This post is about a game I invented called Home in One Piece. I invented it in 2014 specifically to play outside at student barbecues, after years of trying to think of an effective game using play dough. I’ve taken the physical version with me to various places to play it, including to Twitter Math Camp¬†in the USA. It’s time to let it out into the world and allow people to play it for themselves.

Home in One Piece is a game for two players that is played with play-dough. Each player has eight blobs of play-dough and the goal is to join the play-dough together and move it across the board, so that finally you win by having all your play-dough together in one blob in the home zone (hence the name Home in One Piece). The original version was a pure strategy game, but then I remembered that I prefer games with a chance element, so I created three special dice to control how many of your blobs you are allowed to manipulate during your turn.

Picture of Home in One Piece board in action

To make the game you need the following components:

Here is a YouTube video of a game in action, so you can have a better idea about how these rules work in action.

There are so many things I love about this game, if I do say so myself. I love the physical and three-dimensional nature of it, with the manipulation of the play-dough in space. I love the contradictory feeling of freedom and constraint of being able to make the play-dough whatever shape I want, yet having to hold it down in one spot. I love the fact that the movement condition is continuous, whereas in almost all other games it is discrete. I love sitting on the same side of the board as my opponent, so that we are both looking at the same game and reaching past each other to play. I love the fact that I’ve arranged the dice so it’s possible to roll 0. (If you’ve played my other games you might notice that particular fascination of mine.)

Mathematically I love thinking about what it means to be “one blob” and about the interesting topological shapes that happen when people join and rejoin blobs. I am particularly in love with my dice, which have been arranged carefully to give the right probabilities for the different numbers with only one dot per face. (That was a whole wonderful investigation, I can tell you!) I love secret problem zero of dividing your blob into eight equal pieces.

But most of all I love watching people play it. I love the shapes they make, and the realisations they have about what is possible and not possible — I love the walls and lakes and bridges people build, especially bridges over bridges over bridges. I love the cries of despair when the play-dough breaks and when someone rolls zero. I love how lax people are about their play-dough touching at the beginning and how unforgiving they are about it at the end. I love watching people wander over, fascinated at what the hell is going on with the play-dough here.

I hope you enjoy the game as much as I do. Please let me know how it goes!

To finish off, some tweets of the game in action (note this is the first version of the game, which had the rules printed on the game-board.)

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Last week, I had one of those days in the MLC Drop-In Centre where I was hyper-aware of what I was doing as I was talking with students and by the end I was overwhelmed by the sheer volume of things I had thought about. I decided that today I might attempt to process (or at least list) some of it for posterity.

A Real Analysis student was worried about a specific part of his proof where he wanted to show that a function was increasing. The function was a kind of step function with more and more frequent steps as it approached x=1. It was blindingly obvious from our sketch of the graph that it was increasing, but every time we tried to come up with a rigorous argument we just ended up saying something equivalent to “it is because it is”. I remember thinking about whether it was just obvious enough to just say it without proof, but shared the student’s desire to make the nice clean argument. I don’t remember now if he ended up with a neat argument or not, but I do remember finishing with the moral that obvious things are often the hardest things to prove.

Before we even got to this frustrating little bit of the proof, I was reading through what he had done already, and I noticed that he had written “x is < 1″ and commented that this wasn’t a grammatical sentence because the “<” contains an “is” already — it’s “is less than”, which makes his sentence “x is is less than”. This led to quite a discussion about the grammar of “<“, which is complicated by the fact that it can be read aloud in multiple ways depending on context. We investigated some other sentences containing < or > he had seen written before to see how it was pronounced there. In my head I was reminding myself to pronounce written maths aloud more often when I’m with students so they can learn the correspondence between speech and writing in maths.

I was called over to help a Quantitative Methods in Education student with her research assignment because she had mentioned she was struggling with statistics (and I’m the default stats support person). Eventually I helped her figure out that her problem wasn’t statistics at all, but rather that she didn’t understand the wider context of the research, or what the data actually represented, or what the actual goal was. Mainly this came about by me not understanding the context, data and goal and kept asking her questions about those things in order to try to figure out where the statistics fit into what she was trying to do. We talked about how she might go about gaining the understanding she needed in order to come up with the specific-enough questions about variables that statistics would be able to help her to answer. I could see her heart sinking, so I reassured her I would still be here next week to talk about the statistics should she need that support later, but also from our discussion that she would probably be okay with that part when it came.

I returned to the table I came from to discover even more statistics, this time a statistics theory course in second year of the maths degree. This student was struggling to come up with a linear regression proof which was pitched in the assignment using linear algebra. For this small part of the proof, he had to show that the columns of some matrix were linearly independent. I did what I always do with linear algebra proofs and helped him remember various definitions and facts relating to the vocabulary words in the problem, then we chose a representation of linearly independent from the list we had made. He commented that I go about proofs in a very different way to him, and I said I was merely following the problem-solving advice I had already put up in that big poster on the wall, see? In the end we finished with the moral that coming up with connections between things is a great way to solve problems and to study, even if it means ignoring the goal for a moment.

I saw a student on the other side of the room that I had talked to months ago and I went to see how she was doing. When I saw her last, she was questioning her decision to study Statistical Practice I and indeed her whole decision to come to university at all, while simultaneously worrying about letting down her son who was the one who had urged her to come to university. Now she revealed she was systematically working her way through the course content so far, making sure she understood everything before classes returned and the last set of new content arrived. She asked if that seemed like a good plan and I said I was so pleased to see her making plans for her study and persevering with her course. I suggested she do some work soon to connect together the ideas in the different topics, since it’s the connections between ideas that will create understanding. I recommended a mind-mapping idea I had seen on Twitter recently (thanks Lisa).

At the next table I came to, some students and one of my staff were having a discussion about function notation, in order to help a student in our bridging course. They were talking about how unfortunate it is that something like f(x) could be interpreted as multiplication in some contexts and function action in another. I agreed that it was unfortunate, but it was just the way maths language has come to work and it was too late to change it now. I related this to how the word “tear” is interpreted differently depending on context, which seemed to help everyone accept the fact, but not necessarily be happy about it! My staff member brought up how it’s made more complicated by the fact that we then write “y=f(x)” or even “y(x)=…”. I flippantly said that this was making a connection between graphs and functions, which is a whole separate discussion. Of course everyone wanted to hear more about this and suddenly I was giving a little seminar on the fly about function notation and graphs and the connection between them. I was keenly aware of everyone watching me, and of the choices I had to make with my words, especially when I made some mistakes along the way. Here’s more-or-less what I said [with some of my thoughts in square brackets]:

There are many ways to think about functions, but one of them is that a function takes numbers and for each one it produces another number. [I thought about saying it doesn’t have to be numbers, but decided that was just going to distract from the discussion today.] It could be a formula that tells you how to get this other number, or there could just be a big list that says which ones produce which ones. For example, there’s a function which takes 1 and gives you 4, takes 2 and gives you 5, takes 3 and gives you 7, takes 5 and gives you 8. It’s the one that adds 3 to every number. The brackets notation is supposed to highlight that there’s a starting number and the function is a thing that acts on it to produce a result. So we could write something like “addthree(2)=5″. [I’m not sure what motivated me to write this. I’m sure I’d seen someone talk about this on Twitter somewhere, but anyway it seemed right at the time.] Sometimes we want to talk about the function as a thing in its own right so we give it a letter-name like “f” (for function) so that we can talk about it more easily. Or we only have some information about it but don’t know what it really is, so we can give it a letter name until we know its proper name. [I was reminded at this point about a Rudyard Kipling story featuring the origin of armadillos but chose not to mention it. I also thought here that I could have done another example earlier of a function acting like lookupthelist(2), but we’d moved further than that and I didn’t want to go back.]

And what about the connection to the graph? Well you could draw a nice number line and write down what result each number produces. [I started drawing pictures at this point.] You could visualise this by drawing a line of a length to match the result attached to each number on the line. [I accidentally drew the lines representing the input numbers when I drew my picture, and had to go back and change them later. No-one seemed to mind that much.] You could have a ruler to measure how long each of these is when you needed it. Or you could attach the ruler to the edge of the page and line up the drawn lines with it. And now suddenly you’re locating the end of each drawn line by numbers on two axes. This is coincidentally just like the coordinate grid which is something else we’ve learned about before but not directly connected to functions. In the coordinate grid, the y refers to this second number and the x refers to this first number and you can describe shapes by how the x and y are related to each other. It doesn’t have to be y = formula in x, but that is a useful way to do it if you created your shape from a function.

I think it was at about this point that I asked the students how they felt about this. My staff member said he quite liked the “addthree(x)” thing, which he’d never seen before. The bridging course student said that really helped her too, as well as realising that there were two different things happening at once when you write y=f(x). I decided to move on at this point.

The next student was studying Maths 1B and was doing a deceptively simple MapleTA problem which he was stuck on. It listed an open interval — I think his was (-4,5) — and asked him to give a quadratic function with no maximum on this interval, a quadratic function with neither a maximum nor minimum on this interval, and a cubic function with both a maximum and a minimum on this interval. He had answers entered for the first two questions, which MapleTA had marked correct, but he wanted to know why. He revealed a little later he had gotten these answers off a friend and that now he wanted to know how to get them himself. I’m always glad when students do this, because it’s the first step to them really understanding themselves. With the first question, we discussed what it meant to be a maximum, and how that definition played out on an open interval. Once we had done this, I decided that I should take my own advice and make the question more playful, more exploratory. So I asked him what he would need to put in in order to make the answer wrong. He was really intrigued and started thinking through what the function would have to look like in order to have a maximum. After a while, he hit upon the idea that the leading coefficient could be negative, and this is all it would take. This was followed by several attempts to test his theory by putting in bigger and bigger constant terms in order to see if they really didn’t make a difference. They didn’t, and he was most impressed with himself.

In the second part, we talked about moving the function around until the minimum wasn’t part of the domain. I went to get a plastic sleeve and drew a parabola on it in whiteboard marker, then asked him where he could move it to get what he wanted. At first he rotated it. I congratulated him for thinking creatively but did point out that it’s not the graph of a function like it was before. After that he only moved it up and down. Finally I gently moved it a tiny bit to the left, and he caught the idea that he could move it sideways, which he did until the turning point was outside the domain. The issue then came with how to change the function to do this, and we discussed why his friend’s solution was able to achieve it. Again he went playful and started putting in really big numbers and numbers really close to the boundaries to confirm that yes really he could move it anywhere outside that domain.

Now we moved on to the third part and we discussed how cubics look and tried to figure out how to make the two turning points higher or lower than the end points. He wanted to use derivatives, so I let him do that, and it was quite a long journey, though he learned a lot about derivatives and specifying function formulas along the way.

At the end, he asked if he had done all of this the correct way. I replied that it was definitely a correct way. He took this to mean there was a better way, and I said there was possibly a faster way. This meant a further long discussion about specifying zeros of functions and talking about whether we needed them to be zeros or if the function could be higher up or lower down. When we had finished this he asked me to explain the steps of this method again so he could write them down. I said that he didn’t need to remember the method because he’s never going to see this question again. The point is not to remember methods of solving all questions, but to learn something, and look how much stuff you learned today!

Even though it was very close to my home time, I decided to talk to one more student. This one was doing a course specifically designed for the “advanced” maths degree students. He had been set a problem in Bayesian statistical theory. This is not familiar to me, so he spent a lot of time explaining what was going on to me. There were a whole lot of things in his working that I thought could be solved by simply referring to facts he had learned in previous courses, and I asked if he had done those courses. He revealed that they were in fact prerequisites for being allowed to do this course, at which point I said it was okay to use results from prerequisite courses. The biggest problem at the end was that he had assumed he had to integrate the normal distribution density function by hand, because the lecturer had gone to the trouble of giving its formula in the assignment. I assured him that it was in fact impossible to find areas under that distribution by hand, and he was allowed to use a table or technology. Sometimes you just have to tell a student what’s impossible so they can do it another way.

So that was my day. I had three hours in the MLC and talked through so many ideas about learning and studying and maths. I had to make so many decisions about my words and actions and teaching tools. And I was left with a lot to think about. I hope you’ve found it interesting too.

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There is a Twitter account that tweets the prime numbers once an hour in sequence. (The handle is @_primes_.) Since before I joined Twitter, it’s been working its way through the six-digit primes and some of them are very nice. A lot of other people think they’re nice too, based on the fact that they are given likes and retweets. But what is it that motivates people to do this? What is it that makes a prime likeable? Well, that’s what this post is about.

I was planning to do this investigation over Summer, and at the time I got Lewis and Tobin to help me get the data on the prime tweets over the previous few months. Tobin did some analysis of his own, but I haven’t looked closely at it because I wanted to have the fun of doing it myself. Not until now have I had the chance to actually do it. So here goes! What you see here is a record of my thoughts and investigations as I did them.

I have data on primes tweeted between the 11th September 2016 to 26th January 2017, a total of 3217 tweets. I thik there’s a gap of time in there with no data, but for our purposes it should be enough. You can get the raw data here: likeable-primes-data.csv

First up I’ll just have a look at how many likes each prime has gotten and see what we might see.

Oh my! Well one prime in particular is way more likeable than all of the others, and there’s a couple more there that are quite a lot higher, but not nearly in the same league. Let’s have a look at the top ten and see what they are.

Well would you look at that top one?! It’s the first several digits of pi. So it seems that the most likeable thing about a prime is being pi. It seems you have to be right on the dot though – being close just isn’t all that likeable as this list of primes starting in 3141 shows:

I could try to search to see if being the digits of other special numbers is likeable, but it seems the digits of other special numbers just aren’t prime. Pi itself was last prime at 31, which doesn’t stand out as pi-ish, and it’s not going to be prime again for quite some time. Phi and e aren’t going to be prime until seven digits, and the square root of 2 won’t be prime until after 50 digits, so it’s going to be a while before I can check the effect of this on prime likeability. (Check out pi-prime, e-prime, phi-prime and this wolfram alpha search.)

I wonder what it is that made those other highly likeable primes have so much love? I see in the top ten list a whole lot of primes with lots of the same digit, so it seems repeated digits is something highly likeable.

I might come back to that later because I also see 300007 and 299993, which are the first primes before and after 300000. Maybe there’s also something in primes being close to milestones. I’m not sure if there’s any other good milestones in this range of primes to test this theory. Let me go searching for the prime tweets near to other milestones…

Aha! Just a couple of weeks ago we reached the 400000 milestone and there was a spike in likes before and after. There were also spikes when 200000 was passed in 2015 and when 100000 was passed in 2014. So yes it does seem that primes before or after milestones are liked more. It’s interesting that 199999 got so many more likes than 200003. I’m wondering if it’s to do with the repeated digits thing that I mentioned earlier.

So what about these repeated digits then? It seems primes with a lot of repeated digits get more likes. There’s a lot of factors there that might be at play – is repeated digits in a row more likeable than separated? How many repeated digits do you need to get more likes? So many questions!

Well first I’ll count how many of each digit there are in every prime, and I’ll find the maximum number of repeated digits. I’m not looking at repeated digits in a row right now. I’m not sure how to do that yet. Let’s look at the relationship between highest number of repeated digits and likeability.

Oops! That 314159 is making it hard to see what’s going on here! Those other two really big ones could make it hard to see too. I could remove those top three, but I don’t want to lose the fact that they are there. What I’ll do is replace them with something just above the next one down, like 150, 155 and 160. Let’see if I can get a better look at what’s going on down the bottom there after that.

This is much better — there’s definitely something going on there. Primes with five repeated digits are certainly more likeable than primes with less, and four repeated digits definitely seems to increase the chances of likeability.

My attention is drawn to those few extra-likeable ones leaping out of the clump of primes with digits repeated three times. I want to look closer at those.

Some of those are rather nice, but there’s nothing I can see that they share which makes them particularly likeable. Let me widen the search to include the next few most likeable.

Ah! I see most of these have their triple digits all in a row as opposed to separated. A lot of them also have a double-digit too. Other than that, I can see ones with an alternating pattern. Those are going to take a bit of learning for me to figure out how to find them…

Phew! That was some hard work! And unfortunately it doesn’t really tell me anything that much different than the number of repeated digits ignoring the number in a row.

Let’s look at them together: in the graph red is “in a row” and blue is “repeated at all”. If a prime already has repeated digits then having several in a row might make it a little more likeable, but I don’t know if it was worth all the effort to figure out how to get R to count them. The graph is pretty though.

I’m not sure I want to figure out how to search for an alternating pattern. Someone I’m sure will say “just use regular expressions” and I would say in response that I don’t know how to use regular expressions and I’m not sure I care to learn right now. Plus I suspect it probably doesn’t add much more to the likeability compared to just having repeated digits in any order.

Well that pretty much exhausts the things I noticed looking at the most likeable primes. The first thing anyone has suggested when I have mentioned I was doing this was that perhaps time when it was tweeted has an effect, so I might as well take a look. I think the primes with above 50 likes can be attributed mainly to repeated digits and piness, so I’ll just adjust those like I did before. These graphs show the likes on days of the week or time of day, with the mean marked by a red dot.

There’s variation between days/times certainly, but I don’t see that it makes all that much difference to the number of likes compared to the total amount of variation. Actually, my stats-ear is saying I should back that up with some tests. These ANOVAs say that the day of week or time of day don’t really make much of a difference compared to the rest of the variation.

I suppose I could look at the grander sweep of time to see if there’s anything interesting going on there.

Well first I notice a gap in October. Now that I see it I think I do remember Tobin saying something about some missing data. I also notice that in October there aren’t many primes with a lot of likes. I’m not sure what caused that. I do know that most of those highly-liked primes are repeated digits, so what if I colour by the maximum number of repeated digits to see how that relates?

Wow! It seems that almost all of those primes above the river are ones with 4 or 5 repeated digits, and there just happen not to be many of them in October. I can see a few orange dots in the mix there and I do wonder why those ones aren’t very likeable. Interestingly, there’s a lot of orange dots down the bottom there in January. But there’s also a lot of yellow there which means three repeated digits. Maybe in comparison to the general repetition of digits at the time, they just didn’t seem as special as they did back in November when there had been a repeated-digits drought. (Upon closer investigation, that January period is when we were passing through the 330000’s so we were guaranteed to get double 3’s for a while.)

I also notice that there is an upward bend in the river around 314159 in early November and the milestone of 300000 in late September. I think perhaps the twitter account generates higher levels of attention around an important event, which means that likes are more likely. This might explain the high number of likes for 301333, even though it only has three repeated digits a row: it’s the first prime after 300000 with three nonzero repeated digits in a row, so it got more attention because of the afterglow of the milestone.

The final thing to consider is if there is anything that makes a prime specially unlikeable. Let’s have a look at the bottom twenty or so.

I can’t see anything in particular that sets these ones apart, which I is the point I suppose! I do feel sorry for poor 324619, which has the dubious honour of being the least-liked prime in this timeframe. (And as a result, it’s no longer the least-liked prime in that timeframe. )

I suppose it’s time to sum up. What have I found out here?

The following things seem to make a prime number more likeable:

  • Being pi
  • Being close to a milestone
  • Having a lot of repeated digits, especially if not near other primes with repeated digits
  • Being near pi or a milestone

Well. That was fun!

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This post is about my reaction to the book “Choice Words: How Our Language Affects Children’s Learning” by Peter H. Jonston. I was lent the book by Amie and I am very grateful to her because it really is a good book (though it was tough to read with the forest of sticky notes marking […]

Posted in Education reading

Well I did it. I went to Twitter Math Camp 2017 (TMC17) in Atlanta, Georgia, USA.
I found out about TMC last year, when Tracy Zager mentioned me in her keynote at TMC16, effectively yanking me right into the thick of it. I could see that this was one of the things that cemented together the […]

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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 […]

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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 […]

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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 […]

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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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