Mathematician David Strütt, a scientific collaborator at EPFL, worked for four months to develop Matheminecraft, a math video game in Minecraft, where the gamer has to find a Eulerian cycle in a graph. Minecraft is a sandbox video game released in 2011, where the gamer can build almost anything, from simple houses to complex calculators, using only cubes and fluids. These countless possibilities are what lured David Strütt into Minecrafts universe: “the game might be first intended for kids but I was studying for my Bachelors degree in mathematics when I discovered it. I fell in love with the game when I realized there is all the necessary blocks to build a Turing machine inside the game. It was a long time ago, so I have since forgotten what a Turing machine is. But the gist of it is: anything is possible inside the game.”
Matheminecraft, now freely available to everyone, is a video game around Eulerian graphs with a tutorial and four levels. The project was made for the Maths Outreach team with the idea that it should be ready for the EPFL Open days in September 2019. After the success encountered at the Open Days, it was decided that the game will be proposed to classes of the region as a series of ateliers organized by the Maths Outreach Team and the Science Outreach Departement (SPS). During 4 weeks, 36 classes of children—8 to 10 years old– registered to visit EPFL and took part in a two hours matinée where they played Matheminecraft and did various chemistry experiments. Minecraft is a very popular game and has been described as one of the greatest games of all time. Children immediately recognize the game and a growing roar of “are we going to play Minecraft” fills the air as they enter the room. “I think Minecraft digitally plays the same role LEGO did in my childhood. It appeals to anyone who takes a bit of their time to dive into it,” speculates David.
The idea behind the project is the following. Consider a graph: that is a drawing on a board made of dots called vertices which are linked by lines called edges. The question that is asked about graphs is: “is it possible to cross each edge exactly once, pass by each vertex at least once, and end up at the starting vertex?”. The first mathematician to ask that question is the Swiss Leonhard Euler in 1736. Not only did he wonder about that, but he provided the answer, giving an exhaustive description of which graphs admit such a path and which dont.
In the Matheminecraft atelier, we try to answer Leonhard Eulers question. An easy way to introduce Eulerian cycles to schoolchildren is to ask them about figures or drawings that can be done without lifting the pen and going twice on the same line. Triangle, square, star, a plethora of examples comes to their minds. In Matheminecraft each level consists of a graph that admits an Eulerian cycle. The game uses graphs that are easy enough, in the following sense: an Eulerian cycle will be found if the gamers make sure they dont get stuck. Such graphs are quite easy to work with, making the game suited to grade-schoolers.
In the game, each vertex is represented as a large color dot and each edge as a bridge. To keep the video game spirit, and to ensure that one bridge is only crossed once, David Strütt added a “lava condition,” meaning that bridges, once crossed, will turn into lava. That makes them unable to be crossed again. A map of the graph is there to help the children. Famous Minecraft animals were added to decorate the levels, such as skeleton horses and Mooshrooms.
The story of Matheminecraft will not end there, as additional levels are in preparation and new series of ateliers—organized with the SPS—will take place in 2020 and 2021 Furthermore, a Matheminecraft 2.0 will see the day. It will include Eulerian trails, where the gamer will have to choose the starting point of his cycle. This would make the game harder and suitable for older grade-schoolers.
The freedom offered by Minecraft gave rise to other projects in the Maths Outreach Team, as a Summer School is currently in preparation in association with the Education Outreach Department. “Of course, at some point in my childhood I wanted to become a game developer. Only later in my teens did I think I could become a mathematician. Somehow, I became both” concludes David.
The mathematical theory behind the game is vast and well known. Its graph theory and was first mentioned as such in 1736 by Leonhard Euler. Euler laid the foundations of graph theory in his paper about the Seven Bridges of Königsberg (now Kaliningrad in Russia). This is a famous problem related to the urban geography of the city: can we found a walk through the city that would cross each bridge once and only once.
Euler proved that there was no solution to that problem. The graph theory gives us tools to answer our initial question: given a graph, can we visit each vertex, pass by each edge once and end up at the starting point? Let us restrict ourselves to undirected, connected, graphs, which simplifies the answer.
If we can answer “yes,” the goal is reached and the graph admits an Eulerian cycle. Furthermore, the starting and ending point does not matter.
If the answer is “no,” then some of the requirements are not verified. That is the case with the Königsberg bridges. But there exist graphs where we can visit each vertex, pass by each edge once but end up at a different vertex. In such cases, the graph admits an Eulerian trail or path.
If the mathematical proofs might not be suitable for schoolchildren, testing whether an undirected graph is Eulerian (with a cycle or a trail) is easy—depending of course on the graph at hand and ones ability at counting. To know if a graph is Eulerian, we need to define the simple notion of degree or valency of a vertex of a graph. The degree of a vertex is the number of edges that are incident to the vertex—in laymans terms that is the number of edges arriving (or leaving) a vertex.
If each vertex has an even degree then the graph admits an Eulerian cycle. If there are exactly two vertices with an odd degree then the graph admits an Eulerian trail. In the latter case, the starting and ending points are the vertices with odd degree.
If Matheminecraft does not cover Eulerian trails, the theory is nevertheless explained in a very mathematical way, on a blackboard—or on a whiteboard for a lack of better options.
The Math of Minecraft
Now, it’s likely those same children are being homeschooled for the rest of their school year, and Matheminecraft is available to download for both PCs and Macs. The documentation is in French, but chances are good that the children you want to show Matheminecraft to already know how to install Minecraft. If not, maybe someone will make a learning tool inside Fortnite.
In Matheminecraft, players face a graph that has one path all the way through, and choosing a path turns what came before into lava. This is how Strütt says he ensures players can’t double back, and it’s also tapping into shared vocabulary of childhood games (“the floor is lava!”) and video games. Some games that seem simple, like Snake and its many copycats, end up acting like Eulerian cycles because the length of the snake prevents you from doubling over your path.
In just a few examples of Eulerian thinking, it’s easy to see why this one mathematical idea is so influential. Strütt also chose Minecraft very intentionally—the game is popular with children to an extent that rivals LEGO bricks, Strütt says, and yet is a system in which a computer scientist can build a Turing-complete machine. He plans to make a more advanced version of the game where older players can face a bigger challenge.
Matheminecraft puts players in a classic math puzzle called an Eulerian cycle, which you’ve almost definitely tried before. Can you draw a star without crossing your own line or lifting your pen? This means going outside the typical Spirograph-ish way we draw these stars and doing it totally freehand, and indeed, it’s also an Eulerian cycle.
A new game based on Minecraft aims to teach people of all ages about some cool math ideas. By combining the bestselling game of all time with some well-known brain teasers, university mathematician David Strütt hopes to bring math thinking down to (pixelated) Earth.
He presented in 2013 at ICTEV and again in 2014 at SXSWEdu on how powerful Minecraft is as a teaching tool. He is one of the co-authors of Minecraft In the Classroom, Ideas, inspiration and student projects for teachers. In 2015 he presented at TEDx RosalindParkEd on how Minecraft has changed his classroom. More recently has has been seconded out of the classroom, and is now a Virtual Learning Coach, bringing video conferencing and other virtual learning practices to schools across the state of Victoria.
With National Literacy and Numeracy week in Australia in our current thoughts, now is a good time to share some ideas on how Minecraft can support teaching students numeracy. If you play the game yourself and reflect on what things are happening in your head while you are exploring and crafting you will notice that a lot of mathematical thinking happens without active thought. The same thing happens with students while they are playing, all we have to do is call attention to it. Failing that we can leverage their knowledge of the game to support our teaching and their learning in fun and interactive ways. What can you teach in terms of numeracy with Minecraft? In my experience addition, subtraction, multiplication, division, area, perimeter, volume, algebra, graphing, linear equations, the coordinate plane, probability and many more things. This post provides two examples of activities that you can take into your classroom, with little to no time creating pre-made maps for your students. I would like to point out at this juncture, that Minecraft is not doing the teaching, but the discussion you can have with students about the math behind the activity will be an in-depth learning experience. Activity 1: Probability This activity will help students explore the difference between theoretical and experimental probability, as well as working with fractions or percentages if you want them to. There has been a myth around the Minecraft community for ages that dispensers are not truly random in their choosing of which slot to dispense from. Start students in a creative world and ask them to collect a dispenser and a button from their creative inventory. Set this up with the button on the dispenser. Once this is done, get students to collect 9 different stacks of items and place them in the 9 slots in the dispenser. It can be any item they would like, but all 9 need to be different and stackable. Make sure students record what item is in which slot. Once this is all set up, it is time to test out the probability of getting each item. Get students to clear out their inventory and then press the button 10 times, recording what item is shot out each time. This can easily be done afterwards, but it is good practice to get them to record it after each button press. Now it is time to get students to start exploring the numbers, for this first discussion we will focus on slot number 1. Request each student to share how many items from slot number one came out in their first test of 10 presses. Ask students to reflect on whether this was the expected number of items, this is the experimental probability, how did it compare to the theoretical probability. It would be a good idea to have a spreadsheet displayed on the board/projector so that all students can see the results, and you can easily collate them to increase the data pool and explore how more data affects how close the experimental probability is to the theoretical probability. Now this is only half the story, we need to look at the difference between each slot to determine whether Minecraft is truly random when it chooses the slot to shoot out. Get students to compare each slot, and again, collating the results for all students is a great way of making the data set larger without doing heaps of tests. Don’t forget to get students to write down their conclusion as to their thoughts on the myth of the dispenser randomness. Another good discussion point is: is 10 tests per student a ‘good’ number of tests, or should we do more? You can choose to get students to do another set of 10 tests, or even more depending on how much time you have in your class, and whether you feel that the students understand the difference between theoretical and experimental probability. Throughout the lesson you can also introduce students to many of the terms used to explain or describe probability. Activity 2: The Algebra of Crafting This is perhaps my favourite simple activity. The aim of this activity is to get students unknowingly exploring algebra. As a class discuss how many raw resources are required to craft a full set of stone tools. The list of ingredients to craft the full set of tools including stone sword, stone axe, stone shovel, stone pick and stone hoe is actually a tricky question, because you need to include the wooden pickaxe and the crafting bench. (You might want to start using letters instead of full words below depending on the age of your students). Wooden Pick: 2 sticks, 3 planks Crafting Bench: 4 planks Stone Sword: 1 stick, 2 cobblestone Stone Axe: 2 sticks, 3 cobblestone Stone Shovel: 2 sticks, 1 cobblestone Stone Pick: 2 sticks, 3 cobblestone Stone Hoe: 2 sticks, 2 cobblestone For a total of 11 sticks, 7 planks and 11 cobblestone. Now we get to the algebra side of things. Discuss with the students: If 1 log is equal to 4 planks, and 2 planks are equal to 4 sticks, how many logs do we need to get the 11 sticks we need in total? How many logs for the planks required? So how many total logs? (Note: You can break this down into easier steps if need be, you can start with how many planks for the 11 sticks, and move to logs from there if your students are struggling.) Here would be a great opportunity to get them to check this in game. Get them to load up a world, collect only the resources calculated and see if they can produce the complete set of tools. Now to challenge the students a bit more. Instead of stopping at stone tools, upgrade to iron tools, and include iron armour if you think the students will be up for it. Provide the students with the following list of items, and see if they can figure out the basic resources required. An important thing to note here, is that coal is banned, they are not allowed to use coal to smelt their iron. They are only allowed to collect logs, stone and iron ore. Iron Sword, Iron Shovel, Iron Pick, Iron Axe, Iron Hoe. (They will need to include a wooden pick, crafting table, furnace and stone pickaxe as well, but it is up to you whether you include this on your list to students). If you are including armour, Iron Helmet, Iron Chestplate, Iron Leggings, Iron Boots should be included. Give students time to think about it, as they might want to research how many pieces of iron ore each wood type (ie, logs, planks, sticks and charcoal) will smelt. This research could be online, or within the game. Once they have their list of raw resources required to create their full set of iron tools (and armour) let them test it out in-game and report back on their findings. Did they forget anything, what was it, and how does it alter their raw resources? You can take this activity further if you want, perhaps get students to come up with their own list of items that they would like to craft, and work out the raw resources required for this. I hope that these two activities have shown you some simple ways to incorporate Minecraft into your math classroom without taking all of your preparation time. Want some more simple activities? I am happy to share more of my ideas, hit me up on the community site or on twitter (@EduElfie) and I will help you out, but it would be even better if you shared your ideas with the community. Sit down one night, spend a couple of hours playing the game, while playing think about how many options there are for exploring numeracy concepts within Minecraft while just playing. For example, what is the surface area of the floor in your first shelter? If you wanted to paint the interior of your house in Minecraft, and a tin of paint covers 15 block faces, how many tins of paint will you require? These are just a couple of quick ones to think about. Reflect on how many there might be with a bit of preparation time thrown in to create a learning experience for your students like no other. The limit truly is your imagination, or the imagination of your students. If you don’t have time to play yourself, get your students to do it in their spare time- they have more than us teachers, right? Get students to come to you with ideas about how they could demonstrate their numeracy understandings in Minecraft. If they are solid, get them to show your class, make them a leader, and watch them shine. ____
Stephen Elford, aka EduElfie has been a teacher for over a dozen years and an active member of the Minecraft in Education community for nearly half of those years. Since 2011 Stephen has been supporting teachers from around the globe to bring Minecraft into their classes by sharing and showcasing his live lessons on YouTube as well as semi-regular posts on his blog page sharing a ‘warts and all’ view of bringing mainstream games into a traditional classroom setting.
I fell in love with playing Minecraft with my kids. I loved the spatial metaphor of defining each bit of air around you with a block, and allowing the player to redefine that space on the fly. I thought long and hard about how to incorporate this idea into a math game. Unfortunately the web is not ready yet to deliver a high quality real time 3d experience to players through web browser.
After some experimenting I came up with the idea of displaying a three by three block like a Rubik’s Cube. I added to that the ability to remove and add blocks. The blocks have symbols on them, so the player needs to choose the blocks in a pattern to make a math equation that is true, in order to permanently remove the blocks. There are 4 types of gems to discover as the player moves down into and across the blocks.
Admittedly its a bit of a stretch from the original inspiration, but I think it can work for some kids and it also allows for creativity in making equations that evaluate to true.
I started to play with isometric blocks. These are blocks that look like they are in perspective (receding in space) but in fact they are all the same. They can give the illusion of space but can be managed much more easily than real 3D.
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