# Puzzle: How Many Subsets?

I think I’ve blogged this before, but it’s a great problem.

I would not use it as part of a unit on sets. In fact, I used it every year when sets were not in the curriculum. Now that sets have returned, I’ve tried to use the subsets at a completely different time of year (very early). Off-topic problem solving doesn’t have to be completely off-topic, but it feels better when it is.

Now, just a note up front: students are allowed (encouraged) to ask questions to clarify. In this case they would certainly ask what a subset is, and some groups will ask if “order matters.” Occasionally a group has asked whether the whole set is a subset of itself. I don’t recall anyone ever asking upfront about a set with nothing in it… that’s just completely off the radar.

How many subsets does S = { ×, /, ∇, ☺, ☐} have?

Normally I would let a class play with this. I would offer some advice to groups completely stuck. And then we would review solutions and partial solutions – I like to get to it the same period. Generally we start, since everyone wants to speak, with idea that looked promising but didn’t pan out. And then we review approaches that worked. And then I take over, and link different approaches, and maybe offer one or two more.

Challenge for you, readers, how many fruitful approaches can you find? Recall, the kiddies will be high school freshmen with no background in sets (and probably none in counting). Do you have some approaches that work and are good for these kids? that work, but are not reasonable for these kids?

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