Humans, Zombies, & Other Problems crossing the river

This activity takes a couple of classic logic puzzles and adapts them to better suit K-12 audiences. It was designed in the Desmos Activity Builder, so it can be played online and easily given to a classroom of students. The activity as presented was developed at the San Francisco Math Circle, and was recently featured on MathCommunities.org’s Math Monday Live program. 

A town has recently been plagued by an epidemic of zombies! Luckily, the virus has just started to spread and the infected are able to stave off their hunger for human brains… for now. In fact, they’re willing to work alongside the remaining humans to help them get across a river to safety. 

Copy of Crossing the River (2)
Click on the image to open the activity in Desmos.

But it’s not so easy. Rules will be needed to keep the humans safe.

  • The boat can only carry two people (humans or zombies) across at a time.
  • At least one human or zombie has to be in the boat for it to sail.
  • Keep all hands and legs in the boat. The river is too dangerous to swim.
  • If the zombies on one side of the river ever outnumber the humans, even for a moment, they will give into their hunger and attack the delicious humans.

Is it possible to get all the humans safely across the river? If the humans can solve this problem, they will be rewarded with fantastic newsthe town on the other side of the river is able to cure the virus! This leads to a new question: Is it possible to get all the humans and zombies safely across the river? 

Watch students attempting to solve the problem during Math Monday Live.

exploring further:

Three humans and two zombies may be a good place to get started with the problem, but the problem can be explored with many different starting set-ups. Additionally, once students find a solution to the problem, challenge them them to keep track of how many trips it took them and find the fewest number of trips needed to cross the river. (When dealing with zombies, time is of the essence!) Older students can then record this minimum as a function of the number of humans and the number of zombies.

And this isn’t even the only type of problem you can run into when crossing rivers! Readers and students may be familiar with the wolf, goat, and cabbage problem:

Once upon a time a farmer went to a market and purchased a wolf, a goat, and a cabbage. On his way home, the farmer came to the bank of a river and rented a boat. But crossing the river by boat, the farmer could carry only himself and a single one of his purchases: the wolf, the goat, or the cabbage. If left unattended together, the wolf would eat the goat, or the goat would eat the cabbage. The farmer's challenge was to carry himself and his purchases to the far bank of the river, leaving each purchase intact. How did he do it?

Wikipedia
Zombies2
Click on this image for the editable Teachers' Copy of the Activty.

Some Notes on Narratives in problem posing

There is a rich history of interesting math and logic problems students today can puzzle over and learn from, and they often come with striking narratives. However, it is important to consider any cultural connotations associated with how the narratives are presented.  For example, some readers may have seen this activity previously as the “missionaries and cannibals” problem, or may be familiar with the the similar “jealous husbands” problem. We chose to reframe the problem as “humans and zombies” to make the narrative more relatable and, more importantly, to remove the unfortunate colonialist and sexist references that are irrelevant to the actual problem.

In addition, when presenting problems to students, take the opportunity to consider what rules of the problem can be naturally explained through narrative. As an example, when I first presented this activity to second grade students, they immediately asked why the humans couldn’t simply swim across. As opposed to telling them “because those are the rules” and shutting down their question posing, I worked within the story we were telling to find a justification: The river was too dangerous.

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