Two problems have been floating around in my head lately, and it’s only fair to share them, and let you know where I got them.
Imagine this: You have four bottles of liquid. One of the bottles contains a poison strong enough to kill a rat by morning if the poison is taken at night. Only a little bit of the solution is required to provide a lethal dose. if you have two live rats at your disposal and one night to experiment, how can you determine which bottle contains the poison?
This is from the math dept. website of the school I used to teach at in New York: Saint Ann’s. Their problems of the week are available on their site, and absolutely worth checking out. I return whenever I want to get inspired.
Here’s the other:
Several years ago, I think in 2002, there was a Mandelbrot contest problem that asked for the largest number of regions in the plane that could be generated by two triangles. The answer turns out to be 8, and it’s interesting to think about why there can’t be more.
Naturally, this got me wondering about the integer sequence: what is the largest number of regions in the plane you can generate using two n-gons? At first this problem seems very simple: 8, 10, 12, 14, …
But after a little while, I realized it went 8, 18, ?, 38, ?, …, where the number of regions for two 2n-gons is easy to compute. You can probably figure out what I was assuming that wasn’t actually a requirement.
For odd numbers of sides, things are much more complicated. In fact, to this day I’m still not sure of the answer. I have a drawing that I think gives the maximum number of regions, but I haven’t been able to prove that there isn’t a better drawing out there somewhere. Anyone have any ideas for me?
This is our friend Josh Zucker, who just launched a new blog, Uncover A Few. It’s well worth checking out. I’ve already gotten hooked on a couple of his problems.
Both of these sites are now linked in the Math for Love blogroll.