The downstairs toliet had been running continuously for some time. I looked into it over the weekend and discovered the problem. The flap in the tank wasn't covering the hole all the way, because it was warped. It looked like it had been left out in the sun for too long or something. I can't imagine how a piece of rubber inside a toilet tank could possibly end up looking like that. It's not like I have a cold fusion reactor in the tank. I keep that in the upstairs toilet. I'm still completely baffled.
I went and got the replacement. I've decided that toilet flushing mechanisms are like snowflakes. Each one in my house is different, and the ones in my house are different from any other I've seen before. I've got to believe that there's some demented guy in a factory somewhere thinking up crazy arm designs 24/7. "Oooo, I got it, we'll take the floater, attach it to a pully, and then the pully will open a gate which will reveal cheese to a mouse, who will then run on a wheel to get to the cheese, opening the drain and causing a flush. The mouse will eventually drown and then that will stop the take from refilling."
As with all my projects, I had to half-ass the new flap. Works fine, though. Oh, and it rang up as "RED FLAPPY FLAPPER". I sense a new comic strip coming on...