Making Your Own Sense

Reflections on maths, learning and maths learning support, by David K Butler

Category: Puzzles

Descriptions of puzzles I have created or made my own.

  • Jenga Views

    This blog post is about a sequence of visual perception and geometry puzzles I have created called Jenga Views. You can download a file here with 39 puzzles, roughly in order of difficulty.

    In my previous blog post, I showed two variations on traditional Jenga that I think are more interesting and more fun. But long before those were thought of, I had already been doing something non-traditional with Jenga blocks.

    Back in May 2018, I saw a tweet that shared a maths/art activity called Fun With Orthoprojections by JD Hamkins. It was a list of challenges where 1 by 2 by 4 wooden blocks had to be arranged to match views from the top and two sides. I thought it was really cool, but I didn’t have any blocks with those proportions. I did have a quadruple set of Jenga blocks, but Jenga blocks have very different proportions to 1:2:4, so in order to use them I’d have to make my own.

    After a furious effort in Inkscape and Word over one day and night, I had made the first version of Jenga Views, where Jenga blocks had to be arranged to match views of the shape from the top and two sides. There were 20 puzzles in total: the first eleven were based on JD Hamkins’s Fun with Orthoprojections, but the later ones were all my own. (The SVG Inkscape file with the carefully measured block faces is here, if you’re ever interested in making any yourself.)

    A screenshot of four pages from a document. The first is all text and has the title Jenga Views. The other three pages have three elements. They have a number 1, 2 or 3. They have this text: Jenga Views by David K Butler. Build a structure with Jenga blocks that has these three sides. They have a stylised shadowy 3D shape, with three call-out squares pointing at the top and two sides. In each square is a diagram showing blocks in outline.

    I put them out at One Hundred Factorial the next day, where they were an immediate success. Check out these two students working on one of the challenges on that very first day:

    In general, over the years, I have found that the Jenga Views challenges are strangely compelling to anyone for whom they spark interest. Some people aren’t intrigued by them, but those who are intrigued tend to just stick around and do all of them, one after the other.

    One thing I particularly like about them is that you don’t need an answer key, because you can just look at your construction and tell if it looks right or not. There’s something empowering about being able to check it yourself. (I’m sure there’s a lesson there for maths problems we give to students for practice, but I’m not going to explore that thought more right now.) I love watching students crouching down to see their construction from the correct angle and holding up the pictures to compare.

    One of those people who found it compelling was my younger daughter Charlotte. In April 2020, she was 11 years old and only a couple of months into Year 6 (which here in South Australia is the final year of Primary School) and suddenly we all went into pandemic lockdown. My wife, who is a trained teacher, cobbled together her own school away from school curriculum for Charlotte before Charlotte’s school sorted anything out, and I provided some maths activities I had brought home with me from university. I had brought a selection of things from One Hundred Factorial home so I didn’t go insane and so I could try to run One Hundred Factorial from lockdown. One of those things was Jenga Views.

    And Charlotte loved it. Because of the quadruple set of Jenga blocks, she actually had enough to do all 20 challenges separately and lay out all the answers at once.

    On a bench seat next to a table is a blue plastic container filled with Jenga blocks. The rest of the bench and the table are filled with pieces of A4 paper, each with a number in the corner and three designs showing blocks in outline. The numbers go from 1 to 20. In the middle of each page stands a construction made of Jenga blocks.

    (Note that the file has been updated since this image was taken, so don’t use it as an answer key, because almost all the numbers refer to different challenges now! Also you can tell if your solution really is one without an answer key, as I said earlier.)

    Charlotte was so very proud of herself and I was proud of her. I shared the photo and the Jenga Views document on Twitter, and it got a lot of attention. There were a lot of people who were also in lockdown who were extremely grateful for something fun and mathematical to do using resources they had in their house already.

    I was all inspired too, and I created more puzzles, adding five more the next day and then five more the day after that, bringing the total to 30. I also created a print-and-cut net that would allow people to make their own blocks in the Jenga proportions, since someone complained they didn’t have any in their house, as well as a video of how to fold it up into a Jenga block. (Two nets are on the last page of the Jenga Views document, if you want something better quality than this picture.)

    A diagram of a net with tabs. The net has white rectangles arranged in a zigzag, with grey rectangles coming off the sides with an arrow. Where the edges of the grey rectangles run along the edge of other rectangles, there is a stripy shaded pattern. At the bottom right, four rectangles-with-arrows go off in a row from one of the small white rectangles.
    A construction made of home-made paper Jenga blocks. They are stacked in multiple directions with a lot of space between them in the middle.

    Eventually, the interest settled down on Twitter. But over the years I’ve pulled out Jenga Views at One Hundred Factorial regularly, and every time, there’s always people who do what Charlotte did and just work through all of them.

    Now we’re here five and a half years later (and seven and a half years after its original creation), and I’ve finally gotten around to writing all this up as a blog post.

    To reward everyone who waited – including me – I have created nine more puzzles. They are mixed in among the 30 that were already there, so be careful if you’ve downloaded it before because almost all the numbers have changed! Also at the end of the document there are four empty challenges for you to draw your own, and the net of the make-your-own blocks too.

    As an aside, the reason I did nine more and not ten is because then you can print them two-to-a-page or three-to-a-page including the title page and they will line up neatly without annoying blanks. It’s been something that’s bothered me every time I’ve printed them over the last seven years.

    So there’s the whole story of Jenga Views. I hope you enjoyed reading about it, and I hope you enjoy doing the challenges yourself.

  • Gerry-mean-dering

    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?

    You can read the rest of this blog post in PDF form here. 

  • 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:

    1. 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?”.
    2. 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.
    3. 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.

  • Quarter the Cross: Connect the Dots

    This blog post is about a new variation on the classic problem, which I call Quarter the Cross: Connect the Dots.

    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:

  • Twelve matchsticks: focus or funnel

    One of my favourite puzzles is the Twelve matchsticks puzzle. It goes like this:

    Twelve matchsticks can be laid on the table to produce a variety of shapes. If the length of a matchstick is 1 unit, then the area of each shape can be found in square units. For example, these shapes have areas 6, 9 and 5 square units.
    Three shapes made with twelve matchsticks each. The third is a rectangle five high and one wide. The second is a square three wide and three high. The first is an irregular shape like a two-by-two square with two little squares attached near one corner.
    Arrange twelve matchsticks into a single closed shape with area exactly 4 square units.

    I will tell you soon why this is one of my favourite puzzles, but first I want to tell you where I first learned this puzzle.

    Once upon a time, about ten years ago, I met a lecturer at my university who was designing a course called Puzzle-Based Learning. It was an interesting course, whose premise was that puzzles ought to provide a useful tool for teaching general problem-solving, since they are (ostensibly) context free, so that problem-solving skills are not confounded with simultaneously learning content. Anyway, when he was first telling me about the course, he shared with me the Twelve Matchsticks puzzle, because it was one of his favourite puzzles. The reason it was his favourite appeared to be because the solution used a particular piece of maths trivia.

    And the puzzle has become one of my favourite too, but for very different reasons! Indeed, I strongly disagree with my colleague’s reasons for liking the puzzle.

    The first thing I disagree with is the use of the word “the” in “the solution”, because this puzzle has many solutions! Yet my colleague presented it to me as if it had just one.

    The second thing I disagree with is that using a specific piece of trivia makes it likeable. I do recognise that knowing more stuff makes you a better problem-solver in general for real-world problems, but even so I always feel cheated when solving a puzzle requires one very specific piece of information and you either know it or you don’t. It seems to me it sends a message that you can’t succeed without knowing all the random trivia in the world, when of course it is perfectly possible to come up with acceptable solutions to all sorts of problems with a bit of thinking and investigating.

    The main reason I love the Twelve Matchsticks puzzle is because it does have many solutions, and because this fact means that different people can have success coming up with their own way of doing it, and feel successful. Not only that, if I have several different people solving it in different ways, I can ask people to explain their different solutions and there is a wonderful opportunity for mathematical reasoning.

    Another reason I like it is that people almost always come up with “cheat” solutions where there are two separate shapes or shapes with matchsticks on the inside. I always commend people on their creative thinking, but I also ask if they reckon it’s possible to do it with just one shape, where all the matchsticks are on the edge, which doesn’t cross itself. I get to talk about which sorts of solutions are more or less satisfying.

    The final reason is that most of the approaches use a lovely general problem-solving technique, which is to create a wrong answer that only satisfies part of the constraints, and try to modify it until it’s a proper solution. This is a very useful idea that can be helpful in many situations. As opposed to “find a random piece of trivia in the puzzle-writer’s head”.

    Which brings me to the second part of the title of this blog post. Let me take a moment to describe two different experiences of being helped to solve this puzzle.

    When I was first shown this puzzle, my colleague thought there was only one solution, and his help was to push me towards it. He asked “Is the number 12 familiar to you? What numbers add up to 12? Is there a specific shape that you have seen before with these numbers as its edges?” He was trying to get me closer and closer to the picture that was in his head and was asking me questions that filled in what he saw as the blanks in my head. And there were also some moments where he just told me things because he couldn’t wait long enough for me to get there.

    I did get to his solution in the end, and I felt rather flat. I was very polite and listened to his excitement about how it used the piece of trivia, but inside I was thinking it was just another random thing to never come back to again. It wasn’t until much later that I decided to try again, wondering if I could solve the problem without that specific piece of trivia.

    So I just played around with different shapes with twelve matchsticks and found they never had the right area. And I wondered how to modify them to have the right area. And I played around with maybe using the wrong number of matchsticks but having the right area, and wondered how to change the number of matchsticks but keep the same area. This experience where I used a general problem-solving strategy of “do it wrong and modify” was much more fun and much more rewarding.

    When I help people with the Twelve Matchsticks puzzle, this second one is the approach I take. I ask them to try making shapes with the matchsticks to see what areas they can make. I ask what all their shapes have in common other than having 12 matchsticks. I notice when they’ve made a square with lines inside it and wonder how they came up with that. Can they do something similar but not put the extra ones in the middle?  In short, I use what they are already doing to give them something new to think about. I encourage them to do what they are doing but more so, rather than what I would do.

    The approach where you have an idea in your head of how it should be done and you try to get the student to fill in the blanks is called funnelling. When you are funnelling, your questions are directing them in the direction you are thinking, and you will get them there whether they understand or not. Often your questions are asking for yes-or-no or one-word answers to a structure in your head which you refuse to reveal to the student in advance, so from their perspective there is no rhyme or reason to the questions you are asking.

    The approach where you riff off what the student is thinking and help them notice things that are already there is called focussing. When you are focussing, you are helping the student focus on what’s relevant, and focus on what information they might need to find out, and focus on their own progress, but you are willing to see where it might go.

    Interestingly, when you are in a focussing mindset as a teacher, you often don’t mind just telling students relevant information every so often (such as a bit of maths trivia), because there are times it seems perfectly natural to the student that you need such a thing based on what is happening at that moment.

    (There are several places you can read more about focussing versus funnelling questions. This blog post from Mark Chubb is a good start .)

    From my experience with the Twelve Matchsticks, it’s actually a rather unpleasant experience as a student to be funnelled by a teacher. You don’t know what the teacher is getting at, and often you feel like there is a key piece of information they are withholding from you, and when it comes, he punchline often feels rather flat. Being focussed by a teacher feels different. The things the teacher says are more obviously relevant because they are related to something you yourself did or said, or something that is already right there in front of you. You don’t have to try to imagine what’s in the teacher’s head.

    So the last reason the Twelve Matchsticks is one of my favourite puzzles is that it reminds me to use focussing questions rather than funnelling questions with my own students, and I think my students and I have a better experience of problem-solving because of it.

  • 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:

      • 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.

        You can read the rest of this blog post in PDF form here. 

      • 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.

      • Trying maths live on Twitter

        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.

        (Note: You can see copies all these tweet threads in a PDF document here. )

      • 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