1. Oct 3, 2005

mprm86

We have a lamp and a chronometer. The lamp is initially turned off. Then, we start running the chronometer. When it marks 30 seconds, we turn on the lamp. When the chronometer marks 45 seconds, we turn it off. When it marks 52.5, we turn it on. I think you see where am I going. The next interval of time is half of the previous one. The question is: After a minute, how will be the lamp (turned on or turned off)? Does the result depend on the initial state of the lamp?
Thanks.

2. Oct 3, 2005

amcavoy

At first glance, this is what I came up with:

$$\text{Seconds}=\sum_{n=0}^{k}\frac{30}{2^{n}}$$

thus 60 seconds will have passed when k=infinity. That is, you have turned the light on and off an infinite number of times. Because of this, I am unsure of what to do from here.

However, if you have a value less that 60 seconds, you can determine whether the light is on by the following rule:

If k is odd, the lamp will be off. If k is even, the lamp will be on.

I don't know if this is entirelly correct, but it's the best I could come up with quickly.

3. Oct 3, 2005

Tide

At the end of 1 second the light will be on. After enduring extreme relativistic acceleration, the switch undergoes total conversion to energy and ignites the surroundings including the lightbulb. It will take a while longer for things to cool down sufficiently to declare the light off.

4. Oct 3, 2005

nate808

I could be completely off, but i dont know that you could ever reach one minute while following all of the rules. You could flick it on and off an infinately many number of times, but in order to get to 1 minute you would have to stop, becuase the limit only approaches 1 minute, but never actually touches it

5. Oct 3, 2005

amcavoy

Since it is infinite, I would say that the light is on at 60 sec. following what Tide said.

6. Oct 3, 2005

Hurkyl

Staff Emeritus
I'm reluctant to answer, since the original post almost sounds like it's stating a homework problem... I think I'm just getting paranoid, though!

This is another fun pseudoparadox! The mistaken assumption is that it makes sense to ask about the state of the lamp after one minute!

Note that any physical intuition shouldn't even be applicable to this problem since it breaks (at least) one of the fundamental postulates we like to have in our models: that motion is continuous. I'm not just speaking about the jumping directly from on to off and back, but that there cannot be a continuous continuation at the one-minute mark.

(Why do I speak about physical intuition? Well, why else would you think it makes sense to ask about the state of the lamp at a future time, if you weren't applying your physical intuition to the problem?)

(Of course, if we tried this experiment IRL, we'd either reach the limits of our ability, or break the lightswitch, before the one-minute mark)