The back bathroom has an American Standard Champion toilet, and my daughter said she thought it was still running after it flushed. This is symptomatic of a bad shut-off valve in the flushing mechanism. I could not hear it running, but after flushing, the water level should be about an inch below the top of the overflow tube, as it was.
If we wait a day after the last flush, we can see that the water level has crept up to the top of the overflow tube, which indicates that the valve is not completely stopping the flow of incoming water. Time to get a new valve.
Checking the installation instructions for the toilet they describe the valve as a Fluidmaster 400A. Checking the internet, this is still a common device; we bought ours at Home Depot.
The box contains the flush mechanism, plus instructions.
First we turn off the water supply at the cut-off below the tank.
We empty the tank, by flushing, then using a plastic cup to bail what remains, until finally using a sponge to get the rest. Then we can remove the old mechanism.
We put the new valve in place, and attach it by a plastic nut on the outside bottom of the tank.
Attach the water supply line.
And turn the water back on to fill the tank.
Check that it is full to the right level. Check that there are no leaks at the bottom of the tank. And we are done. This took about an hour.