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RIVER CROSSING: Missionaries and Cannibals, Polya

Math Circle 5.19.2015

(May 19, 2015) Only R and J were there;  the rest of the kids were out sick.  We didn’t want to continue with the Dark Bridge/Unicorn problem without the others, so we tackled the famed river crossing problem Missionaries and Cannibals:  In the missionaries and cannibals problem, three missionaries and three cannibals must cross a river using a boat which can carry at most two people, under the constraint that, for both banks, if there are missionaries present on the bank, they cannot be outnumbered by cannibals (if they were, the cannibals would eat the missionaries). The boat cannot cross the river by itself with no people on board. (Wikipedia)

  • The kids named the missionaries - Joe, Tim, Bob - and the cannibals - Marly, Big Jeff, and Nothing Ho Hay. (I abstain from forming any conjectures about what cultural POVs may have been behind this naming convention. )
  • They decided to construct an elaborate diagram, with each person having a distinct appearance. This made the problem come alive. (See photo!)
  • The kids asked lots of questions. A sampling:
    • Can cannibals be girls?
    • Is what the missionaries are doing ethically okay?
    • Is what the cannibals are doing ethically okay?
  • They stated their assumptions, such as
    • None of the missionaries “went native” and became cannibals
  • They made progress, moving everyone across the river in 9 moves, determining that yes, it is possible. They asked a new question:  can it be done in fewer than 9 moves?

 

PROOF

The students were ready to talk about a few other things at this point.  We did several proofs from  Zvonkin’s book Math from Three to Seven:

  • “Prove that we can see with our eyes and hear with our ears, and vice versa.”
  • “Prove that clouds are nearer to the earth than the sun.”
  • “Prove that we think with our head and not with our stomach.” (p77)

The first two were done within 2 minutes by these 9-11 year olds, unlike the much younger kids I did this with last year, who had to struggle for a while.  The third remains unproven – by these kids, the younger ones, and in the book.

 

POLYA 

We moved into a math history interlude.  I told them of George Polya.  Polya had a special interest in problem-solving methods, which we certainly need for river crossing problems!  The kids enjoyed hearing some biographical items from his childhood, such as his dislike of math due to bad teaching.  We discussed Polya’s Ten Commandments of Teaching after trying to predict them first.

The girls saw in the book I was reading from Polya’s famous “Random Walk Problem” and took it upon themselves to spend some time on that:  “In this problem one imagines a modern city with perfectly square blocks with just as many streets running east and west as north and south.  Given any one intersection as the starting point, once could move in any of four directions.  If the choice at each intersection were purely random, what is the probability of returning to the starting point?”*

As the students were forming conjectures from their diagram on the white board, they posed some interesting questions/comments about math in general:

  • What makes a good math question?**
  • The difference between 2-part questions and 1-part questions is interesting.
  • This question (Missionaries and Cannibals) is different from the Dark Bridge Problem. Both however, share some characteristics that make them good math problems:
    • “Has a plot”
    • “Like playing a game,” although “sports and math are really different things”
    • “Less boring”
    • “Hard”
    • “Teasing you”

Then we ran out of time (with no solution to Random Walk, but curiosity aroused).   One more week to go!
Rodi

* Historical Connections in Mathematics Volume III,  p90-91, and  Mathematical Discovery. Vol 2

**In hindsight, I realize that one thing that makes a good math question is that is deserving of a name.

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