I recently gave my juniors the classic Towers of Hanoi puzzle to play with in small groups. It went something like this:
You have three plates, and plate #1 has a stack of 5 pancakes, in order from the largest one on the bottom to the smallest on top. The puzzle is to get the stack onto plate #2 using as few moves as possible.
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Two rules: (i) you can only move the top pancake on a stack, and (ii) at no time can any larger pancake be on top of a smaller pancake.
They spent a couple minutes getting familiar with the mechanics of it, and then settled into working together, shifting pancakes and keeping a count of their moves.
It didn’t take them long to get into a pretty strategic approach: get the smallest two pancakes onto plate 3, then move the third pancake to plate 2, now get the smallest two pancakes onto plate 2 also, move the fourth pancake to plate 3, etc.
A couple of related observations struck me as I watched them do this. First, every group got preoccupied for a while about whether their first move should be to plate 2 or plate 3. And second, even after they’d begun working systematically, they still made each move manually, rather than abstracting the process of moving a substack. For example, even after they got to here…
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…and knew that it had taken them seven moves to move the substack of the smallest three pancakes from one plate to another, they still proceeded to manually make the next seven moves to get to:
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Eventually, each of the groups managed to complete the puzzle using 31 moves, but when I asked each group whether that was the best they could do, they were uncertain. Now, on the one hand, I think it’s pretty likely that if they had made the leap of abstraction to treating moving a whole substack as a “thing”, they would have been able to dispense with their doubts pretty easily.
On the other hand, the fact that they hadn’t made this leap prompted some pretty nice exploration. In trying to decide what was optimal, several groups started looking at the 1-, 2-, and 3- pancake cases and saw the “two times the previous plus one” pattern. Naturally, this did give them more confidence in their answer of 31 for the 5-pancake case, even if it wasn’t a full proof. Realizing that they hadn’t really explained why the 2*previous+1 pattern would hold, one group started keeping track of how many times each pancake is moved in the course of solving the problem. They found a remarkable, delightful result (I’ll not give it away here), but still didn’t quite manage to justify the pattern.
On reflection, I think the reason that the students didn’t naturally make the leap of abstraction is that there were too few pancakes—few enough that it wasn’t actually inconvenient enough. There’s a sort of “sweet spot” to many problems, a perfect size where the problem is small enough to be accessible, to feel like it’s possible and therefore worth trying, but at the same time large enough that it soon becomes impractical or inconvenient to do it concretely. In the end, I showed my class the abstraction in this problem, but I didn’t feel great about having to do that. I think if I’d hit the sweet spot—the range of ideal inconvenience—they very well might have been prompted to make that abstraction themselves, in response to their own authentic desire to simplify and speed up their thinking.
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