Great Halloween Math Lesson: The Count & Vampires

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Guest post by MathaliciousMathalicious is a small curriculum company that’s re-writing middle and high school math around real world topics. All lessons include a student handout, a lesson guide, and a multimedia presentation, and they’re available at www.mathalicious.com.

Vampires are near and dear to our hearts around here. But not too near. On second thought, stay away from our hearts. At any rate, vampires have come to loom large in the popular consciousness over the past few years — books, comics, movies — they’re everywhere. That said, nobody has ever managed to catch one. So, a quick question: Where mah vampires at??

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Our favorite arithmomaniac. (source: wikipedia.org)

Depending on which particular mythology you subscribe to, vampire feeding and reproductive habits vary widely. It’s enough to give a budding vampirologist a serious headache, so let’s settle on two facts, which we at Mathalicious take as dogma:

  • If you are bitten by a vampire, you become a vampire. Do not pass Go. Do not collect $200. Do buy yourself a cape.
  • Vampires need to drink human blood. Animal blood is the tofurkey of the vampire world; only the craziest consider it “food.”

Let’s further assume that vampires have modest hemic appetites, feeding only once per week. How long could the vampire population go undetected by the rest of us human cattle?

If we trace the vampiric lineage back to a single bloodsucking “Adam,” after one week, he would have bitten one person, bringing the total vampire population up to two. After the second week, each of those two vampires would have gorged themselves on neck juice, minting two additional vampires (for a total of four). And so on. Since the vampire population doubles each week, we can model it with an exponential equation: population = 2t, where t is the number of weeks since Adam started feasting.

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Vampire population growth. (source: desmos.com)

While this might seem reasonable at first, our vampires are going to run into a sustainability problem. Namely, exponential growth gets out of hand, fast. Really fast. After three months, for instance, there are just over 4,000 vampires roaming around the countryside. After six months, though, there are almost 17 million vampires. That might make it harder to fly under the radar (ha!). And by only week 33, there would be more vampires than people currently living on the planet. In other words, in about the time it takes to get from 6th grade to 7th grade, the human race would have been eradicated.

Think that’s a little extreme? Even if vampires only fed once a year — even if they only fed once a decade — we’d all have been bat food long ago. So, why aren’t we all vampires? Probably because there were never any to begin with.

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Seems legit. (source: aaronmritchey.com)

Okay, vampires are silly, but they raise in interesting question. After all, if you replace bite with sneeze on, and become a vampire with get a horribly fatal disease, we’re in essentially the same situation. Except this time it’s scary for real. So, the question: Why aren’t we all dead?

For one thing, some diseases are just too nasty for their own good; if I die so quickly that I don’t have the chance to sneeze on some other people, I’m not doing anything to help the disease reproduce. For another thing, the infected population might not be particularly mobile; if I live someplace remote enough, everybody in my community might die, but we’re not likely to spread the disease worldwide. Also, few highly contagious diseases are 100% fatal; even the supremely dire ebola virus “only” has an average fatality rate around 70%. Along the same lines, not everyone I come in contact with will necessarily get sick; some people might have natural immunity.

Besides natural mitigating factors, medical science strives to leverage some of these effects in order to lessen a disease’s devastation. Treatments might lower the fatality rate; immunization programs might lower the infection rate; quarantines might limit the exposed population. In other words, we’re not all dead because diseases don’t actually grow exponentially, at least not on the global scale. So maybe vampire populations don’t either. Maybe there are just enough vampires, in just the right places, biting just enough people, with just enough natural immunity, to sustain a small population capable of going unnoticed for centuries at a time.

Happy Halloween, everybody. And sleep tight.

Teachers: want to have this conversation in the classroom? Check out the lesson materials for our latest lesson, Pandemic.

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