Reflections on maths, learning and the Maths Learning Centre, by David K Butler

Category: Games

Descriptions of games I have invented.

  • Jenga Variations

    In this blog post I will tell you about two variations to traditional Jenga that I find fun, and also reflect on what these say about what “fun” means to me.

    Original Jenga

    It’s best to discuss original Jenga before talking about the variations, just in case there are important things about Jenga you don’t know, but also to explain why I might want variations at all.

    Original Jenga is a family game of dexterity. Rectangular blocks are stacked in layers of three into a tower, with blocks in each layer at right angles to the layer below. Players take turns to pull a block out of the tower from anywhere except the last completed layer, then lay it on the top to complete new layers. If the tower falls during a player’s turn or in the few seconds afterwards, then that player loses and the game is over.

    One thing that is kind of ambiguous is exactly what counts as “falling over”. Does just one block falling to the table count as falling over? If not, then how many blocks does it have to be to count as fallen? If you catch the falling blocks so they don’t hit the ground, does it count as falling? Can you catch the tower as a whole and straighten it to avoid losing? So many questions! But actually I love that there’s so many questions. It’s fun for me to have the discussion and decide together, and sometimes to have to change that decision because a situation comes up that we didn’t think of. In the end, when rules are ambiguous or missing, I follow Dan Finkel’s philosophy: “Choose what seems like the most fun way to resolve questions about the rules”. And having to resolve them is part of the fun.

    Other than that little ambiguity, that’s pretty much it. You choose blocks to pull out, you pull them out and put them on top, and if you’re lucky and/or skilled, you don’t knock it over. The suspense is pretty thrilling near the end of the game, and the tower falling always sparks a cheer. But after a while, if I’m honest, it’s just not that interesting.

    I think I know why: everything’s the same and there’s no creativity. All the layers of the tower are the same as each other and there are only two ways to pull out a block – pull out a side one or pull out a middle one – and putting it back on top is similarly restricted. Yes there’s suspense, but it’s always the same suspense every time, where you fear the tower will fall over. I like games to have a little more variety, and have a little more scope for creativity in the choices you make on your turn. The two alternatives below solve those two problems in different ways.

    Before I talk about variations, I just want to mention two properties of the blocks that are both interesting and relevant.

    The first is how many blocks there are. When it was first released and for a long time, a Jenga set was 51 blocks, which is how I know that 51 is not prime – it has to be a multiple of three since each layer has three blocks. Nowadays a Jenga set is 54 blocks, so one layer higher.

    The second thing is the shape of the blocks. Because the orthogonal layers have to fit together, the blocks have to be three times as long as they are wide. There is no particular need for the thickness to have a special relationship with either of the other dimensions, but it just so happens that the thickness of brand-name Jenga blocks is one fifth the length.

    Jenga blocks arranged on a table, viewed from above. One block is in the middle, with five blocks standing on their edge filling up the length on one side, and three blocks sitting flat filling up the length on the other. The edges of the blocks have the name Jenga on them in two fonts and not all the same orientation.

    (Off-brand stacking block games don’t have this thickness-to-length proportion. Most are three times as long as they are wide, but not all. The ones that aren’t manage it by having gaps between the blocks.)

    It’s these proportions that inspired the first variation…

    Cursed Jenga

    As I said earlier, brand-name Jenga blocks have the property that the longest edge is a multiple of both the other dimensions. That means it’s actually possible to make a square out of any of the three rectangular faces, with the blocks in different orientations. I wondered what it would be like to play Jenga with the blocks in these different orientations, and Cursed Jenga was born. (A student gave it this name during the first ever game at a welcome event for new students in February 2024 where I provided the activities.)

    In the traditional orientation, there are three blocks making up a square. If you put them on edge, you can line up five to make a square. If you stand them on their ends, you need to put them in a three-by-five grid of 15 blocks to make the square. Cursed Jenga alternates between these three arrangements for its layers.

    An important question is whether you can actually alternate the orientations with the 51 blocks you have. If you have all three, that’s 3+5+15, which is 23 blocks, and another set of those is 46 blocks. You can achieve 51 blocks with one more layer of 5. That leaves two options: 5+3+15+5+3+15+5 or 5+15+3+5+15+3+5. You can decide which you prefer, but the first one does begin the game filling in a 3-layer on the top, which is a bit easier. If you want to go to 54 blocks, you’ll need another 3-layer which again gives you two options: 5+3+15+5+3+15+5+3 or 3+5+15+3+5+15+3+5. This is another reason to start with 5+3, because then it will work with either 51 or 54 blocks and you don’t have to change the whole thing if you discover later it’s a different number of blocks after all. (There’s another argument for why there have to be three 5-layers: the 15-layer has a multiple of 3 blocks. A Jenga set has a multiple of 3 blocks. So you must have 3 sets of 5 to make a multiple of 3 in total.)

    Another important question is how to orient two successive layers relative to each other. It matches the original spirit of the game for the long lines between blocks in the 3-layers and 5-layers to be at right angles, but what do we do between a 15-layer and the other layers? Well for structural reasons, you want to avoid lines between blocks in two successive layers being parallel on top of each other. Or stated another way, you want there to be blocks in the 15-layer that touch more than one block in the layer above or below. That effectively makes your choice for you. I very much like that having a principle just decides it for you without having to remember a specific orientation.

    So here’s 51 blocks all set up ready to play Cursed Jenga:

    A tower of Jenga blocks. The bottom layer is five blocks on their thinner face. The next layer is three blocks on their fatter face, so that the long edge is perpendiculer to the long edges of the layer below. The next layer is 15 standing on their edges, with three in a row along the long edge of the blocks below. Then the same pattern repeats again for three layers. Finally there are five blocks on their thin face on the very top.

    You can probably already see one of the reasons I like this version of Jenga – just setting it up involves a whole lot of fun problem-solving!

    And now to play! You can keep the same rules as ordinary Jenga, but you quickly run into an issue caused mainly by that layer of 15 vertical blocks: it’s very easy to make more blocks fall out than you intended! You can’t end the game just because more blocks fell out, because you’d have to restart all the time. My solution to this involved two aspects.

    First, I changed the rule for what counts as “falling over”. The definition is now that the tower has fallen when any block in the top completed layer or above touches the table/floor. You could choose a different rule, and feel free to do so, but most of the time it doesn’t really matter because usually it’s pretty clear to everyone involved that it’s all over. Indeed, another possible rule is just to let everyone vote and if the majority agree it’s fallen, then it’s fallen.

    Second, I made the rule that all blocks that fall out are passed one at a time to the future players in the order of play. Each of those players can choose to just add a block they’re holding to the top rather than take a new block out. Effectively, the goal of a turn becomes to add a block to the top, and if you don’t happen to have one, you have to take it out of the tower. Alternatively, you could leave blocks that fall out where they are, seeing them all as part of the bottom layer for the purposes of the “pulling a block out” part of the turn. They’re not quite the same, because the second one doesn’t allow specific players to save up blocks for later, but still they both mean the punishment for pulling out too many blocks at once is to make it easier for the players after you. It is important to allow people to take a block out of the tower if they want, because that is indeed a fun part of the game and you don’t want to deny them that!

    And that’s it. It’s more interesting during play than traditional Jenga for a number of reasons. There are more different options for the types of blocks you can pull out at any moment, and you can sometimes make the choice of whether to pull out a block at all. More variety for the choices you can make always makes a game more interesting to me. Also, the tower is much more precarious than before, making the end of the game a lot more suspenseful. Plus, it just looks weird so people come to ask what’s going on or at least do a double-take on the way past. I do like giving people something to wonder about.

    A tower of Jenga blocks, with the blocks in each layer in different orientations to the layer above and below. Many of the blocks from the lower layers have been removed, and some of the blocks look very precarious indeed, standing over the edge of the blocks below.

    Ceiling Jenga

    This variant of Jenga was originally suggested by one of my MLC tutors Alexander Mackay in December 2024, and I’ve added some minor tweaks since then.

    You set up the tower as usual, except the top three layers only put two blocks each at the sides, like this:

    (It will become clear shortly why this is important.)

    The rules are the same traditional Jenga except for two important changes:

    1. When you put a block into the tower, you can put it anywhere and in any orientation as long as at least one part of the block is higher than it was before.
    2. The tower as a whole is not allowed to get any higher than it started. (This is the “ceiling” referred to in the name.)

    That second rule is why I set up the top three layers like I did, so that there are places to actually put the blocks that you move upwards! The first time we played, we just pushed other blocks a little out of the way to fit the new ones in, and if you feel comfortable with that, then go for it.

    You’ll also have to decide about what counts as fallen down, because the concept of “completed top layer” ceases to mean quite as much as it did before. My usual rule is that the tower has fallen when any of the blocks at the highest level touch the ground, but as I said, most of the time nobody quibbles about whether the tower has fallen.

    Here’s a picture of a game in action.

    A tower of Jenga blocks, with the blocks in the top several layers arranged very strangely. In particular there is one block sticking out the side with another standing on its end on top of it. There is also a block leaning diagonally. A blue box in the background has a sign on it that says "Jenga Views".

    And here’s a different game in action. This one is with giant blocks that don’t have the thickness proportion of brand-name blocks, but that’s fine because this variant of Jenga doesn’t actually require the thickness to be anything in particular.

    A tower of Jenga blocks on carpet. Higher up the tower, the blocks are at strange angles or hanging out the side. In particular the top layer has five blocks not aligned and with one in particular at an angle.

    I really love this game. One reason is that it just looks so bizarre. Cursed Jenga looks like you’ve set it up wrong, but Ceiling Jenga looks like you’re playing it wrong. Which you are, according to the regular rules. And if people ask, you can tell them all about it.

    The other reason is on every single turn you have scope to do something creative. Want to think ahead to set up gaps to put future blocks into? You can, and feel clever. Want to put a block hanging out the side, or up a different way? You can, and feel inspired. Want to see how far you can push the definitions and argue your case to the other players? You can, and feel empowered. These are feelings you can’t have in regular Jenga, and I love how such a simple-to-state rule can make them possible.

    Conclusion

    So there you go: Cursed Jenga and Ceiling Jenga are two variations on traditional Jenga that allow for more variation in what happens during the game, and so make Jenga more interesting and more fun. Give them a try, and do let me know how they go for you.

  • Introducing Digit Disguises with a small game

    Because [reasons], my game Digit Disguises has been on my mind recently, and reading the original blog post from 2019, I suddenly realised I had never shared my ideas on how to introduce the game to a whole class at once.

    This blog post fixes that.

  • My first Maths Teacher Circle

    Last week I participated in my first Maths Teacher Circle . I just want to do a quick blog post here to record for posterity that I did it and it was excellent. I choose to take the practical approach of just relating what happened.

    I had been interested in somehow going to one since I heard about them a while ago, and then the founder of the Aussie Maths Teacher Circles, Michaela Epstein , contacted me through Twitter back in November to ask if I might like to facilitate an activity at an online session in 2021, and of course I said yes. She invited me to a session about mathematical games, and I was so excited to share some of the games I have invented with some interested teachers.

    Of course, the closer it got, the more nervous I got. When I heard there would be 40 or so teachers ranging all through primary to secondary to post-school teachers, I was rather intimidated! But Michaela and Alex  assured me I would be ok and that what I had planned would work. And they also put up with my scatterbrained discussion of random maths stuff whenever I met with them too. So, feeling a little reassured, but still nervouscited (as Pinkie Pie would say), I dove right in feet first last Wednesday morning.

    To start off with, Michaela invited past Maths Teacher Circles participant Samantha  to  set the scene by sharing what she has gotten out of Maths Teacher Circles in the past. This was a nice way to begin by grounding it in a real teacher’s experience. Then Michaela shared the goals of Maths Teacher Circles, which were exploring maths, strengthening classroom practice, and bringing maths enthusiasts together. I was so glad I had come to a place that resonated with all the things I love. It really matched with the goals of One Hundred Factorial, which is probably why Michaela invited me to present in the first place. This was all a really smart way to begin, because it set the tone for the rest of the session. Even when the housekeeping notes about breakout rooms and whiteboards and chat windows came, it was clear that these were there to support the overall vibe.

    Then we had a very quick chat in breakout rooms with a couple of people. We were supposed to talk about Noughts and Crosses too, but we only just made it through the introductions! But honestly I was happy to just have met a couple of friendly faces to help reduce the nervous part of the nervouscieted.

    By this time, so much had happened already, yet it had only been a few minutes. And now it was my turn. Michaela introduced me and I was now responsible for the journey of these 45-ish hopeful people. I put up the rules for Which Number Where, and asked everyone to quietly have a read, then ask any questions they might have. People had some very useful questions in the chat and out loud, and I felt we were ready to try it live. I asked for volunteers and described how to play the game Mastermind-style, with one player being the Secret Keeper and the other players asking questions. After a couple more questions, we were ready to break into groups to play.

    Michaela put people into groups of fourish, and I popped into about half of them to have a chat. I asked people how they were going and played with them for a bit, seeding a different kind of question than the ones they had been asking so far. I found everyone to be gracious and thoughtful and engaged. Such a thrill to meet such wonderful people and play maths with them. These moments when I was in a small group with people were my favourite parts of the session.

    I brought everyone together into the big group to discuss how the game went. I started by asking people if they had a favourite question that was asked. And then people shared any thoughts they had at all about how to use this in a classroom.

    Suddenly it seemed my time had run out, so I quickly showed everyone my other two games Digit Disguises and Number Neighbourhoods, and encouraged them to go back to their breakout rooms to keep playing Which Number Where or to try a new game instead. I stayed out in the main room where Michaela made sure I was ready to do a wrap-up when people returned. I very much appreciated being able to think in advance about that part!

    One question Michaela asked was why I chose the game I did. I said I chose Which Number Where because it’s about logic, and not any particular maths topic per se. As someone said earlier, it’s about locations rather than numbers per se, which means it’s really about the yes-and-no questions, and about logical arguments and joining information together, and those are skills you use everywhere in maths, which is why I like it so much. Plus I just love to hear how people think and this game gives me a chance to do that.

    And then it was time for me to participate in someone else’s activity. Toby  and James  shared the Multiple Mysteries game and some problem-solving/proving prompts to go with it. I got to play the game with some lovely other people and join in with the play. It really was a lovely thing to just play around with something that someone else shared that they were excited about. I am very grateful to Toby and James for providing such a great game to play and think about, and to the members of my little breakout room who I had such fun with.

    After this, it turned out that Michaela had read the time wrong and had cut short my activity the first time! So I got to have a few more minutes! I decided to share Digit Disguises properly, and instead of using breakout rooms, to play a game as a whole room with me as the Secret Keeper. Some brave souls shouted out questions and I wrote the questions and responses on a Word document on the screen. After a few questions, I decided that I would stop people and ask them what they can figure out from the information we have so far. This part was just wonderful. People had multiple different ways of gleaning new information about the numbers and their letter disguises from what we already knew, and quite a few of the participants expressed a satisfying amount of delight at these fascinating new possibilities. It was extremely gratifying to have people so excited about something that I am excited about (and egotistically, satisfying that people liked something I had invented).

    At this point, my laptop ran out of battery power and I had to scramble to find the power cord. By the time I came back, things were starting to wrap up, with participants filling out a Padlet with their thoughts. And then it was over. It felt like almost no time at all had passed, which is a good sign that I’ve been deeply engaged.

    After all the other participants left, Michaela, Alex, Toby, James and I had a debrief, which was some lovely discussion about how it went and how cool it was to work mathematically with people rather than just present them with stuff, and just some nice discussion about teaching and learning maths with some lovely people. And after that, couldn’t help but keep working on  one of the investigations that Toby and James set me off on, because that’s how I roll and is the sign of a good maths problem.

    So that was my first experience of a Maths Teacher Circle. For me, the best part was the chance to think and play together with other teachers. The environment was so safe to just play and talk, and this was very carefully set up by Michaela in the first place, by discussing what was important and how to keep it safe. Being told explicitly that we were allowed to adjust the activities to match the level of the group made us free to play in our own way. And really, everyone was just so gracious and excited and, well, lovely. I am so grateful to have been a part of it.

  • Changing the goal of the Numbers game

    I conscripted the game Numbers and Letters seven years ago to help promote the Maths Learning Centre and the Writing Centre at university events like O’Week and Open Day. Ever since then, it has always bothered me how free and easy participation in the Letters game is, while the Numbers game is much less so. This Open Day I had a remarkable idea: instead of stating in the rules that the goal is to achieve the target, and trying to encourage people to take a different approach, what if I just changed the stated goal! I don’t know why I didn’t think of it before, to be honest!

    You can read the rest of this blog post, and four other related posts, in PDF form here. 

    The titles of the five posts are:

    • Numbers and Letters (2018)
    • An opening gambit for the Numbers Game (2017)
    • Changing the goal of the Numbers Game (2020)
    • Jack Frost’s centre (2015)
    • The Arts student’s maths brain (2017)
  • Number Neighbourhoods

    This blog post is about a game I invented in February 2020, the third in a suite of Battleships-style games. (The previous two are Which Number Where and Digit Disguises.)

  • Which Number Where

    Last year I invented a game called Digit Disguises and it has become a regular feature at One Hundred Factorial and other events. But before Digit Disguises came along, there was another game with a similar style of interaction that we played regularly, and this blog post is about that game. The game is called “Which Number Where?”

  • Digit Disguises

    This blog post is about a game I invented this week, and the game is AWESOME, if I do say myself.

  • Playing SET

    Amie Albrecht recently posted a most wonderful blog post about SET,  and it reminded me there were some SET-related things I should post too.

    You can read the two posts in this series in PDF form here. 

    The titles of the two posts are:

    • Teaching people to play SET on the fly
    • Team SET

    Resource linked in the blog posts:

  • Numbers and Letters

    Since 2013, the MLC and Writing Centre have been doing a game called Letters and Numbers at Orientation Weeks and Open Days to create interaction with people. I tweeted a photo of one of our sessions during Open Day yesterday and it has attracted a lot of attention, so I thought I might record some details of the game for people to read if they’re interested.

    You can read the rest of this blog post, and four other related posts, in PDF form here. 

    The titles of the five posts are:

    • Numbers and Letters (2018)
    • An opening gambit for the Numbers Game (2017)
    • Changing the goal of the Numbers Game (2020)
    • Jack Frost’s centre (2015)
    • The Arts student’s maths brain (2017)