### From the Homework

In class we did the problem where you have 15 teams and you're organizing a tournament in such a way that every team plays every other team exactly once and you need to calculate how many games are played. We also did the problem where you have five possible pizza toppings and you need to calculate how many ways you can choose two toppings for your pizza. We used the "find a pattern" method of solution. For homework I assigned a problem where there were 15 dots arranged around a circle and you had to calculate how many lines you could draw connecting pairs of dots.

What I wonder is at what point this becomes infinite. With two dots, it's pretty clear that the answer is one. For other relatively small numbers, you can also figure out how many lines there are. But at some point there's a transition and the number of lines just explodes.

Reminds me of some problems involving growth of algebras. Add in or take away some kooky property and suddenly your well behaved algebra with all sorts of polynomial-ish properties suddenly morphs into something that has all of the unfortunate qualities of a free algebra while lacking its simplicity. It's like the countable/uncountable problems that are troubling some of my classmates in the computer science course. Stick your sets together one way, and it's still countable; change things up a bit and suddenly POOF! uncountable.

I'd love to be able to ask my student "What were you thinking?" but I lack the diplomacy to ask the question. When the class I teach reaches the unit on countable/uncountable (at a much gentler level than the cs class), I worry about this student. If your intuition can't tell the difference between 105 and infinity, how can you tell countable from uncountable?

What I wonder is at what point this becomes infinite. With two dots, it's pretty clear that the answer is one. For other relatively small numbers, you can also figure out how many lines there are. But at some point there's a transition and the number of lines just explodes.

Reminds me of some problems involving growth of algebras. Add in or take away some kooky property and suddenly your well behaved algebra with all sorts of polynomial-ish properties suddenly morphs into something that has all of the unfortunate qualities of a free algebra while lacking its simplicity. It's like the countable/uncountable problems that are troubling some of my classmates in the computer science course. Stick your sets together one way, and it's still countable; change things up a bit and suddenly POOF! uncountable.

I'd love to be able to ask my student "What were you thinking?" but I lack the diplomacy to ask the question. When the class I teach reaches the unit on countable/uncountable (at a much gentler level than the cs class), I worry about this student. If your intuition can't tell the difference between 105 and infinity, how can you tell countable from uncountable?