In My Corner

For a while I've been interested in analyzing the game Jenga, since it's very physics-aligned in its design: Maintaining balance under changing forces. I couldn't get a handle on how to look at it though, until I connected it to a tool that's used frequently in my research field: corner plots. Corner plots are a way to display correlations between different variables in a large data set. For gravitational waves, they're used to show how estimates of a source's parameters depend on each other, such as a black hole's mass and spin. I figured I could come up with some measurements of Jenga games, and look at how they relate to each other.

First, we need a way to collect some data. As a reminder, here's what a Jenga tower looks like:

Wikipedia

Each level has 3 blocks that alternate in direction. On a turn, we remove a block from anywhere below the top level, and then add it to the top. The tower will fall and end the game if the center of mass of a subset is over an empty space, and the side is not supported. I was able to make a simulation of this in Python, with the virtual player making a random move on each turn, so long as that move does not cause the tower to fall. Eventually, the player will run out of moves and the game will end, but we can look at what kinds of moves result in longer games. I struggled a bit to find a way to show an example tower being built, and settled on a side view with all the bricks facing out. Remember that the bricks actually alternate in direction, so a hole under a brick is not necessarily a problem:

Now to get our statistics, we want to run a large series of games, and get some measurements from each one. The parameters I chose were max height of the tower, number of turns taken, fraction of turns for which the center block of a level was removed, fraction of turns a block was placed on the center of the top, and the fraction of times a block was removed from the upper half of the tower. After running 500 simulated games, we can make a corner plot with the results (click to enlarge):

This shows us some interesting connections between our parameters. There is strong correlation between the maximum height and the number of turns – This makes sense, since each turn we put a brick on the top level. We can also see anti-correlation (negative slope) between removing the center brick and max turns/height. This is because once the center of a level is taken, we can't take either of the sides, so we run out of bricks faster. Our final two variables, adding to the center and removing from the upper half, don't seem to have much effect on the outcome, indicated by the circular distributions.

Now in reality, Jenga bricks are designed with some irregularities to make the balance a little more difficult to predict, so this is another case where knowing the physics might not help you win, but it was interesting way to explore a tool for visualizing data.