Light switching problem

This is a problem I never fully understood. Maybe someone here can finally explain it to me.

A light starts out off. After 30 seconds, it switches on. After 15 seconds, it switches off. After 7.5 seconds, it switches on. And so on. After exactly one minute from the start, is the light on or off?

edit: I don't know :)

TBQH, you never get to a minute in your scenario because you keep switching it on and off with 1/2 the remaining time left in the minute. For instance you start with a 1 minute switch after half a minute, then you have 30 seconds left, switch after half of that, then switch after half of that. The light will flick on and off at tiny fraction of a second.

Quote: Gabes22

TBQH, you never get to a minute in your scenario because you keep switching it on and off with 1/2 the remaining time left in the minute. For instance you start with a 1 minute switch after half a minute, then you have 30 seconds left, switch after half of that, then switch after half of that. The light will flick on and off at tiny fraction of a second.

This is like Zeno's Paradox with a lightbulb. If it's incandescent, the light will appear illuminated at time t=60s because the power is rapidly flickering and there is a non-zero delay for an energized filament to fully cool off. If it's fluorescent (CFL), it will burn out before you get to t=60 due to the ever-increasing cycles on the ballast, so it will be off.

For a theoretical light which has none of the physical problems above, I don't know. :)

Edit: it turns out there's already a name for this problem. See Thomson's Lamp.

And the answer is "I don't know" so I was right. :)

@ Gabes, The switcher may never get to a minute, but a minute can certainly elapse from the point the switching started from the perspective of another person. So at that person's minute, is the light on or off?

Thanks, ME, I never heard it referred to as "Thomson's Lamp" before. Maybe the real answer is that time is quantized and thus (in our universe) it would be impossible to switch that quickly, therefore the question is invalid.

The "right" answer seems to be that the problem is ill-formed because you're assuming that it is possible to perform an infinite number of tasks in a finite period of time. Time doesn't need to be quantized in order to reach that conclusion either. See the discussion of the Ross-Littlewood paradox.

Another way of looking at it would be that at some point sufficiently close to one minute, the required speed of the lightswitch would be faster than the speed of light...

After about 50 seconds, I'd be getting up and telling the switcher to stop messing around with the lights.

I think attempting to experiment with this problem would cause the wave function of the Universe to collapse out of sheer disgust :P

Quote: jon

This is a problem I never fully understood. Maybe someone here can finally explain it to me.

A light starts out off. After 30 seconds, it switches on. After 15 seconds, it switches off. After 7.5 seconds, it switches on. And so on. After exactly one minute from the start, is the light on or off?

The switching frequency diverges at exactly 1 minute. So your problem description simply lacks the information how the light should be operated at 1 minute (and beyond). There is no answer to your question unless you specify what the dynamics of the "switch" are.

The light is off. Why? It would burn out.

I think the light would be on. The switching would occur so fast that it would appear as if the light were burning steadily.

This is a good example of reality vs fantasy. In reality, there is no infinity, its only something that was dreamed up in the mind of mathematics. In reality, there are a finite number of particles and these are being re-positioned around a finite number of locations, and this occurs in DISCRETE segments of time. Zeno's paradox is also wrong for the same reason...

When we begin to approach the Planck time interval, things dont operate on a continuum, and at that point, the light bulb will begin switching on and off NO FASTER THAN 1 switch per Planck Time. At that point, its just a matter of counting Planck-time intervals until the one-minute elapsed test time is met, and it IS a FINITE count.


This will occur at exactly at "n cycles" when (1 minute - the summation of [ 60 / 2 + 60 / 4 + 60 / 8 ...] ) >= 1 Planck-Time unit.

After this moment, just add (60 * Planck-Time) - n Planck Time Units and that is exactly when the light will cross the 1-minute threshold. One could easily write a computer program that determines the count of "n cycles" and add the count of remaining Planck-Time units; each of these represents a single switch, on or off, so just test if the number is odd, that means the light would be on, because it started out off.