Thomson's Lamp

An impossible gadget:

"Thomson's Lamp" is an example of a "supertask," a category of paradox that involves an infinitely-divisible task. One form of the paradox runs as follows:
You have a lamp that can be turned on and off using a toggle switch. At the start the lamp is turned on for exactly one minute, at which point it is turned off for 0.5 mins., and then turned on for .25 mins., and then off for .125 mins.... and so on. The question is, at the two-minute mark is the lamp on or off? Also, does the answer change if the lamp begins in the off position for the first minute rather than the on position?
Common-sensically and practically it would seem there ought be a simple, or at least a mathematically-calculable solution, to these questions --- afterall, at the two minute mark the lamp MUST be either on or off! But in fact, we are dealing with an infinite sequence (1 + 1/2 + 1/4 +1/8 +1/16 +....), and as such there is no one single right answer --- different arguments/solutions can be logically made, and even semantically the problem is unsettled. In part the answer depends on how fast one assumes the (undetailed) turning on and turning off action itself takes --- is it 'instantaneous' (eating up no amount of time), or does it take some finite amount of time (say perhaps, with the speed of light as a limiting factor)? In short, Thomson's Lamp is a fun thought exercise that oddly evades any proven solution.