Friday, October 23, 2009

Books that my Kids will definitely be reading: Gardner's AHA and Gotcha!

Well they'll definitely be getting a copy. Who knows if they will actually read them. But these were two of my favorite books as a kid. I was reminded by a recent nytimes blog post about their author Martin Gardner. They really got me excited about math (using comic strips) but using serious math, applying number theory and topological fixed point theorems and things that gave me a second Aha when I learned about them again in grad school.

One I pondered for a long time and still occasionally do was the Pop-Quiz paradox. I recall seeing a resolution in college, and recall trying to apply the epistemic game theory I learned in grad school to the problem. The neat thing is that these ideas have stayed with me throughout and my appreciation has only gotten deeper with time.

I also first encountered the Monty Hall Problem there.

Another that stays with me (which in grad school I learned is an application of a fixed point theorem) is to think about somebody hiking up a mountain. She starts a 9am and arrives at 5pm. She camps out on top, and then starts walking down the same path at 9am, and arrives at the bottom at 5pm. The interesting question: Is there a time when she is at exactly the same point on the mountain at exactly the same time of day.

The answer it is revealed is yes. And you can see this by envisioning a video of the mountain as she walks up and a video of the mountain as she walk down. And then projecting the video simulatenously onto the same screen. At some point she will have to intersect herself.

The same idea can be applied to show that if you take a piece of paper that lying flat, completely fills the bottom of a box. Then you can pick it up and crumple it up however you want. There will always be some point of the paper that is exactly above where it was before when it was lying flat. Alternatively, you also know that at any given time, there is some point on the earth that is the exact same temperature as the point directly opposite it on the other side of the planet.

I remember learning about multiple infinities, and the difference between countable and uncountable.

And the best thing was they were all done in cartoon form. I loved the stick figures and the blatant asymmetric shapes. Ah great memories.
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