The Thomson's Lamp paradox:
A mad scientist owns a desk lamp. It begins in the toggled on position. The scientist toggles the lamp off after one minute, then on after another half-minute. After a quarter-minute the lamp is toggled off, then the scientist waits an eighth-minute and turns the lamp on again. The scientist continues toggling the lamp, waiting one-half of the previously waited time between toggles. After a total sum of two minutes of toggling, what is the state of the lamp (on or off)?
Why wouldn't we be able to solve this, since the amount of time that it is in a state for would approach Planck time? Wouldn't we just have to figure out the state that it is in when the time reaches Planck time?
We wouldn't be able to solve this from a physical point of view because once the time goes down past a certain point the velocity of electricity through copper wiring will not be able to keep up with the switching of the (incredibly dextrous) mad scientist. From a more mathematical point of view, the time will keep on getting smaller, and the scientist will never reach the end of the two minutes, as the time between toggles keeps dividing. There is the paradox, as though two minutes is a real amount of time, his system of switch toggling means that the end of those two minutes will never come.