Seeds and Stalks because there are two pieces here… the seed that starts the sequence (the seed) and the sequence that grows out of it (the stalk).

Of course, there might be a seed that leads to 16 in the stalk. And indeed, 8 leads to 16. The most primal seed we could pick for this stalk is 1, since having 1 as the seed leads to the stalk:

1, 2, 4, 8, 16, 23, 28, 38, …

All well and good. But as soon as I thought of the mechanism, I was besieged by questions. The first was:

What’s the smallest collection of seeds that you need to include every number in a stalk?

I can see I’ll need 3 as a seed, since 3 isn’t in the stalk.

3, 6, 12, 15, 21, 24, …

Now 5 isn’t in either stalk, so I’ll need that too. How many seeds do I need to get every number? Or will I need infinitely many seeds?

For me, this is a perfect storm. All I need to start this problem is addition. And yet, I have no idea what will happen. I can feel that there are all kinds of patterns to find. My instinct now is to turn it over to students and see what they can find.

But I’ll look to the internet first.

What questions can we ask about Seeds and Stalks?

What answers can we find?

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Quadrilateral Puzzle

I’ve been immersed in puzzle and lesson creation lately, and I thought I should take advantage and throw some of them out here on the blog. Please take, solve, use in your classrooms or at home, and let me know what you think. If people like the puzzles, I’ll make a point of putting them out here more often.

A Quadrilateral Question for today. This sub-questions goes from easier to harder.

The Big Question: Start with any quadrilateral (Quad 1), label its midpoints, and connect them to form another quadrilateral (Quad 2). When will Quad 2 take up exactly half the area of Quad 1?

Will it happen if Quad 1 is…

1. a square?

2. a rectangle?

3. a parallelogram?

4. a trapezoid?

5. a kite?

6. Nonconvex?

7. Can you find an example when Quad 1′s area isn’t double Quad 2′s? Or will it happen all the time?

You can post in the comments if you’ve got an argument to share…

(Another question is to show that Quad 2 is always a parallelogram. Here’s my proof of that if you get frustrated.)

 

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Twin Prime Hero

I just read this wonderful interview with Tom Zhang, who made recent, important progress on the Twin Prime conjecture.

It’s a strange, quiet interview, and a lovely departure from the world of the fame-obsessed. Another thing I like: he emphasizes the love and the persistence. Here’s how the interview ends:

What would you say to a young student who wants to solve a problem?

Keep going. Do not easily give up.

Where do you suggest they find the motivation?

The most important motivation is to really love mathematics.

Is this a person who would be a hero to young people? I don’t know. Should he be? I think I can say with confidence that he doesn’t need the adulation or admiration of anyone. Isn’t that something we should look for in a role model?

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A spoonful of transgression

I was just observing a third grade class learning/reviewing basic fraction to decimal conversion, and I overheard a great remark. A girl, reading a word problem, said to her table mate, “Jessica ate 6/10 of a cake?! She’s fat.”

There’s a part of me that hates comments like that, and a part that loves them.

I hate the comment because, you know: here’s more evidence of our appearance-obsessed culture getting into the heads of young girls, etc., etc. But I love it because this girl just showed that her relationship with this fraction goes beyond shading in the appropriate portion of the drawing. Six tenths means something to her. Maybe I’m not fully happy with what it means, but at least it’s not meaningless.

My first thought is, why don’t we have more ridiculous math story problems? People eating horrific quantities of food is funny. And what’s funny and horrible has a way of sticking in the mind. Why does Tom always eat 3/8 of a pizza?Why not 49/8? Or 149/8? Save us from the blandness of the unoffensive story problem.

(Of course, we don’t want an unsafe environment for kids. But flatlining all the content is clearly a mistake. Better to talk about issues when they come up.)

Or maybe they’re tiny pizzas, and eating 149/8 is absolutely natural, because each one has a diameter of 1 inch.  I now I don’t even now anymore: is that a lot, or a little? (A new game: I say the number of pizzas Tom ate as a fraction, and you tell me the biggest they could be without Tom suffering permanent damage.)

The point is, being able to tell when something is ridiculous or not is part of understanding math. And straying into the ridiculous is fun, and interesting.

An even more transgressive example happened to me when I was demo-ing an algebra lesson last spring in an all-girls eighth grade classroom. (Herbert Kohl, in On Teaching, remarks that those who work with middle schoolers need to have a high tolerance for the profane. I’ll find the exact quote later. Update: the quote is, “A lack of sexual prudery is almost a prerequisite for junior high school teachers.”) The lesson began with the old magic trick (try it if you haven’t seen it before):

think of a number
add 2
multiply by 2
subtract 2
divide by 2
subtract your original  number.

And then I tell you what number you’re left with, which in this case is 1. (Ta-da!) The trick to it, which we got into, is to let x represent your original number, and keep track of the algebra. In one class, though, someone asked what would happen if you could replace the twos by threes, or fours. I set the class to play around with it, and see if they could predict how changing all the twos to another number would affect the final answer. Is there a pattern?

After they’d worked on it and I was bringing them together again, one group showed me how thoroughly they understood by suggesting we try:

think of a number
add 70
multiply by 70
subtract 70
divide by 70
subtract your original  number.

I saw the answer coming halfway through, but had no choice but to complete the process on the board. It was every middle schooler’s giggly favorite, 69. And again, I hated it, and I loved it. Hated, because that was the last place I want to have anything even remotely sexual suggested to me. Loved, because these girls owned that problem, and they showed it by showing they could hit any target number they wanted. They picked the funniest one they knew.

When we’re trying to interest students, we have to respect what interest them. Reliable standbys are matters of power, death & danger, women & men. Transgressions lie perfectly at the intersection of these topics: they are instances of breaking rules with all the hilarity and danger that involves. I want a math classroom that is safe for all the students. But I don’t want one that’s sterile. A spoonful of transgression helps the math stay memorable.

Let me end with a plug for an underused but incredible educational resource that mines the ridiculous: check out Randall Munroe’s What If? Just look how he seamlessly interweaves the ridiculous and real issues of power into a readable math calculation as he investigates how many punch cards it would take to store all of Google’s data. Anyone else see the class project dying to happen here?

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Teaching Perseverance

Reading an Alfie Kohn’s article on what kids learn from failure made me think of the most common question I hear from teachers about the Common Core Practices:

How can I teach perseverance?

It’s an excellent question, and the answer isn’t necessarily obvious. As Kohn points out, experiencing failure and having a teacher prod you to keep trying isn’t—or letting you hang—isn’t necessarily helpful. He write that

studies find that when kids fail, they tend to construct an image of themselves as incompetent and even helpless, which leads to more failure… if an adult declines to step in and help when kids are frustrated, that doesn’t make them more self-sufficient or self-confident: It mostly leaves them feeling less supported, less secure about their own worthiness, and more doubtful about the extent to which the parent or teacher really cares about them.

This illuminates a real pitfall of the unsubtle approach to teaching perseverance. You can’t simply throw students into a situation where they’re likely to fail and let them founder. When I think about being productively stuck, it’s worth remembering that there’s two ways to get out of balance, and both are pretty unproductive. If there’s no challenge, there’s no real learning. If the challenge is—or feels—insurmountable, there’s no real learning either.

And the best medicine for those about to face a particularly tough challenge, and want to stave off the feelings of failure that threaten to derail them? Start with a few easy successes. Seriously. I’ve realized that I do this myself, and when I help kids do it, they can work longer and with more focus.

I start with easy cases. Ridiculously easy. Say I’m trying to figure out the problem of how many squares are on a chessboard.

There are 64 small squares. Is that they end? No, wait, the whole thing is a square. So that makes 65. But then there are middle sized ones. Suddenly, the problem seems insurmountable. Many kids, once they realize they’re facing a cacophony of counting, give up.

What I do is ask myself how I could turn this into a problem that I could have some success with. How easy could I make this problem? And here’s the interesting thing. I’ll ask kids that question (even prompting them, if necessary, that eight by eight is a pretty big situation to start with) and what they usually say is to try on a four by four board instead. Sometimes they’ll say to start on a three by three or a two by two.

Do you know where I start? Zero by zero.

It’s shameless, I know, but why should I feel shame? I’m just thinking, playing, messing around, and giving myself the gift of instant success.

Because I can see immediately that there are zero squares on a zero by zero board. There’s no board.

And then I go to a one by one board. Which has one square.

So at the point others are giving up, I’ve now experienced two successes. And so have my students, with my help. I didn’t give anything away, didn’t rob them of the challenge. Just modeled how to start simple, bolster yourself up, see what you know. And there’s a way forward, a pathway from simple to hard that seems, maybe, passable. I can see a way to slowly ratchet up the difficulty, going to two by two and then to three by three. Maybe I won’t get where I’m going—believe me, I’ve worked on some problems that go from easy cases to ridiculously hard ones quick enough to give you whiplash—but this is part of what learning perseverance looks like. It’s knowing how to reframe things, make things simpler, change focus to see if you can get new ideas. And the taste of success keeps me going.

Thinking over this, there’s something really interesting happening in the pacing. I could have said

0 squares on a zero by zero chessboard.
1 square on a one by one chessboard.
How many on an eight by eight chessboard?

 

The information and the problem are identical. But the experience is profoundly different. That’s because the act of making a problem simpler is empowering. It shifts the situation from one of impending failure to one of success, even if the success is paltry. We go from hanging from a cliff face to standing calmly at the bottom of the mountain, considering our path forward. And no matter how you slice it, falling from a mountain hurts, sometimes in ways that you don’t recover from. Flailing shouldn’t become failing.

In other words, the actual skill we want to teach is how to shift from the despair of feeling like you’re about to fall off the cliff into ignominious ignorance to being back at the start, thinking about your options, considering your possibilities, and know that you can take at least one step without falling down in your journey of however many miles it will be.

Want to teach perseverance? Teach how to start a hard problem with simple case.

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How to Save Common Core

Those who believe the Common Core Standards are a good thing should take notice. The next year or two will probably make or break them, and the criticism is already starting to mount.

This matters. Reforms only work when the people who have to implement them on the ground—teachers and, to a lesser extent, administrators—buy in. And thanks to a history of bungee reforms that are undone five years after they’re implemented, teachers who aren’t convinced by the reforms can ignore them and be decently sure they’ll go away. It’s tough to win people over in this situation.

So here’s how you can save it, policy people:

  1. Devote the time and resources to training teachers in the new standards, and how to use them.

    Thanks to the sequester, our district has had to scale back training programs in Common Core. Meanwhile, most teachers I talk to haven’t begun to unpack the Common Core Math Practices, which are the lifeblood of the reform as far as math is concerned: the through-line of mathematical habits of mind from K-12 that allows everything else to fall into place. There needs to be professional development that actually helps teachers see how these standards work, why they’re better, how they help, and why the teachers should bother learning them. Because we said so isn’t good enough.

  2. Don’t rush to judgment.

    To change deeply-held habits and beliefs, you need time to experiment and reflect where you won’t be judged. By linking high-stakes tests to the new standards so early, we’re  undermining the entire process of reform. I know you like to talk about accountability and getting the data right away, but if you want a major reform to work, you need a three to five year transition period where people won’t hear that they’ve failed.

Of course, the reason I believe in the Common Core is that I think they’re an improvement over what we had before, and that it will make ultimately make life better for teachers, and give students a better education. Properly implemented, they give teachers more freedom, and at the same time allow students to change schools without getting so lost. They encourage a study that is deeper and more focused (though less broad). The new Math Practices are excellent, and the NextGen Science Standards look to be even better.

I’d like to see this work. But to have a chance, we’re going to need to take a breath and actually put the time in.

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Productively Stuck

When I try to describe great teaching, I notice a certain phrase pops out of my mouth again and again.

Productively stuck.

As in, the goal of the teacher is to get her students productively stuck as soon as possible. As in, we want to hook the students with a compelling question and then leave them productively stuck. As in, when a student is productively stuck, we don’t want to tell them the answer, because that robs them of the educational experience.

You could say that being productively stuck is virtually the same thing as learning. Talk to any teacher worth their salt and they’ll tell you how they draw their students in, how they frustrate them, how they force them to grapple with big ideas. It’s a surprising idea because learning connotes something so positive and frustration  and stuckness something so negative. And yet, they’re intertwined. Frustration just means you have a need that hasn’t yet been filled; learning is taking in (or creating) a new thing to fill the need.

Which means teachers need to be just the right kind of mean. They have to care enough not to be nice and relieve their students’ suffering (a relief that teaches a terrible lesson: that you can complain and the adult will solve your problems for you). No, the caring teacher lets their students suffer in just the right ways and struggle at just the right level: challenging, but not overwhelming.

I’ve noticed that a lot of folks who write about this topic have different descriptions for it.

  • James Tanton: “Let students flail.”
  • Dan Meyer: “Be less helpful.”
  • Paul Lockhart:
    “Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique. But not too much.”

Here’s another coinage from Emily, a teacher we worked with over the last few years: good frustration. I talked to her in the spring, and she shared this story of how starting with a question that her students didn’t have the tools for motivated them to want to learn multiplication and division (skills they would later master). What I love is that she goes seamlessly from talking about teaching math in a way that kids love, to starting your lesson with a question, to getting kids frustrated. She read me her student’s feedback on the class from surveys she had them take, and you know what they said, over and over. They loved the challenge

Here’s the video.


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Can Modesty Cure a Hurry?

ANNOUNCEMENT: Sign up now for our Common Core Crash Course for 1st-5th grade teachers, this August 20-21 in Seattle.

__________________

It’s happened to every teacher. It’s Thursday, but your students don’t seem to remember Wednesday or Tuesday, and you’ve got three times as much material to cover if there’s any chance of Friday’s lesson working. Finally, you gather them together. “Not all of you have figured this out yet,” you say, “But I’m going to show you how this works so you’ll be ready for tomorrow.”

I’ve done this more than I’d like to admit. And every time, I regret it. I see the students slump in a combination of relaxation and helplessness. I’ve just taught them that the teacher’s there with the right answer, and it doesn’t matter if they understand or not. Weeks of work are being undone. And worst of all, it doesn’t work! Friday comes, and the students still don’t know what they needed to know. Now I’ve got to go back to Monday, and start the whole project again, but without the advantage of novelty. “I’ll come back to this topic later,” I think. What a failure.

The best I can say is that it’s my failure, and not my students. I mis-estimated how much time they’d need, and then gave them less rather than more. I succumbed to the old false idol of teaching: if you just tell the students the answer, then they’ll know. It would be so sweet if it worked. But for almost all students, it doesn’t, and then we’ve got to resist the impulse to blame the students. I rushed them. Math takes time, and there are no shortcuts, “no royal road to mathematics.”

No matter how I try to resist, I make this mistake occasionally. But I do think there is a perspective that keeps me honest. A virtue I can cultivate.

Modesty.

A lesson plan is predicated on a lie: that students will “understand” the idea you’re teaching by the end of the lesson. But what we don’t say is that all understanding is partial. No matter how elementary the topic, I guarantee that it’s just a whisper away from unsolved questions in mathematics. It’s not just that I can’t fully understand, or that my students can’t: it’s that most (and it’s possible to make that word “most” quite mathematically precise) mathematical questions are outside of anyone’s reach. The greater the understanding, the more aware of all you don’t know.

And this humility, this modesty, is for me, a cure for rushing. Because where are you going, really? If there’s no end, why are you in a hurry?

This is true even for really simple topics. Consider addition.

  • The question of how many ways there are to break a number up into an addition problem—the so-called partition problem—was unsolved for centuries. It was only just solved.
  • When Russell and Whitehead tried to ground arithmetic in set theory, it took them 362 pages to prove that 1+1 = 2.

Nobody can know everything about anything. There’s no such thing as completely “covering” a topic. And knowing that keeps me honest. I won’t rush my students. I’ll give them time, and with it, the chance to get somewhere real, to attain actual understanding, instead of forcing them to participate in the lie that agrees with my lesson plan. They can’t understand everything, but they’ll have some real understanding, and just maybe, start to know what they know, and what they don’t, and how one becomes the other.

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Proofiness

When Stephen Colbert introduced the word truthiness on his show, he told us to trust our guts.

That’s where the truth comes from, ladies and gentlemen: in the gut. Do you know that you have more nerve endings in your stomach than in your head? Look it up. Now somebody’s going to say, I did look that up, and it’s wrong. Well, mister, that’s cause you looked it up in a book. Next time, try looking it up in your gut.

Truthiness, chosen as the Word of the Year in 2005 by the American Dialect Society, is the quality of certain kinds of baloney that seem true. Technically:

Truthiness, n, the quality of stating concepts or facts one wishes or believes to be true, rather than concepts or facts known to be true.

Colbert’s coinage recognize that for some unscrupulous actors, seeming true is good enough. The genius of truthiness is that it pinpointed how people take advantage of a lack of skepticism. When something seems true, we tend to take it at face value. As soon as it passes the gut check, we take it as a given.

There’s one kind of truthiness that’s particularly subtle and malevolent, and that’s the kind where mathematics gets involved. Author Charles Seife just wrote a book about it, and coined the word to describe it. The book is called Proofiness: the Dark Arts of Mathematical Deception.

Proofiness refers to a particularly devious kind of truthiness: the use (and misuse) of mathematics to give bull the illusion of truth. What makes proofiness so terrible is that numbers glow with truth. Introduce a number into a statement and it just seems truer. (It can be up to 7.3 times as truthy!)

Charles Seife wrote Zero, a decent but not exceptional pop math book. Proofiness is better, and more important. In fact, I could imagine it being the backbone of a (required?) high school class. Seife even raids science, hitting sexy articles from Nature and exposing their errors deducing patterns from noise, an error he terms randumbness. (Seife coins many such terms throughout the book—Potemkin numbers when they’re made up, causuistry for the confusion of cause and correlation, and many more. Normally, I bridle at this kind of cutesyness, and I did here a little, I have to say. In his defense, it was sometimes nice to have terms for some of the specific types of proofiness he discussed.)

Seife’s best chapters have to do with how risk is managed—culminating in a discussion of the financial meltdown that led to the Great Recession, how systematic bias affects polls, and, perhaps most damnably, how faulty probability lead to (probably) wrong outcomes in court. Along the way, he hits some pretty subtle mathematical ideas, and I have to say, he handles them very smoothly. You get a little probability and statistics, a little bit of reading graphs, on proper measurement, estimation, and disestimation—one of my favorite of his coined terms, and an intuitive picture of how gerrymandering works. But more gripping than the strictly mathematical parts is the sense of outrage. You read it and think, I remember when they said that Hummers were more fuel efficient than Priuses. Why didn’t I notice that they made up the numbers? (Advertisers argued that Hummers would last a lot longer, pulling numbers pretty much out of nowhere.) You’ve got to be pretty sophisticated, and pretty vigilant, not to have been taken in by these techniques.

And that’s the point, and the reason the book is important. The issues Seife covers–advertising, our economy, propaganda, science, justice, journalism—are critical to our functioning democracy. We need to understand math, it turns out, to protect ourselves from being manipulated by it. As Seife ends his book,

Mathematical sophistication is the only antidote to proofiness, and our degree of knowledge will determine whether we succumb to proofiness or fight against it. It’s more than mere rhetoric; our democracy may well rise or fall by the numbers.

This book is one of the best articulations of the need for a baseline of mathematical facility in an educated citizenry, and a primer on what kinds of mathematics are most important for everyone to know (calculus, no; statistics, yes). Put it on your summer reading list. And even better, let’s see high school classes that deal with these topics sprouting up soon.

[Note: it's almost inevitable that some of Seife's arguments were problematic themselves, especially since he makes a colossal effort to remain evenhanded when it comes to political issues; for every conservative example of proofiness, he tries to find a liberal one to match it. One of these, a nitpicky jab at Gore's Inconvenient Truth (which Seife takes issue with only on certain details, not in the main) was taken apart here.]

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Inversion Problem Update

I recently posted this interesting inversion problem:

The question is this: in mod n, how many functions f(x)= ax +b are their own inverses?

For example, the function f(x) = 5x + 2, applied twice in mod 12, is equal to the identity. It’s direct to check: f(f(x)) = f(5x+2) = 5(5x+2) + 2 = 25 x + 10 + 2 = x (mod 12).

I offered a conjecture that, a bit later, is almost hilariously false. (Don’t feel bad if you do this: I always try to get my hilariously misperceptions stated, disproved, and out of the way as soon as possible, and nobody gets hurt in the process.)  But I’d like to put forward a few better ones now that I’ve thought about the problem a bit more.

Conjecture 1. Let A(n) be the number of functions ax+b that are their own inverses mod n. If n and m have no prime factors in common, then A(nm) = A(n)A(m).

Example. If I want to know A(35), the number of functions ax + b that are their own inverses mod 35, then I simply need to find the product of A(5) and A(7).

Two quick things to mention. First, if Conjecture 1 is true, then A is an example of a homomorphism, which means that it preserves certain structural elements between where it’s coming from (the domain) and where it’s going (the range). These maps that preserve structure are the lifeblood of mathematics, because they allow you to take what you know about one area and apply it to another; they also allow you to build up to complete solutions from simpler cases. In this case, we just need to know how A acts on primes and powers of primes. After that, Conjecture 1 allows us to multiply them together to get any number, and still keep track of what will happen, as in the example.

It turns out that there are two different cases we need to treat: odd primes (3, 5, 7, 11…) and even primes (2).

Conjecture 2. If p is an odd prime, then A(p^n) = p^n + 1.

Conjecture 3. For positive integer m>2, A(2^m) =  2^m + 2 + 2^(m-1) + 2 = 4(2^(m-2)+2^(m-3)+1). For m=1 or 2, we have A(2) = 2 and A(4) = 6.

Now I got these conjectures by writing out the values of A(n) up to n = 40 in a chart with a student and playing around with them. If you’re reading this without having done that, or having background with problems like this, then it may seem mysterious how I came up with these conjectures, particularly conjecture 3. But it’s really just a matter of spending the time on it, and writing down the algebra to reflect the patterns you’re seeing.

To prove these conjectures, on the other hand, takes a little more subtlety, and more algebra too. First, you have to look at how ax+b behaves when applied twice: it’s a(ax+b)+b = (a^2)x + ab + b = (a^2)x + (a+1)b. For this to equal x in mod n, we must have a^2 = 1 and (a+1)b = 0 mod n. This alone makes the count a lot easier: just look for numbers that square to 1 mod n.

I’m not going to get into full proofs, but I will share Paul Lockhart’s elegant dispatching of the problem. Paul Salomon passed the problem to Paul Lockhart, and seeing the latter Paul’s elegant thinking is a pleasure. However, he pulls out mathematics and shorthand that might be intimidating if you’re not familiar with it. If you are, then you can add some details and prove all three conjectures with not too much more effort.

Here’s the question I’ve been kicking around since: the first example I happened to calculate was A(12), which equals 24. This reminded me of so-called perfect numbers, which have the property that the sum of their factors is precisely twice the number itself, i.e., 1,2,3, and 6 are the factors of 6, and 1+2+3+6 = 12, which is twice 6. Usually perfect numbers are described in terms their proper factors—ignore the 6, and they sum to themselves. However, it’s just as easy to describe them this way, and that makes my thought easier to follow.

Question. How many n have the property that A(n)=2n? Is n=12 the only option, or are there infinitely many? Or perhaps there’s a finite list of number that work?

I haven’t thought about this one at all yet, but it seems like an interesting place to go. If anyone has any breakthroughs, let me know!

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