I have a whole suite of maths t-shirts that I made myself. One of them simply has the number 65536 on it. It’s been getting a bit of attention over the past couple of weeks, so I thought I might write about it.

65536 is my favourite power of 2. More specifically, it’s 216, which means you can make it by starting at 1 and multiplying by 2 sixteen times. Even better…

65536-2222

But this cool stack of powers is not why it’s my favourite powet of 2. It’s my favourite power of two because of its connection to two very cool ideas in maths.

Firstly, 65536 is the last known power of 2 for which the next number is prime. It’s known that if a number one more than a power of 2 is a prime then it must be 2^(2^n)+1 for some n. The first five are all prime

  • 2^(2^0)) +1 = 3
  • 2^(2^1)) +1 = 5
  • 2^(2^2)) +1 = 17
  • 2^(2^3)) +1 = 257
  • 2^(2^4)) +1 = 65537

Fermat apparently conjectured in 1650 that they were all prime, which is why numbers of the form 2^(2^n)+1 are called “Fermat numbers” and if they’re prime they’re called “Fermat primes”. But so far, no more Fermat primes have been found. That is, every bigger number of the form 2^(2^n)+1 that we can calculate has been found to not be prime after all. Yet we haven’t been able to prove that there are definitely no more of them.

Isn’t that amazing? In close to 400 years we haven’t been able to find any more Fermat primes, but neither have we been convinced beyond a doubt that there aren’t any. I think it’s awesome that in maths there are things so simple that at the moment are unknown.

Secondly, 65536 is the only known power of 2 with no powers of two in order among its digits. Every other power of two where we have a list of most its digits, you can cross out some of the digits and have a power of two left behind. But we just don’t know if somewhere out there there’s a really big power of 2 that again lacks any smaller powers of two among its digits. Even stronger than this, every other power of two we’ve calculated has a 1, 2, 4, or 8 among its digits, but again we don’t know if somewhere out there in the distance there might be one that lacks these four digits.

What makes this surprising is that there is a perfectly good pattern to the final digits of the powers of 2. The last digit goes in the pattern 2, 4, 8, 6, 2, 4, 8, 6, … and the last two digits go in the pattern 04, 08, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52, 04, … You’d think that the repeating pattern of the final digits might make it easy to tell what digits were in a power of 2, but it’s not nearly so easy.

What’s even more surprising is that the same concept for prime numbers is completely solved. If you cross out some of the digits of a prime number you might have a prime number left behind. For example, the prime number 16649 leaves behind the prime number 19 when you cross out the 664. So which prime numbers have no prime numbers among their digits? Well there’s exactly 26 of them and they are 2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049. That’s it. All of them. Every other prime number has at least one of these in order among its digits.

This is from a paper in 2000 from Jeffrey Shallit: “Minimal primes,” J. Recreational Math., 30:2 (1999–2000) 113–117. He talks about it here. Within a set of numbers, he calls the “minimial set” those ones with none of the others in the set in order among their digits. He references another author’s theorem which says that given any set of numbers, the minimal set within it must be finite.

Isn’t it amazing that the prime numbers with all their apparent randomness have allowed us to find their minimal set, but the powers of two with their obvious regularity haven’t?

So that’s why 65536 is my favourite power of 2. It represents to me some cool ideas, and more than that, it reminds me that maths is far from all done in the distant past, it’s got unanswered questions alive right now.

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The game of Prime Climb

Prime Climb is a wonderful game by Dan Finkel (aka @MathforLove), which you can find out more about here. The board is a path made of the numbers from 0 to 101, coloured by an ingenous and beautiful system. Each player has two pawns which they move around the board by applying numerical operations to the number the pawn is sitting on. If you finish on a red prime you get an action card to use now or later. If you finish on another pawn, they go back to start.

Picture of Prime Climb from the game's website.

Picture of Prime Climb from the game’s website.

 

The first time I played it at work back in December 2016, we had so many players we decided to play in teams of two. I have to say I enjoyed playing in teams so much more than playing for myself. There was someone to talk to about how we would move our pieces and I have to say the talking about what was happening in the game was the most fun part for me.

The idea of bodyscale Prime Climb

Playing in pairs gave me this most fabulous idea. Each team had two players and each team had two pawns. What if the players were the pawns? What if we could actually walk on the board? That would totally take this game from wonderful to wonderfully awesome!

Unfortunately, the rigours of life and work meant that the idea had to go on the backburner for five months. Moreover, I had a couple of issues to work out with the walk-on version: how would I deal with the fact that I need two 10-sided dice, and what do I do about the deck of action cards you draw when you land on a prime? It occurred to me I could use spinners instead of dice, but try as I might I couldn’t find any that would allow me to make the spinner myself.

But then a couple of things came together that made the dream possible. First, I was in the Reject Shop and found travel Twister for $2 each, which meant I finally had a cheap spinner I could use instead of dice. Also, one of my staff/students played the game at a games night and reported his frustration that you couldn’t move the other players more often. This gave me an idea of an easy way to replace the cards by simply spinning one spinner and allowing you to apply that to the other players. Finally, I was home sick with a chest infection with plenty of time to individually design all the cards and easy access to a laminator. And with that, Bodyscale Prime Climb was born! I was itching to try it out at the next available One Hundred Factorial session.

Playing bodyscale Prime Climb for the first time

And it really was wonderfully awesome when we did play. We laid out all the cards in a back-and-forth line on the floor.

From the outset we had people walking up interested in what was happening and trying to figure out the colouring scheme on the cards. It was one of the best levels of engagement I’ve seen at One Hundred Factorial this year.

We played with three teams of two using the same rules as for ordinary Prime Climb, except for what happens when someone lands on a prime. In that case, the team spins one spinner, and then applies that number with a + or a – to any player on the board (whether in their team or not). We also ignored the usual rule for rolling doubles (which is to actually count it as four of that number) to counteract the extra freedom expected from our new land-on-a-prime rule. Here’s an action shot:

I thoroughly enjoyed the game because of how it felt and because of the talk it generated.

The feeling while playing it was very different from the hand-scale Prime Climb. There was something totally engaging about standing on the board. You could really feel how far you had to move to get where you want to go, and you had to look all around you for numbers you might be able to get to. Bumping people back to start felt so much more intense when you actually forced a person to walk all the way back to zero.

The talk between players was also very different from the hand-scale Prime Climb. I already mentioned how I loved the talk that happened when we played in pairs before, but this was at a whole different level. It’s impossible to hide your discussion when your partner is standing three metres away from you! This meant that all our talk was much more public so that everyone could follow what was happening, even the bystanders trying to figure out what we were doing. Also, because the board was so big and pointing was therefore so inaccurate, we had to be a lot more explicit about our language to each other. On the other hand, having the pawns being different people meant it was easier to talk about where they went. There was a lot of talk like “If we add 2 to me and multiply you by 3, then…”.

Some thoughts

One lesson we learned along the way was that we really shouldn’t have tried to get both players to 101 but instead have the shorter goal of just getting one player to 101. This would have made for a much quicker game and allowed us to play more than one in the time we had, and to more quickly get our bystanders into the game. I’d also like to get some coloured hats or sashes or something to wear so that it’s easier for observers to tell who is on what team.

I’m not 100% sure that our land-on-prime rule was the best replacement for the cards. It would be kind of cool if you could save them up to use on a later turn, but my memory’s just not that good and I don’t want anything that requires us to pass out or hold onto cards. One alternative that just occurred to me is that if you land on a prime, you can add or subtract one of the prime factors of the number where the other player is standing to any pawn on the board. That would make it very highly strategic!

The spinners worked really well. To start with, I held the spinners and did it for everyone. Later when we had an extra person, they became the spinner like in a game of Twister. It worked really well to have this extra person because they could freely move around to help people think and explain what we were doing to observers. I’ll have to appoint an official spinner next time I play!

I do think that maybe I could make some dice instead of spinners using 12-sided dice. I have no shortage of 12-sided objects to use as dice right now! My idea was to have 1-10 on each dice, but then an extra 2, 5 on one and an extra 3, 7 on the other. I think it might work really well. Still, I will miss the spinners, which had a certain charm about them.

The moveable board

One thing I didn’t really expect was the swirling vortex of ideas that were created by having the board made up of cards. The moment we started playing I started thinking about all sorts of ways that I could rearrange the structure of the board and thereby change the way the game felt.

The first thing I want to try is setting it up with one long straight line. Then there would be a real feeling of distance when you multiply or divide. Indeed, you could guess where double or triple your number is by doubling or tripling the distance between you and the 0 card. I also want to try it in the traditional 100’s chart with each row going left-to-right to match up how movement across that chart feels. Adding 10 would be a particularly pleasant experience I’d wager.

I also wonder how the game might work if we arranged the cards not in numerical order. Then to add or subtract, we’d have to actually calculate where to go rather than just step it out. I’m not sure I want to lay them out completely randomly, because it would be very hard to find the number you wanted. What if you laid it out like the 100’s chart but started a new line every time you got to a red prime? What if you had all the composite numbers in order in one row and the primes in order in another? What if you made a big Venn diagram of the multiples of 2, 3, 5, 7?  I want to feel the feeling of what it’s like to adding or multiplying in these situations and move from one collection of cards to another. Not to mention the very interesting task of simply arranging the cards themselves!

On that note, I actually reckon I might make a small version of the Prime Climb cards to use in the original game. How cool would it be to sit and sort the coloured numbers to see what sorts of patterns you could find before actually playing the game?

Stay tuned!

So stay tuned! At the July One Hundred Factorial session I want to try all the variations we have the energy to try and see how they turn out. And then a week later I’ll be at Twitter Math Camp and I’ll be trying it again there. I’m going to have so much to write about so check in in a couple of months to see how I went ok?

The resources

If you want to make your own bodyscale Prime Climb board (or indeed a set of Prime Climb number cards) then you can access a PDF of the cards and the spinner template at the link below. (I don’t recommend printing it at home because of the density of the ink! Best to use your workplace’s laser photocopier/printer.)

PDF Prime Climb cards and spinner template

If you do play it then please let me know! I’d love to see how it turned out for you.

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Yesterday I talked about one of the common responses to people finding out I am a mathematician/maths teacher, that of saying, “I’m not a maths person.” The other common response I get is, “I don’t have a maths brain.” (John Rowe mentioned this in his comment on the previous post.)

This is how I reacted last time someone said this to me:

It may not have been the best response, but I stand by the sentiment. I strongly believe there is no such thing as a maths brain. Or at least, that all brains are maths brains. I believe all human brains are wired in such a way to be able to learn and do maths, not least because I observe babies engaging in mathematical thought long before they can talk, so that capacity is there in all of us from the beginning. But more than this, I believe that the skills that I use to be good at maths are the same skills that other people use to do other things that they wouldn’t call maths.

I have one specific story to tell about how I helped an Arts student to believe that maybe she did have a maths brain after all.

Earlier this semester (a few months ago now), several student services were invited to an orientation event for Arts students, to make sure they knew about what was available for them. So I went along with a Writing Centre staff member to do our usual joint activity of Numbers and Letters.

A student came along to see what we were doing and happily engaged with the Letters game. She then glanced over at the Numbers game and I asked if she’d like to join in. With a rather green look on her face she said, “I don’t have a maths brain!”

I said, “I’m not sure I believe there’s such a thing as a maths brain.” Then I asked her what she was studying, and she revealed it was mainly poetry. “That’s really cool!” I said. “I reckon the skills you use to analyse and create poetry are tha same ones I use to do maths. Did you want to try a different sort of activity?” She graciously agreed and so I wrote this haiku on the board:

Word points, letter lines:
Ute fur you oft try fey roe.
My geometry.

“What do you notice?” I asked.
“It really is a haiku. And there’s a lot of really interesting words in that middle line.”
“Yeah I know right? I particularly like the concept of fey roe. What else do you notice about the words in that middle line?”
“Well, they’ve all got three letters…”

Following this was a most wonderful conversation about the letters they start with and end with, and which letters appear and how many times, scribbling notes on the board. This all culminated in the beautiful moment where the student realised the symmetrical nature of the words and the letters here and made an “oh!” of satisfaction.

“That was cool,” she said, after declaring she had to go. “Maybe I have a maths brain after all.”

This was one of the greatest moments of my entire teaching career, right then.

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I am a mathematician and a maths teacher. Therefore it is an occupational hazard that any random person who finds out what my job is will respond with “I’m not a maths person.” The most frustrating people are my own students who I am trying to tell that my actual job is to help them […]

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Two books I’ve read recently have encouraged me to investigate my memories from childhood. In Tracy Zager’s “Becoming the Math Teacher You Wish You’d Had“, she urged me to think about my maths autobiography to see what influenced my current feelings about maths. In Stuart Brown’s “Play“, he urged me to think about my play […]

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Looking back at my blog over the past few months, I’ve done a lot of these “book reading” posts. I really did mean to do some more on other ideas, but I felt I had to get these thoughts out of the way first. So here’s another book reading post, this time about the book […]

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Here is another post about a book I’ve read recently. This time, I’m writing about the book “Making Number Talks Matter” by Cathy Humphreys and Ruth Parker.
In Cathy and Ruth’s words, number talks are “a brief daily practice where students mentally solve computation problems and talk about their strategies”. I had heard people talk about […]

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This post is about Tracy Zager’s most excellent book, Becoming the Math Teacher You Wish You’d Had. I actually finished reading it back in January, and I live-tweeted my reading as I went. The process culminated with this tweet:
I've just finished reading your #becomingmath book @TracyZager. This is the bit I liked: pic.twitter.com/nWHp9mHUgt
— David Butler […]

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In the online resources for Becoming the Math Teacher You Wish You’d Had, Tracy Zager provides information about the benefits of writing a “math autobiography”. I really have tried to do this, but I am having a lot of trouble organising my thoughs and memories. However, I reckon I can track some of my memories […]

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It was O’Week a couple of weeks ago, when new students arrive on campus to find out how uni works and the services they have access to. Our tradition for the last several years is to play Numbers and Letters on a big whiteboard out in public as a way to engage with students.
In case […]

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