This game is designed for two players, or two teams.
Setting up:
Each player/team has two grids with the letters A to J, one labelled MINE and one labelled THEIRS, like the picture below.
Each player/team writes the ten digits 0 to 9, each in a separate box in the MINE grid, keeping the grid where the other player/team can’t see it. (I actually have a printable version with the rules on them that can be turned into battleships-style game stands here.)
The goal:
Each number has been disguised as a letter. You need to find out which number each of the other player/team’s letters is, by finding out what calculations with the letters produce.
Your turn:
On your turn, ask for the result of a calculation involving exactly two different letters and one of the operations +, -, ×, ÷. Some examples are A – B or C÷J or H+D or E×G.
The other player/team answers your question truthfully, either telling you which of their letters is the result, or telling you the result is not a letter. To be clear: players do not ever say a number in response to a question. They only ever say a letter or “not a letter”.
You can write notes to help you figure out what you know from the information you have so far.
Now it is the other player/team’s turn.
Ending the game:
Once during the game, instead of asking a calculation, you can say you are ready to guess. Then you say what number you think goes with every letter. The other player/team tells you if you are right or wrong.
If you are right for all letters, you win! If you are wrong for any letters, you lose and the other player/team wins! Either way the game is over.
Thoughts from Digit Disguises games I have played
This game was inspired by a puzzle I wrote called “The Number Dress-Up Party“. In that puzzle, all of the numbers are at a dress-up party and you have to find the identity of just a few of them by asking them to perform operations. I was reminded somehow of the puzzle and I was wondering about modifications of it. One thing I was wondering was if I had only a few numbers at the party, how long it would take to find out what all of them were, and so this game was born. Later that same day I played it across Twitter with a friend, on either side of a whiteboard at One Hundred Factorial, and with my daughter at home. I was then completely obsessed.
I learned a lot from these three early games.
In the first game with Benjamin on Twitter, I was struck by how quickly the logic got complicated, and then how quickly it all cascaded into finding everything when I finally got a few different numbers. It was interesting thinking about what it meant to get the response “not a letter”. I loved finding ways to keep track of the information I knew so far, and making sure I was using all the information I had so far. The presence of the 0 really made for an interesting ride on Benjamin’s side. (You can see the original twitter chain here or see screenshots of the whole chain in this PDF document.)
…
When I played at One Hundred Factorial, we played in teams on either side of a whiteboard and it was way more awesome! Firstly, it was heaps more fun to play in teams — talking through the logic so far and finding ways to represent it so the rest of the team understood was a really pleasurable experience. I loved hearing other people’s thoughts about what we knew so far and what we should do next. Not to mention having people notice when we had made a mistake in our logic. Secondly, having a huge space to write all of our reasoning was really nice. It was fascinating to see the other team’s approach, which was to have a big grid of which letter could be which number and slowly cross off the possibilities. I had never even considered doing something like that!
When I played with my daughter C (she’s just turned 11 years old), it was a whole different experience. C had no trouble answering my questions using the letter/number correspondence. In fact, she really enjoyed that part. (Indeed, when I offered to play with C the next day, she was happy just to be the keeper of which letter was what number and let me play Mastermind style.)
I found it rather fascinating that even though they haven’t really done algebra at school, it was perfectly natural for her to refer to these numbers by their letter disguises and to write stuff down on the page in terms of the letters. That is, she didn’t need to be taught about using letters for numbers to really get into the idea of the game. I think that is pretty awesome, actually!
What C was having trouble with was making sense of the information she was getting. She immediately realised that getting something like A-B = “not a letter” meant that A was smaller than B, but she didn’t really know what to do with the information that B-A=C. A little discussion helped her realise that this meant B had to be A+C and both A and C had to be smaller than B. But then it seemed like a huge task to find any letters!
It took a few tries to come up with a representation that was helpful, and a bar model really worked to make sense of it, and even allowed us to pull out information such as I=2F. It was still a little difficult for her to understand how knowing I=2F and I/F=C meant she could know that C=2.
So I reckon that for kids her age, definitely playing in teams is a good idea, or having the whole class be a team against a mastermind, or even the very first time with the teacher against the class as the mastermind with the teacher describing their thought process, to get an idea of the strategies involved. I’m also imagining a version much more like the original dress-up party puzzle where a collection of students are the numbers and they know which student is which number and the rest of the class have to figure out which student is which number by asking them to combine with operations.
Mathematical wonderings
The other thing that is so very awesome about Digit Disguises is how many mathematical wonderings flow out of it so very easily. On top of the usual things that come up during the game, such as what getting an answer of “not a letter” tells you about the numbers involved, you can go a long way wondering about the game as a whole and what happens if you change it.
I have wondered all of these things, but only investigated some of them. I won’t ruin the answers for you for the ones I have thought about already.
How many questions do you need to finish the game? Is there an algorithm that will finish in the smallest number of questions possible?
What if the goal was to correctly identify just one number? How quickly can you find the first number?
Which number is the easiest to find? Which number is the hardest?
Can you identify all the numbers using only one operation, such as only using subtraction? What is the smallest number of questions to finish the game if you’re only using each operation?
What if the operations were mod 10? You wouldn’t be able to use subtraction to tell you greater or less than in this case, but would it still be possible to find everything?
What if you had 0 to some other number, like 0 to 25 or 0 to 5. How long would the game be then? Would your strategy be any different?
What if you didn’t have 0 or 1? What if you had some negative numbers? What if you had some completely other collection of numbers?
Is it even possible to find all the numbers if you have a different collection? Are there collections of numbers any size you like where you can’t find any of them? Are there collections of numbers where you can find some but not others?
For the previous question, if there are only a small number of numbers (like two or three), what are all the sets of numbers that can all be identified?
Have fun!
I hope you like my game of Digit Disguises. I think it’s AWESOME! If you do play it yourself or with your students in a classroom, or you have thoughts about the answers to my wondering questions, or have anything you are wondering about yourself, please do let me know.
These comments were left on the original blog post.
Lisa Winer 22 September 2019
I am going to try this when we have a few minutes left in class! How long do you think it will take?
David Butler 22 September 2019
How exciting! At One Hundred Factorial it took us about 20 minutes per game. We were playing with teams of two and three.
James Cleveland 20 August 2020
I played this today with a student (rising 6th grade), as well as Numbers and Letters. Both went quite well! He was a rockstar at Numbers, and I was surprised at how often we could hit the target exactly, not just within 10.
For this one, I definitely feel like they picked up some things from asking questions (like they figured out where my 0 was), but agree that they didn’t know how to work through the other information given. But they were able to follow my logic as I talked out loud and solved theirs, so it was still a good time overall.
I have an algorithm that will solve a game in a max of 17 questions, but I’m sure there’s a better way – this one is pretty simple and just relying on finding the 1 using multiplication, then using that to determine all the rest using addition.
David Butler 20 August 2020
Thanks so much for sharing! It’s good to know your experience because I can only get so much from playing myself.
I think I remember at One Hundred Factorial we were made an algorithm that was guaranteed to finish in 14, using multiplication to find 1 and 0 first. Most games seem to finish in about 11 turns when I play.
Context fatigue is a particular kind of mental exhaustion that happens after having to make sense of multiple different contexts that maths/statistics is embedded in. I feel it regularly, but I feel it most strongly when I have spent a day helping medical students critically analyse the statistics presented in published journal articles.
The problem with maths in context is that the contexts themselves require understanding of their own in order for the maths to make sense. This is nowhere more true than in statistics, where you have to use your understanding of whether you expect the relationship to exist, what direction you expect it to be, and whether you think this is a good or bad thing. The classic one in my head is an old first year statistics assignment where they used linear regression to investigate the relationship between manatee deaths and powerboat registrations in each month in some southern coastal American city. You have to know what a manatee is, what it means to register a powerboat, and why those things might possibly be connected in order for the statistical analysis you’re asked to do to make sense, not least because at least one part of the question will ask you to interpret what it means. When helping students read published articles recently, I’ve had to find out what’s been done to the participants in the study, how things have been measured, what kind of measurements those are, why they’ve been measured that way, and all sorts of little details to decide how to interpret the numbers and graphs that are presented.
Even ordinary everyday word problems are a minefield. Across two recent assignments, some financial maths students had to cope with album sales for AC/DC, flooding of the land a factory is built on including insurance, bull and bear markets, machines in a mining operation, committees with various named positions, road testing electric cars, contraband being smuggled in shipping containers. This is a lot of context that has to be made sense of before you can get a handle on the maths, and there is nothing in the question itself to tell you what any of this context means if you don’t already know. Even if you are already familiar with the context, you actually have to suspend some of your understanding in order to do the maths problem, because it’s much simpler than the actual situation any of the questions are talking about.
All of this interpreting is exhausting stuff! It just tires you out if you have to even a moderate amount all at once. You just feel like you don’t have any more energy to deal with any more today. That feeling there is context fatigue. Yesterday the first year maths students were doing related rates and every question was a new context with little nuances created by the context that had to be dealt with. Those poor students were exhausted after just one problem, letalone three or four.
As teachers, we need to realise that as the people writing the assignment questions, or at least people who have dealt with them before, we are much more aware of the details and nuances of the context than the students are, so we don’t have to work so hard to make sense of them. Not only that but we’re usually simply more experienced in both life and language than most of our students so it’s easier for us. Imagine the context fatigue you would get reading ten research papers in an unfamiliar area in one day (I feel this in real life regularly). That’s the sort of context fatigue your students have just from your assignment questions. Cut them a little slack, and make sure there is adequate time to process the context with appropriate rest time between context-interpretation. Also it wouldn’t be the worst thing to explicitly teach them strategies for making sense of context, such as ignoring the goal, and finding out about what some of the words mean. Strategies can make the work less intimidating, especially in the face of knowing how tiring it is already!
PS: If you’re in charge of tutors in a drop-in support centre, especially one that deals with statistics, please be kind. Context fatigue is real and tends to wear us down some days!
I am often saying how important it is to listen to students, and that I am fascinated by student thoughts and feelings. When students say I am a good teacher my usual response is to say it’s because I have spent the last eleven years in a situation where I get to listen to lots of students.
But there is an important thing I almost never talk about, which is that sometimes listening is actually awful. I can think of not many more debilitating curses to lay upon people than to wish them the ability to listen.
Because listening is exhausting.
Because I am listening to students, I know an explanation doesn’t work, so I have to come up with new ones, usually on the fly. Because I am listening to student thinking, I come across new ways to think about all sorts of things that I had never considered before, which I then have to process. Because I am listening, I am faced with people’s feelings and stories, which I have to process emotionally. Because I am listening, I can easily become fascinated with new ideas and problems which take up my mind. Because I am listening, I hear things that need changing in teaching methods or university systems, and either try to work to change them or worry that I can’t. In short, because I am listening I am constantly processing information and emotions both in the moment and later on. It’s exhausting.
I don’t always cope well with it. In person with students I can just deal with who is in front of me and it’s ok, though there are times I need a break and just walk away for a few minutes. Unfortunately when I’m apart from the students, I can’t leave my brain behind and I carry with me the swirling thoughts in my head all day long caused by the listening to students. One way I have to cope with this is to talk with people about those thoughts, in person or on Twitter. But I actually can’t talk about all of them, so I have to choose one thing to think about and ignore everything else. There are times I have to say to students or my tutoring staff that actually no I can’t think about that right now, which is really really hard. And there are times I can manage to do an activity like origami or folding or watching tv to turn off my brain for a while to give it some rest. Still the call to listen is back again soon enough.
This isn’t a whinge session to get sympathy, it’s a warning. Be warned that if you choose to listen, you too will have to find ways to ignore some things, to find moments of brain-calm, and to find ways to process the thoughts you do choose to entertain.
Was my aim to scare you off? Certainly not! I wouldn’t ever give up listening and sacrifice the pleasure and learning I get from it, or the benefit it has for students. The blessing far outweighs the curse.
Three years ago, my university’s Student Engagement Community of Practice collectively wrote a series of blog posts about various aspects of student engagement. I thought I would reproduce my blog post here, since it is still as relevant today as then.
There is a lot that staff can do to engage students in the university community and in their learning, and a lot of these things have to do with the staff being engaged with the students. One way that any staff member can show their own level of engagement with the students is to learn the students’ names.
Names are important. Your name is a part of your identity, and not just because it is what you call yourself. Your name may tie you to the culture or the land of your ancestors, or it may speak of your special connection to those you love. You may prefer to be called by a different name than your official one because your chosen name is more meaningful to you. What all of these have in common is that your name is an important part of your identity.
For myself, my name is David, and I don’t like to be called Dave. I grew up in a community with several Davids and other people were called Dave, so being David kept my identity separate to theirs. Yet many people give me no choice and call me Dave without asking for my permission, despite me introducing myself as David. I find it intensely rude that someone would choose to call me by a different name than the one I introduce myself. On top of this, I am a twin, which means as a child I was forever being called by the wrong name entirely. We are not identical twins, and yet this still happened, because we were introduced as PaulandDavid, without an attempt to give us a separate identity. The fact that I was called Paul, or “one of the twins”, meant that I had no identity of my own separate to my brother. Being called David means that I have an identity of my own and this is important to me.
For many students, these and worse are their daily lives. Imagine a student who no-one at university knows their name. They have no identity at university, can feel very alone and can quickly disengage. Yet according to “The First Year Experience in Australian Universities” by Baik, Naylor and Arkoudis , only 60% of first year students are confident that a member of staff knows their name.
Not having your name known at all is one thing, but being called by the wrong name can be worse. An international student has to deal constantly with being different to other students, and in the community at large has to deal with a lot of everyday racism. To have your name declared “difficult to pronounce”, or to have it declared as not possible to remember, is just another one of these everyday racist events. The person doing so may not be meaning to be racist, but it adds up to the students’ feeling of not belonging, to their feeling that they themselves are not worth remembering. Similar to me and my twin brother (only worse), they may have the feeling that others believe all international students are the same, so why remember them separately. In “Teachers, please learn our names!: racial microagression and the K-12 classroom” by Rita Kholi and Daniel G Solórzano , there are many examples of the hurt that such treatment of student names can have.
So what can we do to learn our students’ names? Members of the Community of Practice suggested several strategies.
One idea is to spend time talking to them and ask them what they would like to be called. You can’t learn their names unless you find out what they are! Be visible in your effort to pronounce it correctly, be adamant that you want to call them by the name they ask to be called. If you get multiple chances to talk to them one-on-one, ask their name again if you can’t remember and try to use it as you talk to them.
Another idea is to print out photos of your students and to practice remembering their names. If you don’t have access to their photos, then it should not be hard to find someone nearby who can. (Though of course it would be excellent if there were a simple system whereby anyone teaching a class — including sessional staff — could get photos of their students!) Even if you can’t get their photos, simply working your way down the roll and remembering how to pronounce those names, or what the students’ actual preferred names are, is good exercise. The students are likely to appreciate the effort you put in here, even if they can’t know how much time you did put in!
You may have your own ideas on how we can make sure we know students’ names. I’d encourage you to share them in the comments, along with any stories of how it made a difference to student engagement.
I would like to work in a university where 100% of the students are confident that someone knows their name. We have hundreds (possibly thousands) of staff in contact with students on a regular basis. If each of us only learns a tutorial-worth of names, then we can surely meet that goal easily!
I firmly believe that all students deserve to play with mathematical ideas, and that extension is not just for the fast or “gifted” students. I also believe that you don’t necessarily need specially designed extension activities to do exploration – a simple “what if” question can easily launch a standard textbook exercise into an exploration.
This is lovely, but one problem is those students who on the face of it don’t want to play. The majority of students I work with in the MLC are not studying maths for maths’s sake, but because it is a required part of their wider degree. That is, I am helping students who are studying maths for engineering, or calculations for nursing or statistics for psychology. A lot of these students just want to be told what to do and get the maths over and done and don’t like “wasting” time playing with the ideas.
Or so I thought. I have realised recently that actually they do like playing with the ideas. I just couldn’t see that this was what they were asking for.
One of the questions I like to use to play with maths is “what if?” To ask it requires the asker to notice a feature they think might be different, and wonder what would happen if it was, and so investigate how this feature interacts with the other features of the situation. This investigation of how ideas interact is exactly what mathematics is, to me. I am very very used to doing it at extracurricular activities like One Hundred Factorial, and it’s easy to do with students who are doing very well with their maths and have the breathing space to wonder about this stuff because they’ve finished their work. For students working on an assignment they are really struggling with that they wonder about the usefulness of for their degree and which is due in the next 24 hours, it’s not so easy.
Only maybe it’s a little easier than I thought, because I started to notice that the students were already asking questions about the connections between things.
A very common question students ask around exam time is “What would you do if the question was like this …?” as they suggest a change in a small detail that makes it more similar to the past exam question they have at hand. I used to get really annoyed at this sort of question, but then I realised that in order to ask this question they had already noticed that the two problems were similar in most respects and different in this one. Sure, they may be motivated by a belief that success in maths is about a big list of slightly different problems and remembering ways to deal with each, but on the other hand they have noticed a relationship and are trying to exploit it, This is a mathematical kind of thought, to look for similarities and differences and relationships, and we can hang on tight to it and actually learn something!
Another kind of question students ask is “Why is this here in this course?”. I used to get annoyed at the whinging tone of voice here, but then I realised that a student is begging for a connection between this and the rest of what they are studying, and connections is precisely what understanding maths is all about. I can respond to it by saying that yes I am also confused about the curriculum writer’s logic for including it, but then we can search out connections together to see if we can find them.
A third kind of question is the one where the student looks at the lecture notes or solutions and asks “How did they know to do this?”. Sure, it’s usually motivated by wanting to be able to successfully do it in their own exam in restricted time, but on the other hand it does recognise that there ought to be reasoning involved in that decision, as opposed to just guessing. This expectation of reasoning is the beginning of believing that they themselves could reason it out too.
My second-last kind of question is “Why is this wrong?” as the student points to the big red cross on their MapleTA problem on the screen. Sure they just want to make it all better so they can submit the damn thing. But they are also recognising that there must be a reason why it’s wrong, and so are looking for meaning. These students are often ripe for the experience of looking at the information and how it’s related to what they’ve entered, thus doing exploration. They are also usually ready to try various different ways of entering the result, or different strategies of getting a right answer to see where the edges of the idea are.
The final kind of question I want to mention isn’t even a question, it’s an exclamation of “It doesn’t make sense!” Even this is telling me that the student thinks it ought to make sense. They are crying out to do something to make sense of it. Often they describe being almost there and needing something to push them over into understanding. This student is ready to explore the edges of the understanding they do have to see where the nonsensical stuff can fit, or where their ideas need some tweaking to fit together better.
It is very helpful to me as a one-on-one or small group teacher, or even a lecturer with a big room of students in question time, to be able to see the questions struggling students are asking as cries for sense-making exploration. It doesn’t matter that they are struggling or don’t understand things. Indeed, being in a situation of not knowing is exactly what we are doing for the fast students when we give them extension activities, so why is the everyday maths the slower students are not understanding any different? In my experience students like being treated as if they are behaving like mathematicians when they have these struggling sorts of questions. And it was so much easier to treat them that way after I realised they are behaving like mathematicians when they have these kinds of questions.
Once upon a time, I decided I would be vulnerable on Twitter. As part of that, when someone posted a puzzle that I was interested in, I decided that I would not wait until I had a complete answer to a problem before I responded, but instead I would tweet out my partial thinking. If there were mistakes I would leave them there and respond with how I resolved them, rather than deleting them and removing the evidence that I had made a mistake. I wanted the whole process of solving problems to be out there in plain sight for everyone to see.
One reason I wanted to do this was because too often we give the impression that maths is all so slick and easy, when really it is actually messy and hard. This issue is exacerbated by the nature of puzzles, which often have clever simple solutions, and when people share only that, we give this impression that maths is done by some sort of divine inspiration of the clever trick. I wanted my responses to problems to show the honest process of muddling through things, and getting them wrong, and only seeing the clever idea after I’ve done it the unclever way, and even if a clever way did occur to me, what it was that made me think of it.
Over time, this has grown into me doing epically long tweet threads where I live-tweet my whole solution process, including all the extra bits I ask myself that are sideways from or beyond the original goal of the problem. I have to say I really really love doing this. I love puzzles and learning new things, and it’s nice to do it and count it as part of my job. I also get to have a record of the whole process for posterity to look back on. Even in the moment, it’s very interesting to have to force myself to really slow down and be aware of my own thought processes enough to tweet them as I go. In my daily work I have to help other people to think, so the sort of self-awareness that this livetweeting of my thought process brings is very useful to me, not to mention fascinating.
In the beginning, I had some trouble with people seeing my rhetorical questions as cries for help and jumping in to try to tell me what to do. I also had people tell me off for not presenting the most elegant solution, saying “Wouldn’t this be easier?” Every so often I still get that happening, but it’s rarer now that I have started to become known for this sort of thing. I am extremely grateful that a few people seem to really love it when I do this, especially John Rowe , who by rights could be really annoyed at me filling up his notifications with hundreds of tweets with a high frequency of “hmm”.
I plan to continue to do this, and so if you want to keep an eye out for it, then I’ll happily let you come watch. From this point forward I will use the hashtag #trymathslive when I first begin one of these live problem-solving sessions, so I can find them later and you can see when I’m about to do it. (I had a lot of trouble looking for these tweet threads, until I remembered that a word I use often when I am doing this is “try” and I searched for my own handle and “try”. In fact, that’s the inspiration for the hashtag I plan to use.)
Note that I am actually happy for people to join in with me – it doesn’t have to be a solitary activity. I would prefer it if you also didn’t know the solution to the problem at the point when you joined in, but were also being authentic in your sharing of your actual half-formed thoughts. I definitely don’t want people telling me their pre-existing answers or trying to push me in certain directions. If you feel like you must “help” then, asking me why I did something, or what things I am noticing about some aspect of the problem or my own solution are useful ways to help me push my own thinking along whatever path it is going rather than making my thinking conform to someone else’s. I will attempt to do the same for you.
To finish off, here are a LOT of these live trying maths sessions.
UPDATE: With the demise of the Twitter maths community, I’m now on BlueSky, but I still use the #TryMathsLive hashtag there too.
Sometime in the past, I was approached by academics in the Faculty of Arts to discuss the numeracy skills of the students in their faculty. They wanted to discuss how they might include numeracy skills in some of their courses across all the degrees they teach. It was a lot bigger than the MLC could reasonably do, but I said I would certainly be able to do a small thing in a few courses, and certainly help their students in the MLC itself when they came to talk.
Then in January 2019, almost out of the blue, I was sitting down at a meeting with the Faculty of Arts Associate Dean Learning & Teaching, and the course coordinator for their core first year course called “The Enquiring Mind”. We were talking about how I might run a workshop for their students to introduce the importance to numerical skills for Arts students. We agreed on Week 4 of semester, and then I walked away into the Summer School exam period and O’Week and the crazy beginning-of-semester rush, with ideas percolating in the back of my mind for what I could possibly do in an hour.
As the weeks went on, ideas began to crystalise, but only really came together during the week before, and specifically in the weekEND before. After a late night Sunday night with a few hours put in by my wonderful wife with the printing and laminating, we were ready, and we launched into the tutorials the next day.
It’s been an interesting week. Across the week, we did 19 tutorials, with me (David) doing 14 of them, Nicholas doing four and Ben doing one. (Meanwhile at home I had parent-teacher interviews for both daughters, and one of our pet chickens died, and we were preparing for our older daughter’s 15th birthday.) We had some very rude pushback from some of the regular tutors and the students, and some mild eye-rolling from several students in every tute. But some of the other tutors raved about what we did, and I could see students taking photos of the slides and learning terminology, and they all certainly know they can talk to the MLC. Overall I think it went okay, but it still might not be quite the thing the students needed.
The activities
Anyway, let me tell you what we did. I didn’t want to swoop in and tell these students that they needed maths skills and beat them with some activities that would teach them those skills, especially when several of them probably didn’t need us to teach them those skills and others would just have paralysing fear. I very much wanted them to hear a message of confidence, so I decided to tell them that the very thing they need to make sense of number is the same skills they use for their Arts courses. I used a couple of things from the #MTBoS and designed a couple of my own things to tell a story around this message.
I asked the students to pick something they can say “how many” about and to write just the number on the board. When I first started, I just asked them “How many?” but students universally thought it was a puzzle and all figured out the number of lemon slices, as opposed to getting a number of different responses. So I changed it to highlight that they could count many things.
After getting several different responses, I asked students who had written them to tell me what the number was counting, sometimes asking how they counted them. My favourite response of the week was “12 lemon-diameters around the edge of the big square”. But I did love just as much the variety of things that were counted, such as seeds, artists, the letter e, likes, empty spaces and cups.
The point of the activity was that numbers are meaningless without knowing what they are counting or measuring. This might seem like it’s a really basic message, but it really isn’t. These students will need to be interpreting information presented in articles and government reports and there are many a number in there that simply won’t make any sense without knowing how it is measured. For example, if a report mentions “reading ability” with a score, you have no idea what that means unless you know how it was measured.
I didn’t give this specific “real world” example until a couple of days in, when we got an irate email from a student saying the activities weren’t applicable to his degree. So I went into the reading list for the course’s assignment and specifically chose something that was mentioned there. I also started explicitly mentioning how the tutorial will continue in the same fashion, with activities that are a little frivolous, but will help us to think about concepts we can later apply to things more serious. This seemed to get students on board a little more than the very first tutes.
2. Big, small, middle
The second activity was one I designed myself, and its point was to highlight that while knowing what a number measures, it is still not enough information to interpret what the number means. I tried it out on Twitter a few days beforehand to see how it might work, and people seemed to like it, so I decided to include it.
I put the number “100 people” up on the slide, and asked the students to discuss among themselves in what situations they might hear this number and think it was a big number, what situations it would be a small number, and what situations it would be a number in the middle. After they had discussed for a bit, I took suggestions from each group and wrote them on the board. There were some good suggestions and some pretty silly ones like “100 people is big for people in an elevator”. It wasn’t until the second tutorial that I started getting deeper into that and saying, “Do you think it’s possible for that many people to be in an elevator?” (The consensus is that it might be possible if you lie them down and stack them.) Anyway, answers like this allowed me to make the point that it might be better to give answers that are possible, but still big or small.
Now that they knew how the game worked, I gave them a couple more to discus (2 kg and 5 hours) for a few minutes, and afterwards I took answers from the groups that they thought were the “best” by whatever method of choosing best they liked. Several tutes mentioned that 2kg would be a lot of drugs to find, and several also said 2kg would be big/small/usual for a newborn baby. This particular one we googled a couple of times to see where it did actually lie.
The overall point was that even if you know what a number is measuring, it’s only meaningful by comparison. For the purposes of their Arts courses, this means that if they want to interpret a number in an article or report, they might need to do some research to find out what the other possibilities are for what it could be, in order to know whether it is big, small or in the middle. (The newborn baby helped with this point!)
This was the point where a few people in the tutorials really looked like they were listening, probably because I had made a point specifically about their process for attacking assignments in their degree!
Before I moved onto the next resource, I showed them the website Is That a Big Number? which allows you to put in a number with units and it will tell you how big it is relative to big things (for example it will tell you that 500kg is just a bit lighter than a thoroughbred race horse). [UPDATE: Unfortunately in 2026 it seems this website no longer exists.]
3. The Squeetles
The people who ran the course wanted something to get students thinking about numerical summaries like means and percentages, and I had had a lot of success in other courses with doing that using data cards, so I set about creating a set of data cards I could use to do this with. I do have a set of cards I use in the sciences and health sciences about transport use and weight and height, but knowing the kind of students I was working with, I was afraid of the philosophical details getting too much in the way of the mathematical point, so I wanted to create something that was a bit more frivolous. It wasn’t until quite close to the classes that I hit upon the idea of making little animals, which came to be called squeetles after a discussion with my wife and kids.
I also needed to make something that allowed me to form groups of students easily, without it being really fiddly or the point being messed up because we didn’t use all the data from a specific group. This meant the numbers and categories had to be carefully spread across the cards. All of these constraints were tough, but in many ways a fun little puzzle to solve, though a bit stressful doing the final design, print and laminate on the night before the first tutorial!
Anyway, I got them made in time. Each squeetle has either spots or stripes, and is shown jumping to a specific height. You can download the squeetles data cards here, if you want them. They are designed to hand out to a large class and form groups of size about 12 using the numbers in each corner. It doesn’t matter if the groups aren’t exactly 12 each — they’ll work with a bit of wiggle between groups. Also it doesn’t matter if you only have two or three groups, using only 24ish or 36ish of the cards, though it’s nicer with at least three groups.
I got them to do several activities with the squeetles cards, to get at the idea of trying to represent a group using numbers like averages and percentages.
3.1 Choosing a representative number
First, I asked them within their big group to choose one squeetle from among the group who they think best represents the whole group’s jumping height. This was very interesting. A couple of groups across the week decided that they should choose the highest-jumping squeetle, thinking that in a jumping competition they would win. A couple of groups across the week decided that they should choose the lowest-jumping squeetle, because that would mean if a decision was made based on squeetle jumping ability, then all squeetles would be included. These did highlight that the word “representative” is not actually meaningful without knowing the context, which played into our overall theme. Unfortunately most of the groups that chose these were more or less deliberately trying to come up with things that threw a spanner in the works, and were a bit miffed when I asked them to consider how they might choose a representative if you didn’t know whether jumping high was good or bad.
Anyway, the majority of groups chose a representative that was in the middle in some way. Most groups either chose the mean or the median, going to great effort to do the mean calculation when they did. A few groups found the highest and lowest jumping squeetle and chose the squeetle closest to halfway between them. Whatever they chose, we talked through why they made that choice. Interestingly the ones who chose the mean all said it’s representative because it includes all of the people in the calculation, but didn’t mention anything about it being the middle. We talked through how the median still needs the whole group to exist, and also talked through why you might expect the mean to be in the middle.
3.2 Comparing the representative number to the individuals
After this discussion, we did some walking activities to see how averages are used to compare groups. I had laminated a set of numbers, which I used to set up an axis on the floor and asked the groups’ representatives to come and stand near their number. In a couple of classes there wasn’t room to do this in the classroom, so we went outside into the hallway, which they thought was pretty cool. (You can get my axis labels here.) I asked the whole class what they could say about the groups based on their representatives, and we agreed that in some sense this group jumps higher than this group, but these groups were about the same.
Then I picked two groups whose representatives were furthest apart, and asked all the squeetles to come and stand next to their number, one group on one side of the line and one group on the other. There was definitely not enough room for them, so I had to move them outwards a bit to fit in, turning into something like a histogram. The other groups were asked to observe and comment on what they see. I asked the representatives to put up their hands so we could see where they were. The final conclusion was that yes one group was generally higher than the other, but that this wasn’t true of every person in every group. There were a lot of squeetles in the lower group that were higher than squeetles in the higher group, so averages do help to compare groups, but don’t help to compare individuals.
3.3 Choosing a representative category
I asked them to go back to their groups and this time focus on the patterns. They needed to choose a squeetle which was most representative of the whole group’s pattern, as opposed to number. In some groups, there was a clear majority, and those groups always chose the majority, but in other groups it was more even (or exactly even) and there was some argument in those groups as to who they should choose. In those classes where a group had a lot of trouble choosing, I stopped them and asked them to share with the rest of the class why they were having trouble. Whether or not this happened, I always asked all the groups how they chose their representative, and they always said it was the majority (and sometimes how they chose exactly which person from the majority would be it, such as “because they went last time”, or “because they jump the highest”, or “because they are the only Saggitarius”.) I’d say that was pretty logical, and then ask the people in one of the groups with the other pattern to put their hands up, asking them to look how many people would be left out by choosing one category. In some classes, a student would say it doesn’t really matter, and I would say sarcastically, “Ok, well only 3% of Australians are of Aboriginal heritage. We won’t worry about them never being represented because they aren’t in the majority.” And then I’d remind them that the frivolous helps us to think about serious things.
3.4 Percentages help to keep the whole group in mind
After this little bit of seriousness, I got everyone up to organise themselves based on category. I put pairs of spots and stripes markers (which you can find in the axis labels file) on the floor and asked the groups to line up next to the markers. I asked the students to tell me what they noticed and they said it was much more obvious the similarities and differences between the groups, and everyone was represented. I told them that usually you’d use percentages to present this information, to tell about what’s going on in the whole group, because there is no such thing as a representative category.
4. The Three Reads
At this point I summarised what we had seen so far:
You have to know what a number measures/counts for it to be meaningful.
Numbers are only meaningful by comparison.
Averages help to compare numbers between groups, but don’t necessarily represent individuals.
Percentages help to compare groups in terms of more or less in a category.
That is: THE STORY IS WHAT MAKES THE NUMBERS MEANINGFUL
And then I had one activity left to do.
When designing the tutorial, I really wanted to do something that drew all of this together, and gave students a strategy they could use when they came to face numerical information during the course of their degrees. There is a lot of stuff across the #MTBoS about sensemaking and I hoped to use something I had seen there. My first thought was the “How old is the shepherd” problem, as described by Tracy Zager in her book (I wrote a review of this book here) and described in a video by Robert Kaplinsky here. In the end I decided not to use it because I didn’t want to make any of my maths-anxious students feel stupid, or begin a conversation about making sense by presenting something that doesn’t make sense. I was also inspired by the numberless word problems championed by Brian Bushart, which are designed for making sense of context before numbers, but his routine involves rewriting problems from scratch so that the numberless version is the only thing the kids see, whereas I wanted to teach my students to be able to make sense of situations where they had an existing piece of writing and no choice but for there to be numbers in it. And then I heard about a routine called “The Three Reads”, which seemed to be the right thing if I gave it a bit of a tweak. Only later did I hear that there are multiple versions of the Three Reads, with differing usefulness beyond school word problems. Below you will see my version.
So I was finally in the tutorial, and after the squeetles, I told them I wanted to show them a strategy for making sense of writing that has numerical information, and one type of writing with numerical information that they have all seen is the classic word problem from a maths class at school! I revealed the problem for only a few seconds, and then we worked through it according to the Three Reads routine.
Read 1 is about the story. In this part of the routine, we are looking to understand what’s happening. I revealed the first sentence, and asked them what their thoughts were about it. (In later tutes, I said that I knew the story was pretty stupid, so they couldn’t mention that!)
In most classes, the students mentioned that an online poll is not necessarily representative, which led me to ask what group of people you would hope they are representative of, which meant that we needed to think about who would like to know this information. Students mostly thought squeetle breeders or pet shop owners would like to know this information. Another option was just a person who wanted to buy a squeetle and didn’t want to be shamed by friends or strangers for buying the wrong type. I also asked them what other information they might like to know, and apart from wanting to know more about who and why the poll is being run, they also wanted to know what the results were.
After this discussion, I revealed the second sentence with the numbers covered. I asked them to think along the lines of our earlier activities and think about what would be a big, middle or small number for the number of respondents or the percentage. There were various responses for at what point they would think it was enough people to be representative, but this was tempered by the knowledge that squeetles are apparently a “craze”, so you’d want there to be a lot of people. For the percentage, people mostly said not until 80% would it make them commit to a specific type of squeetle as a breeder/seller. For squeetle-shaming, they thought a higher percentage would be ok, because then there would still be some people who would agree with them. I loved this discussion. It was so powerful to see them think about how they would interpret the numbers before they actually arrived.
One shocking moment in the earlier tutes was when one student said it didn’t matter who thought they were cuter because they were all ugly. The rest of the 40 or so students collectively gasped at this shocking statement. I’m glad my little creatures had endeared themselves to the students so much. It did also bring up an important point that the survey won’t tell us whether people think the squeetles are necessarily cute per se, just which people think are cuter.
Read 2 is about the details, in this case, the numbers themselves.
Now that we had the numbers, we could easily decide what to do with them, because we had already thought about it. It was awesome to have people so easily say that maybe it wasn’t the craze the writer said it was, and also that 60% wasn’t nearly far enough from half to make you commit to anything. I pointed this out here, that you were ready to make your own conclusion about the numbers here and weren’t swayed by whatever the writer might have been wanting you to think. Humans are easily swayed by numbers and we’ve shortcutted that by waiting to look at the numbers until we thought about the context first.
Before we moved on to Read 3, I asked the students to discuss what we could figure out from this information. I took multiple responses and multiple ways of figuring them out. People variously came up with 40% said striped squeetles were cuter, 150 people said spotted squeetles were cuter, and 100 people said striped squeetles were cuter. The 150 people was done in multiple different ways in each tute, and I celebrated all of them, saying that we love all student thinking in the Maths Learning Centre and that the key is the reasoning. In most tutorials, at least one student piped up and said we don’t actually know whether there was a “neither is cuter” option, so we can’t be sure it was actually a whole 100 people who said that striped squeetles were cuter. I loved that, and pretended it was a new idea in most of the tutes that people brought it up.
Read 3 is about the goal. I revealed the final sentence, and lo and behold we had already calculated it. In fact, we even had a better answer than the original question-writer had in mind, which is that there are up to 100 people who thought striped squeetles are cuter, depending on what the other options in the poll were. I noted how easy it was for us to just do this, since we had already thought deeply about what was going on before we even read the goal. I also commented that without the earlier focus on the story and the details, the goal wouldn’t have made any sense. If we had started with the goal we would have been playing catch-up the whole time, going back and forth to fill in things we needed for it to make sense and probably getting confused in the process.
At this point I made a big deal of how you can apply this strategy to anything that has numbers in it. When you read a government report or a newspaper article, you can make sure you understand the context, and you can look up the general sizes of the numbers involved before you read what they say about them, so that you can have your own opinion and decide your own conclusions before you read the writer’s conclusions. Then you are able to evaluate whether you agree with their conclusion or not.
I also said that the Three Reads is a useful strategy for reading anything, even if there are no numbers involved. For example, a History essay question. If your History essay asks you to discuss a particular figure in History, then you can think about the general story of the historical period and what’s going on, and then think about the details of the particular person and think about what things you could say about them, and then you can read the actual goal of the essay. By the time you’ve done all that thinking, the goal will make sense and you will probably already have an outline for your essay. Many students really liked that particular idea.
And this brought us to the end of the tute. I emphasised again that it was their Arts skills of understanding context and story that help them to make sense of number, and if they want to fill in any of the details about specific things, then they could always come and talk to the Maths Learning Centre. And I walked away, usually into the next tutorial where I would do it all again with a new set of students.
Reflection
It was a rollercoaster of a week. The regular tutors ranged in attitude from Tigger-like enthusiasm to open hostility at my presence, and ranged in participation from joining in with the activities and helping to draw connections to the rest of the coursework to actually leaving the room to avoid participating entirely. The students generally got into it, with various degrees of apathy, but generally I got the feeling they weren’t sure what to do with these weird activities so different to the usual methods of just chatting. One student emailed the executive dean of the faculty to complain about me doing patronising “primary school activities” with them (but I later learned this was not unique to my week of class). Still to be fair to that student, I think many students in this course would have liked it to be more boring in the sense of feeling like they got work done.
The original plan was to have the second half of the tutorial be about applying the ideas they had seen to their actual course materials, but due to me not being ready until the weekend before, and also the course being basically written as it was being taught anyway, this didn’t work out. My experience tells me we really need to make sure this happens in future semesters. In the future, I think I will likely create a whole new set of activities about making meaning of a real article or even just a graph (like I do with the Health Science students), though I reckon I can still focus on the Three Reads because they did find that useful for their Arts stuff too.
When all is said and done, I am still proud of the activities I made, and I hope to use them again, but maybe with a different cohort, or at a different time, or with a bit of tweaking.
Resources
To finish off, I thought I would collect together the resources if you want a closer look at them.
These comments were left on the original blog post:
Amy Zimmer 27 June 2019:
Brilliant David! So much good discussion! (Sorry for the booger students). I love part 2, and am totall inspired by the Squeetles discussion. I applaud you for keeping the discussion moving. I will have to reread all the maths in part 3 to get a better picture if what the students were seeing when they lined up on the axes. A HUGE thank you for all the time but into writing about this project. Yours, Amy
David Butler 5 July 2019:
Thank YOU for the encouraging words Amy, and the encouragement while I was planning this in the first place. I really appreciate it.
While I am thinking about SET, it is high time I wrote about a version of the game SET that was invented at One Hundred Factorial back in 2017, but has never been recorded anywhere for posterity. It is prosaically named Team SET.
In case you don’t yet know, the game SET is a game of visual perception where from an array of twelve cards, each with various attributes, you need to find three with a certain condition, which form what is called a SET. A good place to start to learn more is at Amie Albrecht’s excellent blog post about SET and you can find out how I teach people to play SET in my previous blog post.
Usually in SET, it’s everyone for themselves and you call out “SET!” as soon as you see one, thus claiming the SET for yourself. There are times when I decide I don’t want to get SETs myself (for example, when I’m helping new people learn how to play and want to give them some success finding SETS for themselves), so instead of claiming them for myself, I just tell everyone I can see one. If they can’t find it, I tell them one card involved in the SET to help narrow the search. One day in 2017 I was doing this at One Hundred Factorial while we were playing with the giant SET cards and other people joined in, each saying in turn they could see the SET until everyone had found it, and I had a rather wonderful idea: What if this was the actual game?! What if we played in teams and you only claimed the SET when all of the team members had seen it? And so, Team SET was born!
We had two teams of three people, each standing on opposite sides of the table. The team only claims a SET when each player on the team puts a hand on a different card in the SET. Players on the same team aren’t supposed to tell each other what cards to put their hands on, trusting their team-mates to find the SET themselves. Of course, once two cards in the SET have been claimed, there is only one card that could complete the SET and it’s a race for both teams to claim it. Competitive games can turn me off sometimes, but this one I found extremely fun!
I’ve played this game several times over the last couple of years, and it’s really very fun every time. I love watching people get all excited about the game, and the frenzied suspense created as people wait for their team members to find the set they can so clearly see. One thing that makes it fun is simply the reaching across the table making it rather like Twister. (To make it even more Twister-like, we did try once putting the cards on the floor and standing on them, but an excited run to claim a card resulted in a pretty spectacular slip, so we never did try that again!) I’m pretty sure the game wouldn’t work with ordinary-sized SET cards, because it wouldn’t be possible to see what the cards were if people’s hands were on them, especially through the tangle of arms once some cards had been claimed already! On that note, here is a PDF to print the giant cards for yourself .
To sum up, here’s the rules again:
Choose two teams of three people and stand teams on opposite sides of the table.
Deal out 12 SET cards in a grid.
Players look for SETs and when they see one, they put one hand on one card of the SET.
Each player is only allowed to put one hand on one card at a time. You can move your hand to a different card, but can’t touch two at once.
Two players from opposite teams are allowed to touch the same card.
Players are not allowed to talk to each other about which cards to touch.
The first team to have their three players touching the three cards of a SET claims the SET.
Once a SET is claimed, deal out three new cards. If everyone agrees there is no SET, or can’t be bothered looking for one anymore, deal out three new cards.
The game ends when no more cards can be dealt and no SETS can be found.
The winning team is the one with the most SETs.
And to finish, here’s some videos of a game in action at a recent Adelaide University Maths Student club games night, just so you can see how much fun it really is. (Click on the images to be taken to the video on YouTube.)
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.
The first is this little reflection on how I go about teaching people to play SET. Amie talks here about a very excellent way to do this, which is to get people to look at the whole deck of cards and organise them all to see if they can be sure they have them all, then notice the patterns there. But what do you do if you’re partway through a game in a public place and someone comes over to ask what you’re doing? Well, this is how I go about explaining it in that situation…
Imagine we have a game of SET going on the table, which means there are twelve cards laid out face up on the table and a group of people looking at them.
(You may notice these are not the same as the usual SET cards. For a start, they are much bigger, and secondly the colours are different. The size is because I wanted cards you could see from a long way away. The colours are because the original cards are red, green and purple, which are not good for colourblind people, so I chose primary colours in different tones. Also I have the attributes written on each card. If you want my versions of the SET cards for printing, you can find them here. )
Someone walks over and asks what we’re doing. I say, “We’re playing a game called SET, would you like to join in?” They say, “But I don’t know how to play.” And I reply, “I can teach you. Have a look at the cards and tell me what you notice.”
People almost always notice that there are several shapes or colours, and very rarely mention the patterns or number of shapes. Some people notice that there are twelve cards in rows. Whatever people notice, I ask them if they notice anything else before moving forward, and within the first couple of things they say, they always mention some attribute the cards have. Then I will say something like, “You said you noticed different colours. How many colours do you see?” I’ll do this for each attribute they have noticed, and they will notice three types of each.
Then I’ll ask them what else they notice, but I’ll be more specific this time. I’ll say, “You noticed that there were different shapes and colours. What else makes cards different from each other?” At this point they’ll start to notice the patterns and number of shapes. (I find it very interesting that people usually don’t talk about these until you specifically mention comparing cards to each other. The number of symbols per card in particular is very rare for people to mention without explicit prompting.) And I’ll ask them how many options there are for those too.
Now I sum up where we are up to: “So you’ve noticed there are three different colours, three different shapes, three different patterns, and three different numbers. That’s important to how the game works, and actually every combination of those is somewhere in the deck of cards.”
Now we are ready to move on to how the game works. I will pick up two cards and ask them to compare the cards. That is, I’ll ask what’s similar and what’s different about the cards. They’ll tell me, for example, that they’re both blue, they both have two, they’re different colours, and different patterns.
Now I ask, “If you had to pick a third card to go with these two, what would you pick? It may or may not be on the table — just describe what the card would be.” People invariably always pick the right card to complete the set. I ask them why they chose that card and they’ll say something like, “Well those two were both blue, so I thought I should choose blue, but they were different shading, so I chose the missing shading we didn’t have yet.”
At which point I say, “That’s exactly what a SET is. When you pick two, the third one has to match the things that match or be the third option for the things that are different.” We try it with some other pairs with different levels of similarity and difference so that they can get the idea.
And only now do I explain how the game works. I explain how we will all stare at the twelve cards, and call out SET when we see one, and whoever sees it first will claim it. Then I invite them to join in.
While the rest of us play, I always make sure that anyone who claims a SET explains why it is a SET to the group, for the benefit of the newcomers. We also explain why things that aren’t SETs aren’t. This helps to solidify the rules as the game progresses.
Note the usual rule of a SET where each attribute in turn is either all same or all different doesn’t seem to work so well for on-the-fly teaching, whereas the rule I said earlier that “the third card has to match the things that match and be the third option for things that are different” seems to make a lot more sense to people. The traditional rule does tend to emerge naturally as the game progresses, interestingly.
The reason I go through this process is that just trying to explain what a SET is to people almost never works. This method always ends up with people constructing for themselves how a SET must be, without me having to explain it. If I tell them about the attributes and tell them the rule for being a SET, they always have to keep clarifying over and over. This method is based on their own noticing and reasoning and so it just sticks better sooner.
Let me summarise my method again:
Ask people what they notice until they mention at least one attribute of the cards like colour or shape.
Ask how many options there are for those attributes.
Ask what else makes the cards similar or different, so that the other attributes appear.
Ask how many options there are for those attributes.
Summarise the options for the four attributes and tell them that every combination appears in the deck.
Pick two cards and ask what the third card might be that completes the set.
Ask why they chose that card, and confirm that yes this is how a SET is made. “If something matches the third card has to match too, if something is different, the third card has to have the third option.”
Explain how to claim a SET.
Make sure players explain why things are or are not SETs as the game is played.
There you go. I hope this is useful to you for those times when you need to explain SET on the fly. It happens to me a lot, but then I deliberately put myself in such situations by bringing this game out at One Hundred Factorial or at games nights or orientation activities!
The four fours is a rather famous little puzzle which goes something like this:
Using exactly four of the number 4 each time, write calculations using +, -, ×, ÷ that produce each of the natural numbers from 0 to 20.
It’s a classic 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. The majority of people start branching out into other operations like concatenation (writing 44), decimal points (writing 4.4 and .4), factorials (4! = 24), and square roots (√4=2). Basically anything that doesn’t require the use of another digit tends to be fair game. Indeed, some presentations of the Four Fours puzzle explicitly allow these operations from the outset, and I have been told off for “forgetting” to include them when I present the puzzle myself.
The reason I find this fascinating is that nobody ever seems to ask the question that to me is the most natural question to ask about the Four Fours. The first question I asked myself when I was first presented with the Four Fours was “are they all possible with just +, -, ×, ÷?” However, I have yet to meet any students who ask this question or try to figure it out even if they don’t voice it. No, they always ask, “Am I allowed to use [insert operation here]?” I find it fascinating that the first response is to seek out other operations to use rather than to see what can be done with the operations you do have. 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.
I also find it fascinating that once other operations are allowed, it suddenly becomes a game to use the fanciest operations the students know. This aspect in particular is what I find frustrating because for many it’s now a way to show off, and people get praised for a solution using a fancy operation seemingly because it uses that fancy operation. 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. I do get the fun of using All The Things, really, but it does seem to me to go against the spirit of the original problem which is all about how much is possible within constraints.
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?
The two-part Four Fours problem
And so, I have come up with a two-part version of the Four Fours problem. It goes like this:
The Four Fours, Part 1: The goal is to write whole numbers from 0 to 20 as calculations, each using exactly four of the number 4, and as many of the operations of +,-,×,÷ (and brackets) as you need. For example, 8=4+4-4+4. Six of the whole numbers from 0 to 20 are not possible using these rules. Which numbers are they?
The Four Fours, Part 2: If you were allowed to also use one other symbol or operation or function as many times as you like (along with +,-,×,÷ and brackets), which one could you choose so that you could write all six of the missing numbers as a calculation using exactly four fours?
The first part is presented this way to encourage people to try as much as they can with just the basic operations before trying something else. I deliberately chose to reveal that there were six impossible numbers to remove the need to prove that they were impossible, but If proving was important to you, then you could instead say that some of them are impossible, rather than specifically six of them. Also I wanted it to be very clear when you could move on to Part 2, because in my setting where I work with other people’s students or with strangers walking up in a public place to join in, I can’t always have the luxury of negotiation, which relies on a relationship I don’t have time to form.
It is worth saying that yes presenting it this way does make people curious and keeps them working on it until they have found 15 numbers that are solvable. Well, more people came over and joined in than with some other puzzles we’ve tried at One Hundred Factorial, and more persisted for longer, anyway. Also, when we did it at One Hundred Factorial, people were more systematic than I usually observe with the Four Fours, seeming much more likely to modify existing solutions than to just try random things. Your mileage may vary, of course.
The second part is presented this way to still include the idea of constraint. Sure, you can think of a function/operation that can subvert the rules to get the answers you are missing, but can you make it get all the missing answers? Now you still have to stop to think about the range of possibilities, and evaluate your idea. Plus, I am sharing one of the questions I have had about the Four Fours from very near when I first heard it — I get to see how others respond to a question I’ve always wondered about. It’s particularly fun to me to look at the list of operations people do tend to allow (concatenation, factorial, roots, decimal points) and see which of them alone allows you to complete the missing six numbers. Some of them actually do allow you to do it, which I find amazing since usually they are all allowed, even though sometimes just one of them is enough.
Some solutions
If you would like to see some discussion of the solutions to Part 2 of the Four Fours, then check out the replies to this tweet. Of course, you may want to try some things yourselves, in which case don’t check out the replies yet!
I’ll also include on the following page a list of solutions that have been found. Don’t scroll down if you don’t want to see them…
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Solution to part 2 of the Four Fours
Who found it
Date
f(x)=14 for all x
David Butler
7 Nov 2018
f(x)=x+10
Daniel Ty Brown
8 Nov 2018
τ(n), the number of divisors of n
Daniel Ty Brown
8 Nov 2018
π(n), the number of primes less than n
@literallyjustq (on Twitter)
8 Nov 2018
Γ(n)=(n-1)! (at least it’s that for natural n)
@literallyjustq (on Twitter)
8 Nov 2018
log₂(x) the logarithm base 2
Lyron Winderbaum
8 Nov 2018
f(x)=⌈log₁₀(x)⌉, the number of digits in n
Student at One Hundred Factorial
8 Nov 2018
Factorial
David Butler
8 Nov 2018
Decimal point (allowed to not have a leading zero)
Dab Finkel
8 Nov 2018
f(x)=1/x, the reciprocal
David Butler
9 Nov 2018
count(X), which counts how many elements are in the set X
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![A powerpoint slide with text. A word problem in the classic style: Squeetles are the latest pet craze! An online poll asked people to choose which of striped or spotted squeetles were cuter. From 250 responses, 60% said that spotted squeetles were cuter. [Redacted] Three reads: READ 1: The story. READ 2: The details.](https://i0.wp.com/davidkbutler.xyz/wp-content/uploads/2026/10/three-reads-2-arts.png?resize=1024%2C557&ssl=1)
