Clarifications, Flaws, Assumptions, and Codes
November 6, 2012: “You and your parents are in your coastal house at a time of great danger. A person who can end the danger is in a ship. At some point in the night, the ship’s captain hops into a rowboat and comes close enough to shore to see the upstairs windows in your house. It is your responsibility to signal to the captain when it’s safe to come ashore. The safety code is “27.” We have 5 visible windows (2 in the children’s room, 2 in the parents’ room, and 1 in the hall) and 5 candles, and nothing else we can use to signal. How can we do it?”
We began session 1 of our new Math Circle for 9-10 year olds with this question. Immediately, the students began spouting clarifying questions:
What is the danger? Are there any other windows? Any doors? Can you get onto the roof? How many matches do you have? What’s between the house and the water? Does the enemy know the code?
U, X, and D stood up with excitement. Then A and J (and others) took over the storytelling:
“There was a 6-year-old princess who was kidnapped. Her kidnapping started a war. Enemies (on the kidnapper’s side) are surrounding your house and you can’t get out without losing your life. Spies are in the water and the bushes in front of the house.”
Possible solutions (and their flaws) started flowing:
- We could strike the match, or light the candles, 27 times (but you don’t know when the captain is coming) – S and A
- We could melt the candles and shape them into the number 27 like birthday number candles (“But the enemy can’t see the signal because then they would get prepared,” countered C) - A
- We could make shadow puppets and form the number 27 with our hands (a very popular idea, but foiled by the idea of enemies, again) - X
- We could distract the enemies and jump out of the house and get a raft and go get the captain (spies – “you can’t leave the house!” reminded V) - A
- We could watch for the captain and see when the captain comes and then point a laser pointer into a 27 (too dark, and besides, spies again) - J
Students started seeing more flaws than possible solutions – an important math skill. We can’t say with certainty what is possible until we know what is impossible. V pointed out that “you have to know if, one, the captain is loyal, and two, that it’s really him.” This was met with discouragement until I gave the group permission to make assumptions. As the Circle evolves, they’ll get to a point where no permission is needed; they’ll develop the math skill of making assumptions and, very importantly, stating them. Our group assumed that the captain is loyal, the person in the rowboat is truly the captain, and that we can’t know for sure whether the enemy knows the code.
M then made the suggestion that, “since this is a Math Circle, the answer probably has to do with math.” This comment moved the discussion in another direction. Someone mentioned using a code for 27. Students wondered whether the enemy could crack the code. U proposed adding in a false code. D disagreed, saying, “But a false code might not necessarily be the best way to distract the enemy.” While not everyone agreed on the need to distract the enemy, everyone liked the code idea. Discussion turned to possible codes. Someone suggested using 2 candles (“the number 11”) to represent 27 (“safety”). Another suggestion was to value the candles in each room differently: candles in the children’s window were worth 10 and in the hall worth 7. Then the captain could add up the value of each lit candle to know when it’s safe to bring the “VIP” (as named by D) to shore. The class seemed happy with this solution, so I threw a possible wrench into their plan: there are other messages you may need to signal to the captain. What might they be? The group suggested “SOS at the house,” “enemy sending ships out,” and “general danger to the ship.” The kids assigned each of these messages a numeric value: 44, 13, and 16. We were out of time as I stated that I was confused and needed help seeing how these numbers would work with the code. People were still calling out ideas (seagulls, distractions, false codes, and so on) as I was trying to walk out the door. I promised that this will be continued next time.
Thanks to Bob Kaplan for this problem. Also, thanks to the kids for telling part of the story. When kids compose part of the problem, their interest level soars.