So here’s some of the stuff I’ve noticed while being on the Internet and sometimes noticing interesting mathematical stuff.
Here from the end of January is a bit of oddball news. A story problem for 11-year-olds in one district of China set up a problem that couldn’t be solved. Not exactly, anyway. The question — “if a ship had 26 sheep and 10 goats onboard, how old is the ship’s captain?” — squares nicely with that Gil comic strip I discussed the other day. After seeing 26 (something) and 10 (something else) it’s easy to think of what answers might be wanted: 36 (total animals) or 16 (how many more sheep there are than goats) or maybe 104 (how many hooves there are, if they all have the standard four hooves). That the question doesn’t ask anything that the given numbers matter for barely registers unless you read the question again. I like the principle of reminding people not to calculate until you know what you want to do and why that. And it’s possible to give partial answers: the BBC News report linked above includes a mention from one commenter that allowed a reasonable lower bound to be set on the ship’s captain’s age.
In something for my mathematics majors, here’s A Regiment of Monstrous Functions as assembled by Rob J Low. This is about functions with a domain and a range that are both real numbers. There’s many kinds of these functions. They match nicely to the kinds of curves you can draw on a sheet of paper. So take a sheet of paper and draw a curve. You’ve probably drawn a continuous curve, one that can be drawn without lifting your pencil off the paper. Good chance you drew a differentiable one, one without corners. But most functions aren’t continuous. And aren’t differentiable. Of those few exceptions that are, many of them are continuous or differentiable only in weird cases. Low reviews some of the many kinds of functions out there. Functions discontinuous at a point. Functions continuous only on one point, and why that’s not a crazy thing to say. Functions continuous on irrational numbers but discontinuous on rational numbers. This is where mathematics majors taking real analysis feel overwhelmed. And then there’s stranger stuff out there.
Here’s a neat one. It’s about finding recognizable, particular, interesting pictures in long enough prime numbers. The secret to it is described in the linked paper. The key is that the eye is very forgiving of slightly imperfect images. This fact should reassure people learning to draw, but will not. And there’s a lot of prime numbers out there. If an exactly-correct image doesn’t happen to be a prime number that’s all right. There’s a number close enough to it that will be. That latter point is something that anyone interested in number theory “knows”, in that we know some stuff about the biggest possible gaps between prime numbers. But that fact isn’t the same as seeing it.
And finally there’s something for mathematics majors. Differential equations are big and important. They appear whenever you want to describe something that changes based on its current state. And this is so much stuff. Finding solutions to differential equations is a whole major field of mathematics. The linked PDF is a slideshow of notes about one way to crack these problems: find symmetries. The only trouble is it’s a PDF of a Powerpoint presentation, one of those where each of the items gets added on in sequence. So each slide appears like eight times, each time with one extra line on it. It’s still good, interesting stuff.