A recent video from Howie Hua showed how if you split a collection of numbers into equal-sized groups, then find the mean of each group, then find the mean of those means, it turns out this final answer is the same as the mean of the original collection. He was careful to say it usually does not work if the groups were different sizes. Which got me to wondering: just how much of an effect on the final mean-of-means can you have by splitting a collection of numbers into different-sized groups?
Tag: one hundred factorial
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Making the lie true
We at my university regularly sell quite a big lie.
At Open Day and the Ingenuity STEM Showcase and any number of outreach activities, students do puzzles and play with construction toys and walk around with ropes and draw curves on balloons. Whether we say it explicitly or not, there is a message there that says: here at this University, maths is fun. This is a lie.
Maths at university is not fun. There are hours of video content to watch where the presentation is basically slides or handwritten examples. The classes are presentations, possibly with little quizzes breaking them up, or they consist of doing maths problems similar to the relentless weekly quizzes and assignments. Pictures are rare, making sense by manipulating something with your hands is much much rarer, making sense by moving your body is non-existent. The chances to chase your curiosity are few. The chances to have your own thinking validated and celebrated are fewer. It is very far removed from the experience of university maths the prospective students get when they visit us.
We are lying to our prospective students. The experience they have of university maths at our events is a lie.
I do understand that learning does not have to be “fun”, and expecting it to be so all the time is unreasonable and unhealthy. I also understand that ordinary everyday problem-solving and figuring out can feel fun. I understand as well that play, which is essential to learning deeply, is not the same thing as fun. But there is no denying that the activities we do with prospective students are indeed fun, and that experience is not what it will be like at university.
Do I want to change the activities we do with prospective students to look as boring as life will be at uni? Of course not. But there is another way to not lie, and it’s to make your lie true.
One way I make the lie true is to provide One Hundred Factorial, a weekly games, art and puzzle session where students can experience mathematical play without having to be assessed on it. The sorts of things that happen as a one-off at outreach events happen every week at One Hundred Factorial, and I think it would be a good thing to tell prospective students that this exists. (Writing this blog post is partly to help myself pluck up the courage to suggest to the academics in Maths here that they can do so.)
Another way is to actually include some of the features in your outreach activities actually in your teaching. I’ve seen the maths academics do an awesome job of running engaging activities and helping students feel like their efforts are meaningful and valued. They’re good at it. What I want to say to them is this: Perhaps you can actually include some whole-body movement or physical models in your university classes, or at very least in your videos. Perhaps you can actually have some free exploration of new ideas without having to immediately write an assignment about it. Perhaps you can keep the idea of celebrating students’ mathematical thought in the very front of your mind more often when they are doing everyday maths problems or answering questions in the lecture. Even just a little more of any of these things might make university maths a little more like the outreach activities you do so well.
The experience prospective students have in your outreach activities doesn’t have to be a lie. You can make the lie true.
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Running out of puzzles
Because people know I run the One Hundred Factorial puzzle sessions, they often ask me if I have a repository of puzzles they can use for their classroom, enrichment program, maths club, or their own enjoyment.
Sometimes I feel embarrassed because I don’t actually have a big repository of puzzles. Surely since I am a person known for promoting problem-solving and puzzles, I should have such a thing. At the very least I should have a record of the puzzles we did do. But I don’t.
It turns out my scatterbrained tendency to forget record-keeping is not the main thing that caused this lack of puzzle repository, but only in the last few weeks did I realise what the main cause actually was. It’s that I don’t feel the need for lots of puzzles. A person recently asked for my advice on where to find puzzles and told me the reason was they were worried their maths club would tear through them and so have nothing to do. Only when they gave this reason did I realise I don’t worry about this at One Hundred Factorial. But why?
Firstly, puzzles are not the main food at One Hundred Factorial. I usually have exactly five activities available: a logic puzzle (eg sudoku), a word/geometry puzzle, an art activity/construction toy, a game, and the Numbers Game. If people get to the end of the puzzles, there is always other stuff to do instead.
Secondly, and much more importantly, the whole vibe of One Hundred Factorial means that puzzles do not end. I have carefully cultivated a culture encapsulated in the mantra:
The goal is not the goal.
The end is not the end.What “the goal is not the goal” means is that the stated goal of a puzzle or problem is not the actual goal. The “goal” might be to find the area of a shape, or the probability of some event, or count how many of something there is, or whatever. They are not the goal. The real goals are to learn something, or understand someone’s thinking, or make something beautiful, or find a connection to something else.
What “the end is not the end” means is that even if you do get to the stated goal of a problem, it doesn’t mean the thinking stops. You can ask if there’s another way, or what the problem would be like if you changed this aspect, or look for a connection to something else, or build something cool out of the answer or process. The truth is there is no end.
The mantra of “the goal is not the goal, the end is not the end” means that we can get by at One Hundred Factorial with just one puzzle. In fact, we can get by with no new puzzle at all. Maybe someone was at the previous session and we want to continue with the non-end of last week’s puzzle. Or someone saw a random thing during the week that inspired their thinking and turn up ready to include others in their thinking or find out what thinking it might inspire in others. Or someone pulls out a puzzle that’s been done before and wants to find out how other people might think about it.
As far as I can see it, my approach to cultivating a “goal is not the goal, end is not the end” culture had three aspects:
- Constantly ask goal-free, non-end questions like “what are you thinking?”, “is there another way?”, “what would happen if?”, “what can we make?”, “what is this connected to?”.
- Notice when other people ask those sorts of questions and run with it. I found that once I became attuned to them, I noticed people asked them a lot more often than I realised.
- Provide open-ended things other than just puzzles, like construction toys or art activities. There is nothing like an activity with no goal to foster a more goal free attitude. Even just puzzles with more than one solution foster a more open-ended attitude.
So that’s how I don’t run out of puzzles: I don’t only use puzzles, and when I do, we go further or in different directions than the puzzle says to.
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My first Maths Teacher Circle
Last week I participated in my first Maths Teacher Circle . I just want to do a quick blog post here to record for posterity that I did it and it was excellent. I choose to take the practical approach of just relating what happened.
I had been interested in somehow going to one since I heard about them a while ago, and then the founder of the Aussie Maths Teacher Circles, Michaela Epstein , contacted me through Twitter back in November to ask if I might like to facilitate an activity at an online session in 2021, and of course I said yes. She invited me to a session about mathematical games, and I was so excited to share some of the games I have invented with some interested teachers.
Of course, the closer it got, the more nervous I got. When I heard there would be 40 or so teachers ranging all through primary to secondary to post-school teachers, I was rather intimidated! But Michaela and Alex assured me I would be ok and that what I had planned would work. And they also put up with my scatterbrained discussion of random maths stuff whenever I met with them too. So, feeling a little reassured, but still nervouscited (as Pinkie Pie would say), I dove right in feet first last Wednesday morning.
To start off with, Michaela invited past Maths Teacher Circles participant Samantha to set the scene by sharing what she has gotten out of Maths Teacher Circles in the past. This was a nice way to begin by grounding it in a real teacher’s experience. Then Michaela shared the goals of Maths Teacher Circles, which were exploring maths, strengthening classroom practice, and bringing maths enthusiasts together. I was so glad I had come to a place that resonated with all the things I love. It really matched with the goals of One Hundred Factorial, which is probably why Michaela invited me to present in the first place. This was all a really smart way to begin, because it set the tone for the rest of the session. Even when the housekeeping notes about breakout rooms and whiteboards and chat windows came, it was clear that these were there to support the overall vibe.
Then we had a very quick chat in breakout rooms with a couple of people. We were supposed to talk about Noughts and Crosses too, but we only just made it through the introductions! But honestly I was happy to just have met a couple of friendly faces to help reduce the nervous part of the nervouscieted.
By this time, so much had happened already, yet it had only been a few minutes. And now it was my turn. Michaela introduced me and I was now responsible for the journey of these 45-ish hopeful people. I put up the rules for Which Number Where, and asked everyone to quietly have a read, then ask any questions they might have. People had some very useful questions in the chat and out loud, and I felt we were ready to try it live. I asked for volunteers and described how to play the game Mastermind-style, with one player being the Secret Keeper and the other players asking questions. After a couple more questions, we were ready to break into groups to play.
Michaela put people into groups of fourish, and I popped into about half of them to have a chat. I asked people how they were going and played with them for a bit, seeding a different kind of question than the ones they had been asking so far. I found everyone to be gracious and thoughtful and engaged. Such a thrill to meet such wonderful people and play maths with them. These moments when I was in a small group with people were my favourite parts of the session.
I brought everyone together into the big group to discuss how the game went. I started by asking people if they had a favourite question that was asked. And then people shared any thoughts they had at all about how to use this in a classroom.
Suddenly it seemed my time had run out, so I quickly showed everyone my other two games Digit Disguises and Number Neighbourhoods, and encouraged them to go back to their breakout rooms to keep playing Which Number Where or to try a new game instead. I stayed out in the main room where Michaela made sure I was ready to do a wrap-up when people returned. I very much appreciated being able to think in advance about that part!
One question Michaela asked was why I chose the game I did. I said I chose Which Number Where because it’s about logic, and not any particular maths topic per se. As someone said earlier, it’s about locations rather than numbers per se, which means it’s really about the yes-and-no questions, and about logical arguments and joining information together, and those are skills you use everywhere in maths, which is why I like it so much. Plus I just love to hear how people think and this game gives me a chance to do that.
And then it was time for me to participate in someone else’s activity. Toby and James shared the Multiple Mysteries game and some problem-solving/proving prompts to go with it. I got to play the game with some lovely other people and join in with the play. It really was a lovely thing to just play around with something that someone else shared that they were excited about. I am very grateful to Toby and James for providing such a great game to play and think about, and to the members of my little breakout room who I had such fun with.
After this, it turned out that Michaela had read the time wrong and had cut short my activity the first time! So I got to have a few more minutes! I decided to share Digit Disguises properly, and instead of using breakout rooms, to play a game as a whole room with me as the Secret Keeper. Some brave souls shouted out questions and I wrote the questions and responses on a Word document on the screen. After a few questions, I decided that I would stop people and ask them what they can figure out from the information we have so far. This part was just wonderful. People had multiple different ways of gleaning new information about the numbers and their letter disguises from what we already knew, and quite a few of the participants expressed a satisfying amount of delight at these fascinating new possibilities. It was extremely gratifying to have people so excited about something that I am excited about (and egotistically, satisfying that people liked something I had invented).
At this point, my laptop ran out of battery power and I had to scramble to find the power cord. By the time I came back, things were starting to wrap up, with participants filling out a Padlet with their thoughts. And then it was over. It felt like almost no time at all had passed, which is a good sign that I’ve been deeply engaged.
After all the other participants left, Michaela, Alex, Toby, James and I had a debrief, which was some lovely discussion about how it went and how cool it was to work mathematically with people rather than just present them with stuff, and just some nice discussion about teaching and learning maths with some lovely people. And after that, I couldn’t help but keep working on one of the investigations that Toby and James set me off on, because that’s how I roll and is the sign of a good maths problem.
So that was my first experience of a Maths Teacher Circle. For me, the best part was the chance to think and play together with other teachers. The environment was so safe to just play and talk, and this was very carefully set up by Michaela in the first place, by discussing what was important and how to keep it safe. Being told explicitly that we were allowed to adjust the activities to match the level of the group made us free to play in our own way. And really, everyone was just so gracious and excited and, well, lovely. I am so grateful to have been a part of it.
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Quarter the Cross: Colouring
Quarter the Cross is one of my favourite activities of all time, whether in maths or just life. I learned about it way back in 2015 and have been mildly or very obsessed with it ever since. This blog post is about one particular version of the Quarter the Cross problem you might like: the colouring version!
You can read the rest of this blog post, and four other related posts, in PDF form here
The titles of the five posts in the series are:
- Quarter the Cross (2016)
- A Day of Maths: Quarter the Cross (2016)
- David Butler and the Prisoner of Alhazen (2016)
- Quarter the Cross: Colouring (2020)
- Quarter the Cross: Connect the Dots (2020)
Some resources linked from this post:
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The MLC Date Blocks
This blog post is about a piece of the MLC learning environment which is very special to me: the date blocks. It’s a set of nine blocks that can be arranged each day to spell out the day of the week, the day number, and the month. I love changing them when I set up the MLC in the morning, so much so that since the face-to-face MLC closed due to COVID-19, I brought them home and have been changing them each morning here in the dining room. The story of how this object came into the MLC is the reason it is so special to me.
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One Hundred Factorial – the puzzle and the event
The weekly puzzle session that I run at the University of Adelaide is called One Hundred Factorial. In the middle of the night, I suddenly realised that I have never written about why it is called One Hundred Factorial, and so here is the story.
The very beginning
Once upon a time I was a PhD student in the School of Mathematical Sciences at the University of Adelaide. Sometime during the third year of my PhD program (2007), I was asked to give a talk to the first year undergraduate students as part of an evening event where the goal was to hopefully convince them to keep studying maths at a higher level next year. I titled my talk “How to Tell If You Are a Mathematician”. I don’t remember any of the things I spoke about, except for one thing. Before I started talking, I put a puzzle up on the document camera. I did not mention the puzzle in any way or look at the screen at all. I just did my little talk as if it wasn’t there. But right at the end of my talk I said this:
The final and truest way to tell that you are a mathematician is that you haven’t been listening to any of what I just said, and instead have been trying to solve this puzzle.
Cue guilty looks and nervous laughter from all of the academic staff in the audience, which successfully proved my point. Anyway it worked. Several students came up to me to talk about the puzzle, and I was able to direct them to lecturers who could talk to them about their study options. Yay for puzzles, right?!
This was the puzzle I used so neatly to make my point about the mathematician’s mind:
The number 100! (pronounced “one hundred factorial”) is the number you get when you multiply all the whole numbers from 1 to 100.
That is, 100! = 1×2×3×…×99×100.
When this number is calculated and written out in full, how many zeros are on the end?I don’t remember where I got the puzzle from, but it is a pretty famous one that’s been around for some time. I actually hadn’t even thought through a solution at the time either. I just knew that it mentioned a concept that had been in the first year lectures recently.
The puzzle sessions begin
The other thing that happened that night was that a group of students and staff stood at the blackboard in the School of Maths tea room to nut out a solution to the 100! puzzle. I can’t even remember if we finished it or not, but we did decide that we should get together regularly to solve puzzles together, and a weekly puzzle session was born. At the first session, we started with the 100! problem again, and an extension of it, which is to find out what the last digit is before all those zeros start. Then as the weeks went on, we would do puzzles that I would find and bring to the sessions.
When I finished my PhD in mid-2008 and took up the job in the Maths Learning Centre, I took my little puzzle session with me, and was able to invite more students to come along, and it slowly morphed into a student event more than a staff event, which really pleased me. In fact, a regular at these puzzle sessions for years was that first student who had come up to me after my talk at the first-year event, and he eventually became one of my tutors at the MLC.
The name of the event
Over the years the puzzle session has had many names. We started out calling ourselves “People with Problems”, and then simply “Puzzle Club”. For a while it was called “The Hmm… Sessions” after the sound we made very often while thinking about puzzles. Indeed, there is a reference to the Hmm Sessions inside this very blog. But in 2012 after the website where I was hosting our online discussion was decommissioned, I decided it was time to change the name. I was also starting to think about moving the sessions out of the MLC itself and into a public space, and to match with this move I wanted a new name. I thought long and hard, and decided to name it after the first puzzle we ever did, the puzzle that first inspired staff and students to talk and think about maths together, the puzzle that helped students decide they really were mathematicians after all.
The legacy
So the regular puzzle session of the MLC became One Hundred Factorial at the end of 2012, and here we are in 2020 still going, so that now it’s been One Hundred Factorial longer than it’s been any other name. It’s been my testing-ground for new puzzles and games and teaching ideas, a place where I have made friends and welcomed people from around the country and the world. And it has become a glowing island of mathematical play in the middle of the stressful university life, and indeed the middle of a stressful life generally. In recent weeks it is a glowing island of community in a world of pandemic-induced isolation.
One Hundred Factorial reminds us that there is always something joyful to think about if you are looking for it, and that it’s okay to pause and ignore your responsibilities for a while to think about it, and that doing this with people is a source of shared joy. I hope the puzzle and the event can keep reminding us of that for a long time yet.
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Playing SET
Amie Albrecht recently posted a most wonderful blog post about SET, and it reminded me there were some SET-related things I should post too.
You can read the two posts in this series in PDF form here.
The titles of the two posts are:
- Teaching people to play SET on the fly
- Team SET
Resource linked in the blog posts:
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The second part of the Four Fours
The four fours is a rather famous little puzzle that requires some creativity and also gets people thinking about how the operations interact with each other. One thing I find both frustrating and fascinating is what happens when people come up with numbers that are very hard to produce with the standard basic operations of addition, subtraction, multiplication and division. People seem to be focused on producing the results in any way they can, rather than asking whether it’s possible to produce the results. You also start getting solutions using All The Things, even though it’s totally possible to get the answer for some of them just using the most basic of operations.
So here’s the question: how do I arrange the Four Fours puzzle to make it more natural for people to consider what they can or can’t achieve using just the basic operations, and if new operations are allowed, how do I prevent it from becoming All The Things?
You can read the rest of this blog post, and two related blog posts, in PDF form here.
The titles of the three posts in the series are:
- Four alternatives to the four fours
- A day of maths: Zero Zeros
- The second part of the four fours
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The Number Dress-Up Party
I created the Number Dress-Up Party puzzle way back in 2017 and every so often I stumble across it again when searching Twitter for other stuff. When I stumbled across it today, I decided it was time to write it up in a blog post.
The puzzle goes like this:
The Number Dress-Up Party
All the numbers have come to a dress-up party in full costume. They all know themselves which costume everyone else is wearing, but you don’t know.
If you pick any two of them and ask them to combine with +, -, × or ÷, they will point out which costume is the correct answer, and they’ll happily do it as often as you want. For example, you might ask for robot + bear and they will point to unicorn. (If the answer isn’t at the party, they’ll tell you that too.)
How do you correctly identify the numbers 0, 1, 2 and 3? How do you do it in as few steps as possible?
(Photo from http://www.worldrecordacademy.com/mass/most_mascots_to_do_the_same_danc… )
It is worth clarifying now that when I say “all” numbers I originally meant it to be all the real numbers. (But it is very interesting to think about how the problem is different or even possible if it means all the rational numbers or all the integers or all the natural numbers or some other set of numbers entirely.) It’s also worth pointing out that it is actually possible for you to guarantee that you actually find these numbers – you shouldn’t have the possibility of having to go through all the infinitely many costumes to be sure of finding 0, for example.
It’s also worth clarifying that the rules say you have to ask two different costumes to combine with an operation. If you can see how using two of the same costume might help you identify actual numbers, then you are thinking along some helpful lines. However, the puzzle is much harder and much more interesting if you have to use two different costumes every time.
I love this puzzle so much! The reason I love it is that it forces you to think about numbers and algebra in a completely different way to any other puzzle or problem I have seen. You really need to think about how numbers are related to each other and how they behave under operations in order to figure out a way to correctly identify these numbers from the way the costumes interact.
Also, in order to tell someone about your solution, or even figure it out in the first place, you need to find a way of talking about or writing about this, which is a lot more difficult than it might seem at first glance. I find it really interesting to see how people attack the problem of describing what they are doing in this problem.
The feel of this puzzle to me is the feel of an abstract pure maths course like abstract algebra or number theory or real analysis, where you are digging deep into how numbers work without reference to specific numbers per se. I would love to go into such a course and use this in the first lecture/tute to get students in the right sort of mindset to attack the rest of the course. I’d also love to do an extension to this as an investigation into how the Euclidean Algorithm for natural number division works. To all those people who haven’t had that sort of training I say you are doing what a pure mathematician does when you think about this problem! You never thought it would be so much fun, did you?
Anyway, there is the Number Dress-Up Party puzzle for posterity. There are several different solutions and they are all lovely. Have fun!
PS: If you feel like seeing how people have thought about this problem, and are ok with spoilers, then check out the replies to this tweet .
These comments were left on the original blog post:
mau 4 February 2019:
I did not see much of a difference between real and rational numbers: at least, the solution I devised is the same (and no real number can be generated in a finite number of steps, unless we stumbled into a costume specially related to that number: but maybe I am wrong). Integer and natural numbers need a different approach, indeed.
David Butler 11 February 2019:
Yes I didn’t see much of a difference between the real and rational number parties. Or indeed the complex number party either. I think once they’re closed under all four of the operations we’re allowed to use, you don’t get much extra.
And yes I agree the natural numbers/integers are very interesting!