Idle Math Games


With the invention of smartphones has come a new genre of video games commonly referred to as “idle games”. Frankly, I love them. The term “idle” comes from the fact that most of these games are designed to progress regardless of whether the player is logged into them (or really, they just recompile based on time elapsed when you do log in). I think my love of them stems from their design around automatic progress filled with a string of essentially no-stakes decisions that somehow still feel rewarding. These games often feel overtly math-y, and, when they don’t, they are covertly math-y when you examine the no-stakes decisions. And the examples in the post are all free to play.

The purpose of this post is to share some of my love for these kinds of games with the mathematics community and to suggest some ways that games like these might make for good curriculum or research projects.

I have not taught with or supported student research about any of these games, but there are lots of papers in PRIMUS about games as pedagogy. For example: Martha Byrne used SET, SIX, SPOT IT!, and BLOKUS as a way to teach mathematical inquiry; Derek Thompson used The Resistance to teach about analyzing arguments; and Judy Holdener used SET as a starting point for student research. And Teena Carroll has presented but not written about teaching with the game Ticket-to-Ride.

You might think of this post as a call for papers like the examples above using idle games.

Idle Mechanics

How do these idle games work? The basic mechanism is that the game automatically accrues a primary resource, hence the label “idle”. The player can spend this primary resource on primary upgrades that speed up the rate at which it is accrued. Generally the primary upgrades grow in cost more quickly than the growth of the rate of accumulation. At some point, the player has the option to reset the whole cycle, starting over on the accumulation of the primary resource and removing the upgrades; however, this reset generally produces a secondary resource that can be spent to speed up the whole process.

First a metaphor, then a mathematization. Metaphorically, let’s imagine that we’re trying to feed Leela and the Genestalk (Futurama) the whole world with beans. Once planted, an individual bean plant grows over time, converting nutrients into plant structures and beans. Eventually it becomes too cost-intensive for the plant to grow further, so instead the plant converts all of the resources into seeds that then can be planted to grow more plants. In this metaphor, the nutrients are the primary resources, and the beans are the secondary resource. One could imagine either planting the beans to increase the number of plants or using them to feed the community who are then better able to cultivate beans. [Of course, in a physical space, there would be constraints on how this grows, constraints that are largely avoided in the games. See for example this sadly un-ironic tweet.]

Mathematizing this a little more, the primary resource might be something like an exponential function of time:

P(t) = a*bc*t.

The primary resource can be spent making it so that time increments more quickly (raising c), essentially moving the value more quickly along the same exponential curve, and the secondary resource can be spent to permanently increase the coefficient a so that the resource value is moving along a greater function, allowing the player raise c more quickly in subsequent cycles. Many of these games have more layers with similar structure. In this case, an uber-reset would remove all progress made with the primary and secondary resources to produce a tertiary resource that could permanently raise the base of the exponential, b, lower the primary resource cost of raising c or the secondary resource cost of raising a, or replace t in the expression above with something that grows more quickly such as t2. Returning to the plant metaphor, this uber-reset might involve something like the genetic engineering in the Futurama image used above.

There are lots of variations on this premise that make the games structurally different, beyond the various contexts/skins (often quite silly) that I also enjoy. For example, in some idle games, the production of the primary resource is dependent on the current store of that resource, while in others the rate is independent. Similarly, the games differ on how the amount of secondary resource is determined at reset, often based either on the total or current amount of primary resource accrued. Dependency on the current amount of a resource means that players must weigh the costs and benefits of upgrades carefully.

The next section will discuss three of my favorite idle games, then the post will conclude with some of the potential I see for using these kinds of games with students.

My Favorite Idle Games

Exponential Idle

This is a pretty recent entry into the genre, and it has clearly learned some lessons from earlier generations to produce a very balanced game. The premise is right over the plate for this blog. Initially, the player is an undergraduate student invited to do some research on the convergence of the iterative process:

f(t+dt) = f(t) * eb*x*dt.

It certainly seems like the process “converges to infinity”, but you work on this question throughout this program. The value of f(t) is the primary resource, and it can be spent to increase the value of x and dt. The value of b is determined by the accumulated total of f(t), and mu is the secondary resource, which can be spent to improve the ways that primary upgrades improve x. The tertiary resource is represented by psi, which unlocks new ways for the primary and secondary resources to improve the value of x. Later in the game, when you have your own PhD students, the graduates determine the values of phi and tau, and there are some late-game challenges that sort-of address the problem of “proving” that a process converges to infinity by running it, at least narratively. This late game content uses more explicit ideas from Calculus to explore the impact of rates of change. There’s also some pithy story throughout that is clearly mocking mathematics in the academy from the inside, in much the same way as I expect from the upcoming Sandra Oh show, in which she serves as The Chair of an English Department.

Two aspects of this game jump out to me for teaching and learning. First, the numbers are HUGE. In the image, you can see scientific notation and double scientific notation (does anyone else call it that), in which ee46190.5 means 10^(10^46190.5). As a result, this game really demands logarithmic and multiplicative thinking. Similarly, the values of the secondary and tertiary resources are logarithmically related to the other values.

Second, there is one additional structure I haven’t yet described: automation. This game allows the player to auto-buy the upgrades and to code rules for when to perform the resets and uber-resets. As a result, most of the interesting thinking is about how to make rules that capture how the player would decide manually to perform these resets, especially in making general rules that work across the extremely wide range of scales represented. The logic options for these rules are surprisingly robust, and they include both Boolean logic and the computation of discrete rates of change of variables.

Clicker Heroes

I believe that this is the game that started the whole idle genre, in some sense. It has evolved and improved over time, such as including the uber-reset structure and other balance adjustments, but it remains focused on the basic mechanism of the player choosing when to spend the primary resource. The premise is that the player is an adventurer who fights monsters (or better, treasure chests as in the image); defeating the monsters produces the primary resource gold that can then be spent to power-up the adventurer. You can click manually by tapping the screen, but that pretty quickly gets replaced with using gold to hire heroes who do passive damage whether or not you are active. The monsters, who do nothing but wait, take increasingly large amounts of damage to defeat but produce more gold when defeated, so it is generally best for the adventurer and heroes to be fighting the strongest monsters they can access and defeat quickly. There are periodic monsters that have a time limit, and failing to meet that time limit means the adventurer will get stuck below that level of monster.

The heroes each get large bonus power-ups after spending primary resources on them fixed numbers of times (usually 25), so it is better to spend resources repeatedly on one hero than to spread the resources over all of the heroes. Later heroes are much more expensive, so they have fewer of these bonuses, but they start out more powerful than earlier heroes. As a result, the decision-making process focuses on which hero is the most efficient conversion from gold (primary resource) to power (rate of accumulation of primary resource). Beyond this, there is a decision about when to reset in order to achieve a high rate of accumulation of the secondary resource and when to uber-reset to achieve a high rate of accumulation of the tertiary resource. The online community of people playing this game contains fascinating graphs that attempt to optimize resource allocation and efficiency for choices that increase the rates of resource accumulation.

AdVenture Communist

Adventure Communist is a spin-off from AdVenture Capitalist, but the spin-off is much more interesting because the producers of primary resources are inextricably linked to each other, rather than being an individualist free-for-all. I think the name and this structure is a critique of the unbridled consumption sometimes built into idle games, but the game also has a willfully silly 80s-era USSR skin. And while I’m critiquing these games, some of them have absolutely horrible contexts, like managing the finances of for-profit prisons, and many of these do not engage the horror or dehumanization of these contexts.

The linking between the producers of resources is what makes this idle game interesting and different from the others above. In the image, you see the tab for the production of pills (called placebos in-game). The first line shows 3.35 NN (I think this means 3.35*10^129) nurses, which produces 2.56.35 OO placebos/second. The second line shows 2.16 HH clinics producing 18.94 JJ nurses/second. In other words, each row in this column produces more of the row above it, and the top row produces the primary resource. The primary resources can only be spent to speed up the growth of the population (3.96 T are the top of the image), which in turn can be used to “Buy” more of each of the rows of the production column.

Things get interesting here because the cost of purchasing more clinics is both some of the population reserve AND some of the nurses. So building new hospitals initially reduces the rate of production of the placebos, but it increases the growth of the rate of change of that production, essentially sacrificing some of the first derivative in order to increase the second derivative. While there are some other mechanisms, I think this means that the user is spending the primary and secondary resources in order to manipulate the coefficients and degree of a polynomial that controls production of resources.

The blue flasks and numbers are called “science”, and they can be spent to permanently improve the rows and columns, by automating, accelerating, or improving the outputs from them. Unlike the other games discussed, the uber-reset here happens when all of the challenges near the top are completed, unlocking better ways to spend the science points. Science points are most closely associated with in-game purchases that cost out-of-game money. Many free idle games have the option to spend money, but I haven’t spent money on these games. Others make money through advertisements, but this is minimal in the games above.

Teaching with Idle Games

I see four ways someone might teach with idle games.

First, I think these games are a lot of fun, and games of this type generally do an excellent job of teaching the player how the features work. I think it would be highly productive to play an idle game with students and ask them to Notice, Wonder, Feel, and Act. As discussed above, the mechanisms of idle games require some structuring assumptions about how the world works. I expect that students would respond to these assumptions with Notice and Feel in ways that lead to good discussions about topics around economics, justice, and the ways that math is used to model the world. There are also lots of interesting things to notice in these games, especially place value, order of magnitude, multiplicative and logarithmic thinking, and exponential/polynomial/linear growth.

Second, these games clearly run on quantitative mechanisms, but sometimes the details of these mechanisms are hidden from the player, either for simplicity or to create some ambiguity. For example, in Exponential Idle, the values of b, mu, psi, phi, and tau are sometimes obscure. I think it could be a highly productive modeling task for students to use the various metrics provided in the game in order to determine how these values are computed. Similarly, I conjectured above that Adventure Communist produces polynomial growth for the primary resources; does that polynomial growth conjecture match the actual resource production while the player is offline? If yes, how good an approximation is this continuous model for something discrete, and if not, what would be a better model for the discrete recursion?

Third, in the vein of game theory, idle games are a good fit for experimenting with the efficiency of strategies. As mentioned above, there are serious questions about how to measure efficiency, and this work seems to demand deep thinking about graphical representations of relationships between quantities. In my estimation, there is not a “right” answer to which stances toward efficiency is most appropriate, so this work leads to game theoretic issues that are more aligned with daily decisions than those connected to other classic games.

And fourth, while idle games are built to require some “grinding” while players are offline, the games that thrive are the ones that balance the benefits of spending resources with the time it takes to accumulate those resources as well as the need many players have for both familiarity and novelty in the game play. Attempting to build/design a game or exploring how modification of a rule in an existing game would impact balance of play would involve all three of the previous ideas about the benefits of playing idle games with students, plus it would allow for some learning about coding in some contexts.

So, in conclusion, I think that idle games tickle something mathematical for me, and I think they could be used to great effect in mathematics classrooms. I look forward to hearing about the other math-y games readers love, and I encourage those of you who have taught with games to share your work trough PRIMUS.


I really love the Disgaea series of games that shares a lot with the idle games above, in which a team of demons saves the world while learning lessons about love, family, justice, responsibility and other supposedly un-demonic themes. [Caveat: The games include some content that walks both sides of a line between critiquing sexism and just being sexist in places.] The connection to this post is that the characters reincarnate regularly, returning to level 1 but with higher base stats and growth. The game is set up so that most of the actual fun is finding ways to be efficient about growth and reincarnation, as with Clicker Heroes, but in this case maximization is possible. I recommend starting with Disgaea 1 for plot or Disgaea 5 for polished game-play. Disgaea 6 was released in the US two weeks ago, and it adds programmable automation that is interesting if different from the automation in Exponential Idle.

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