a couple of math & gaming articles
Feb. 4th, 2019 10:17 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
ursulaRecently, I came across a couple of fun articles about math and computer games in MAA (Mathematical Association of America) publications.
The MAA was founded to support the American Mathematical Monthly, an expository math journal (disclaimer: I'm one of the Monthly's many associate editors). All of the MAA publications are dedicated to explaining mathematics, rather than recording research mathematics, though individual articles do often include new research! Both of these articles should be accessible to someone with a solid undergraduate math background: one mostly uses calculus (and is based on research done by undergraduates) but is rather densely written, while the other draws on some graph theory and number theory/abstract algebra, but is more conversational in tone.
When I say "accessible", though, I'm thinking like a mathematician. These articles do a nice job of setting up their problems and explaining the kinds of tools used to solve them, but understanding the details probably requires sitting down and working out lots of examples. And if you find yourself sitting down and working out examples inspired by recently published math papers, you're basically doing research mathematics, which might not have been on your agenda for today (or which might have been, but not with these papers). This is all to say, if you find yourself skimming the beginning of one of these articles and then saying, "Hey, cool idea, but my brain is full now!" that's a normal response for both mathematicians and non-mathematicians, and you should still feel chuffed about learning a thing.
Anyway, on to the articles!
Aaron M. Broussard, Martin E. Malandro, and Abagayle Serreyn, Optimizing the Video Game Multi-Jump (American Mathematical Monthly).
This article is about platformer video games where a character can jump, then magically start another jump in mid-air. The question is how to time the jumps so the character lands at a specific point (on a floating platform, for example). There are explicit solutions if the path of each jump is part of a parabola, as well as a discussion for more general, perhaps-fantastic jump shapes. I enjoyed the comments in this article about the way AI-controlled characters in specific games fail to make optimal decisions.
Tom Edgar and Jessica Sklar, A Confused Electrician Uses Smith Normal Form (Mathematics Magazine).
This article is about the type of puzzle where flipping a switch or pressing a button can turn multiple lights on or off, or rotate them through different colors. The goal is to find the right combination of button-presses that will turn all the lights on or off simultaneously. The analysis starts with graph theory, converts it to a problem involving matrices of integers, invokes the SageMath computer algebra system and a little bit of number theory, and ends with Smith normal form. Smith normal form is a beautiful way to factor matrices, but if you never restrict yourself to integer matrices, you probably haven't heard of it. It was the solution to a problem I ran into when I was writing my dissertation, and I've had a soft spot for it ever since.
The MAA was founded to support the American Mathematical Monthly, an expository math journal (disclaimer: I'm one of the Monthly's many associate editors). All of the MAA publications are dedicated to explaining mathematics, rather than recording research mathematics, though individual articles do often include new research! Both of these articles should be accessible to someone with a solid undergraduate math background: one mostly uses calculus (and is based on research done by undergraduates) but is rather densely written, while the other draws on some graph theory and number theory/abstract algebra, but is more conversational in tone.
When I say "accessible", though, I'm thinking like a mathematician. These articles do a nice job of setting up their problems and explaining the kinds of tools used to solve them, but understanding the details probably requires sitting down and working out lots of examples. And if you find yourself sitting down and working out examples inspired by recently published math papers, you're basically doing research mathematics, which might not have been on your agenda for today (or which might have been, but not with these papers). This is all to say, if you find yourself skimming the beginning of one of these articles and then saying, "Hey, cool idea, but my brain is full now!" that's a normal response for both mathematicians and non-mathematicians, and you should still feel chuffed about learning a thing.
Anyway, on to the articles!
Aaron M. Broussard, Martin E. Malandro, and Abagayle Serreyn, Optimizing the Video Game Multi-Jump (American Mathematical Monthly).
This article is about platformer video games where a character can jump, then magically start another jump in mid-air. The question is how to time the jumps so the character lands at a specific point (on a floating platform, for example). There are explicit solutions if the path of each jump is part of a parabola, as well as a discussion for more general, perhaps-fantastic jump shapes. I enjoyed the comments in this article about the way AI-controlled characters in specific games fail to make optimal decisions.
Tom Edgar and Jessica Sklar, A Confused Electrician Uses Smith Normal Form (Mathematics Magazine).
This article is about the type of puzzle where flipping a switch or pressing a button can turn multiple lights on or off, or rotate them through different colors. The goal is to find the right combination of button-presses that will turn all the lights on or off simultaneously. The analysis starts with graph theory, converts it to a problem involving matrices of integers, invokes the SageMath computer algebra system and a little bit of number theory, and ends with Smith normal form. Smith normal form is a beautiful way to factor matrices, but if you never restrict yourself to integer matrices, you probably haven't heard of it. It was the solution to a problem I ran into when I was writing my dissertation, and I've had a soft spot for it ever since.