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A note from the future: I've got to say that as I'm writing these problems, I'm both tempted and reluctant to talk about the solutions. I'm tempted to talk solutions because I find myself wanting to be specific about the kinds of conversations I like to have with students about these problems. I'm reluctant because I, personally, would (almost) always rather explore a problem on my own, at least for a few minutes, before reading someone else's ideas about it... and maybe some of you are like me in that way.
I've chosen to avoid solutions, though I think it might make this post sound a bit vague and hand-wavy. I welcome your thoughts in the comments.
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You may already know the first one. It's a classic:
The Handshake Problem. If everyone in our class shook hands with everyone else, how many handshakes would occur?
Then, once we find the pattern (and perhaps even an equation for the number of handshakes given the number of people), I like that we can explain that pattern in a number of different ways. We could explain it purely using the context, or using arithmetic in a clever way, or using algebra, or by making a geometric argument... And, even better, we can connect all of these different explanations.
I like to follow-up the handshake problem with a seemingly unrelated problem:
The Polygon Diagonals Problem. A convex polygon has 170 diagonals. How many sides does it have?
Students realize immediately that 170 diagonals is ungrokable; we have to start smaller. But then students usually start by drawing polygons and counting their diagonals, without realizing that they have inverted the problem. That is to say, the problem gives you the number of diagonals and asks you for the number of sides. Students usually start by fixing the number of sides (for example, by drawing a pentagon) and counting up the diagonals. I like that the strategy of solving a simpler, related problem emerges so naturally.
Plus, the polygon diagonals problem has all the great features of the handshake problem when it comes to explaining the pattern and the equation that are eventually revealed.
Then, I like to conclude with a problem that seems both (a) unrelated and (b) totally tedious:
The Multiplication Table Problem. What is the sum of all the entries in the standard 9-by-9 multiplication table?
In fact, if you haven't ever explored the multiplication table problem, I encourage you to kick it around for a few minutes. I was surprised at the number of cool connections hiding in there.
I look forward to hearing from people in the comments!