Lines Beyond y = mx + b

What are my professional goals? Many days, I’ve got precisely three: I want kids to feel curious, then frustrated, then ohhhhhhhh.

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Math is pretty great for this. It’s full of puzzles and mysteries. Why do the angles in a triangle always sum to the same thing? How many moons would fill the sun? Why is it so hard to roll double sixes? There’s plenty here to excite curiosity, to elicit frustration, and to satisfy the intellectual itch for ohhhhhhh.

But there’s a common problem: too often, kids can beat the puzzles without feeling any of those things.

I find this, for example, with lines in the coordinate plane.

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The Island of Democrats and Republicans

On February 10th, the world lost Raymond Smullyan: logician, puzzlemaster, and blue-ribbon Gandalf lookalike.

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Even if you don’t know his name, you’ve probably wrestled with his logic puzzles. They share a whimsical sense of rigor: “You come to an island where there are two types of people: knights, who always tell the truth, and knaves, who always lie…”

They’re silly and frustrating and fun; everything mathematics should be. I love this origin story for how Smullyan first got into such puzzles:

On 1 April 1925, I was sick in bed… In the morning my brother Emile (ten years my senior) came into my bedroom and said: “Well, Raymond, today is April Fool’s Day, and I will fool you as you have never been fooled before!” I waited all day for him to fool me, but he didn’t.

Or did he?

Young Ray had spent all day expecting to be fooled. But the fooling had never come. Didn’t this constitute the greatest fooling of all?

I recall lying in bed long after the lights were turned out wondering whether or not I had really been fooled.

In Smullyan’s honor, I wanted to offer up my own amateur variant on his knights-and-knaves puzzles.

I call it: the island of Democrats and Republicans.

Now, Republicans and Democrats look identical to an outsider like you. But they always recognize one another immediately. And because of their mutual antipathy, they follow this strange custom:

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So, here comes your puzzle. Ten of them, really.

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