The glass bridge game
Playing the infamous glass bridge from Squid Game as an introduction to the binomial distribution
2021 saw the Netflix original 'Squid Game' break all kinds of viewing records. Whilst the show may divide opinion, the 'glass bridge game' gave us a really popular example of how the binomial distribution plays out in context and it seems an unmissable opportunity for this course. On this page we will talk you through playing a (much less sinister) simulation of the game and then using it to get a firm understanding of the binomial distribution. This is about a weeks worth (3 to 4 hrs) of activity depending on how much you use and how quickly you go.
Activity
The following runs through the three parts of the activity. Stage 1 - play the game. Stage 2 - glass bridge to binomial distribution. Stage 3 - Variations
As much as possible, it is important for students to experience mathematical simulations for themselves. These concrete experiences are hugely important starting points on which theory can build. Also, it can be lots of fun. So here is an outline of how you can play.
What is the game - In the show, the game plays out like this. 16 players are lined up in order to cross a bridge with 18 steps. for each step them must choose between 2 glass squares. One of the glass squares is made of tempered glass and will support their weight. The other is not and will shatter and the player will fall from the bridge. (The outcome is a bit sinister in the show, so I imagine the bridge about 2 metres above a foam pit or river!). You can't (in theory!) tell the difference between the two glass squares and so, at each step there is a 50% chance of surviving or falling. There person at the front takes all the risk because those behind them can simply follow. If the person at the front falls then the second person becomes the new leader. The goal is obviously for as many people as possible to make it across the bridge.
Setting up the bridge - This is key of course, but fairly easy and possible in a number of ways. Here is a suggestion. I used a big hall/auditorium space and 40 chairs. I wanted a 20 step bridge. Each step needs two choices, so I set out two columns of 20 chairs with space between the two of them. On each chair there was a piece of paper, blank on one side and on the other it said either 'Win' of 'Lose' (here is one you can use - note that the font text is light so it can't be seen through). For each step, the player at the front of the queue must jump either next to the chair on their left of the chair on their right and then turn over the piece of paper. If it is 'Win' then they can take the next jump. If it is 'lose' then they sit down on the chair and the next person becomes the leader. Clearly, teachers will want to think in advance about how many players they will have and how many steps they want their bridge to have. Repeating the game simply involves redistributing the 'win/lose' sheets randomly.
Player order - In the show, before the players know what game they are going to play, they are asked to choose a number from 1 to 16 which will be player order. Once you know the game of course then you know that you are best off going towards the end! This can be easily simulated by laying out numbers and asking students to choose. Here is a list of numbers to save a little bit of time.
Playing - so by now you have your bridge set up and your players have chosen their number order. It remains only to play. The first time through I suggest just playing without too many interruptions or discussions. It is fun. Those that know the show are likely to be instantly engaged and others will catch on pretty quickly. Thereafter there are lots of opportunities for teachers to ask questions and repeat the games with different numbers of players and/or steps. Here are some thoughts.
What are the chances? - After the first game, if some people make it across the bridge, ask students to reflect on how likely that was? What needed to happen for somebody to make it across the bridge? Of all the things that could have happened, how likely was this one? This is the beginning of thinking about counting possibilities.
Different numbers of people - One idea is that on the second play, the survivors watch whilst all those that fell get another chance. They will already have the right numbers and can quickly shuffle and deal these out whilst the survivors redistribute the 'wni/lose' sheets. This will likely be a much smaller number of people and so the game could get a bit more tense. This time you can stop the game for some discussion about probabilities and what needs to happen for different players. These kind of conversations set the seen for 'counting successes'. If you get mor survivors you can repeat again until you have a game where no one makes it across the bridge.
Looking at outcomes - In the picture below you can see a finished game. The people in the seats are the ones that fell and the survivors can clearly be seen at the other end. This is a freeze frame of one possible outcome that invites us to think about how many their might be.
This is a lovely visual of an outcome that starts LOSE (L), WIN (W), W, W, W, L, W, L etc.... and you can walk down the middle here and point this out. Then you can explore the question of how many correct choices were needed for the last survivor to make it which gets you neatly in to the idea that you are counting the number of successes and the number of different ways you could have had that number of successes.
Exploration - at this stage it is a question of how much time you want to spend with this but there is potential here for some investigation with different numbers of players and steps. There is potential to collect lots of data here too. At some stage, students suggested that they might be able to tell which seats was 'lose' because it looked like it had been sat on. Introducing bias here cold be fertile ground for exploration too.
The following Glass Bridge activity is designed to lead you through the theory of binomial distribution in the context of this glass bridge game. You can work through this individually or as a whole class and you will need your GDC!
In this next stage 'A bridge with a difference' we look at a glass bridge where there is a choice of three glass squares at each step, which changes a lot about the problem. This eventually leads us to the idea of 'probability of success' which is the last variable to address. Then we are fluently calculating probabilities with the binomial distribution.
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A ToK moment
This is, of course, riddled with ToK moments. On top of those we are already know about probability, this is a lovely expose of how probability can go from simple and intuitive to complex and counter intuitive. To work out the probability of the 7th person in the queue of a 16 step bridge surviving involves an awful lot of steps. To do it by hand you would have to establish a tree diagram that showed 216 paths. You would need to understand that fot the 7th person to survive, that no more than 6 people could fall and, as such, the need to be at least 10 sixes. Then you would need to know how many of the 65536 possible paths involve at least 10 successes. You need to establish what the probability of each of those paths are and then add them together. This is an epic task and is a lovely example of how mathematics can build with the combination of things we need to understand to do this calculation. Along the way we do some great mathematics. As the number of steps on the bridge increases it becomes more and more labour intensive to do this by hand. Enter the wonder of Pascals triangle, the conjecture that it will successfully predict the number of paths with a given number of success and then the knowledge that it can be proven. Then the wonder that someone has programmed our calculators to look up the different numbers on the triangle, so we don't need to write it out to see the numbers in the 20th row. We cross an important bridge (!) in recognising that binomial events don't have to be 50/50 and then discover that our GDCs can do this complex calculation for exact and cumulative numbers of successes in just a few pushes. I think it is a significant mathematical knowledge journey!
A global view
Perhaps less related to the mathematics and more the cultural impact of the show, there is lots to discuss. Although opinions are divided, there is no doubt over the shows main theme about how societies can can cast off their poorest and least fortunate while the rich play games. So much of this is down to chance as well which is what makes the game so fitting. This has lead to significant discussions and diversions.
Teacher Notes
Although we try to write as many of these pages as we can for a combined audience of students and teachers, I imagine that this page will be largely frequented by teachers only. As such, much of what is written above is advice for teachers about how to approach this activity. As I have taught binomial distribution in the past I have struggled to find a truly engaging context that sustains an interest in the counting of possibilities. My eyes lit up when I saw the glass bridge (independent of the context of the show of course) and all the articles and videos that followed as mathematicians latched on to it as an opportunity to explore binomial probability. For this reason I recommend strongly that you make time in your schedule to actually play so that the paper based exploration that follows is rooted in a concrete experience. Then the only choice you have to make is how much you want students to work on themselves and how much you want to lead it. The activities allow for a bit of both which I think is a good compromise!
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