The squirrel and the switch
A hyperactive squirrel has drunken far too much coffee and sits now in front of a switch for a light bulb and is tasked to the press the button several times over a duration of 2 minutes.
After 1 minute he presses the switch and the light bulb goes on.
Then it wait for 30 secs and presses the switch again, so the light goes out.
After another 15 secs the switch is pressed again and this scheme continues until the 2 minutes have passed.
The question now arises: After 2 minutes, is the light bulb on, out, on and out at the same time or is it neither of these choices?
Details and Assumptions :
- Physics shall be ignored for this question as there would arise far too many problems (especially with relativity!), if the squirrel started to press the switch with a speed > c .
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3 solutions
I understand the concept, but got a question: why 0.999999... is equal to 1, and 1 +1/2 + 1/4 + 1/8... can't be equal to 2? Seems to me it's the same principle, and if it was then the answer to this problem would be "light on and off at the same time".
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The two examples you give are of convergent series, whereas 1 , 0 , 1 , 0 , 1 , 0 , . . . . , perpetually oscillates between two distinct values and hence does not converge to one or the other value, (or "state"). The option for the light to be both on and off at the same time does not exist, so we are forced to conclude that there is no way to know the final state of the bulb. (There is a concept known as Cesaro summation which would assign a value of 2 1 to this limit, but this is purely abstract and could not be applied to a physical scenario such as is dealt with here. Conceptualizing a light being both on and off at the same time is better left to the realm of a Zen koan than an actual physical analysis. :) )
This problem could also be compared to lim x → 0 + sin 2 ( x 1 ) , which does not exist as the curve oscillates more and more rapidly between 0 and 1 as x approaches 0 from the left, but does not settle at either (or any) value in the limit.
Can't this be considered as another version of schrodinger's cat? The lamp would be in a super state at the end of two minutes ie both on and off
What I understood that this is impossible case having no result as it can't happen at all. Am I right?
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The scenario described in the question is physically impossible, but Alisa asks us to ignore the rules of physics and just look at the question conceptually. With this in mind, since the final state cannot be determined, (other than noting that the light cannot be both on and off at the same time), we must conclude that "none of those choices" is the correct option.
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We were asked to ignore conventional physics, so it seems that the "light cannot be both on and off" argument is null. At the two minute mark where we find our switch at infinite oscillation, couldn't both states coexist? Barring physics from the problem seems to both open this possibility while elimination the only logic, the logic of impossibility, that makes the "neither on nor off " argument valid.
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@Joe Waller – We are suspending the laws of physics, but not the laws of logic, nor the passage of time. Two minutes will pass, so we can't have a state of infinite oscillation to the end of time. For the duration of the two minutes, there are only two possible states, namely on and off. So once the two minutes have passed and the squirrel scurries off to gather some nuts, the light should be in one of these states. Before the two minutes have passed there is no way to predict what the end state might be, since no matter how close we are to the two minute mark, there is still time for an infinite number of flicks of the switch.
After the two minutes have passed, if the light is on, then there must have been a point in time just prior to the two minute mark that the squirrel flicked the switch on. But since the rational numbers are dense, (i.e., between any two rational numbers there exists another rational number), between the moment the squirrel last switched the light on and the two minute mark, there must have been a moment closer to the two minute mark that the squirrel subsequently switched the light off. Thus the light cannot be on. A similar line of reasoning tells us that the light cannot be off, either. So the light can neither be on nor off after two minutes are up, which is absurd since it can only be in one of these states. So by default the answer is "none of these choices".
This conclusion that the light cannot be on, off or in a state of simultaneous, infinite oscillation at the end of the given time period is actually an argument in support of the statement that such a task, referred to in philosophy as a supertask , is not only physically impossible, but logically impossible as well. As discussed in the link, one could argue that if supertasks were possible then motion would be impossible, which clearly is not the case, unless we are all in fact just dreaming our collectively interwoven life experiences.... Which brings me back to your point, which could be interpreted as saying that the final state of the light need not necessarily be determined by the previous states available to it. We are told what states the light could be in for times prior to the two minute mark, but there were no conditions set on what state the light could be in at p r e c i s e l y two minutes. In some universe it could just vanish; in another it could turn into a frog. While this may sound silly, it is meant to drive home the point that we are not logically compelled to conclude that the light must be either on or off after the given time period - or at least that is the way some philosophers deal with the contradictions presented by this thought experiment. The value of this question is not in the answer itself, as it may not really have a "correct" answer, but in making us consider the logical and philosophical issues outlined above.
OK I understand.
2 ≠ 1 + 1/2 + 1/4 + 1/8... The example shown here halves the duration the squirrel presses the button as shown (1min, 30 sec, 15 sec and so on). Thus 2 minutes would technically never pass.
It can be said that the state at the end of 2 seconds cannot be determined. There would be infinite number of toggles happening before two minutes pass. However, 2 minutes would pass regardless of whether the squirrel toggles the switch or not. Hence, it is incorrect to say, "2 minutes would technically never pass"
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I agree. The light can only be in one state or the other. 2 minutes is a determined amount of time and will occur (this is not a half of a half of a half issue which will approach but never end). There is a definite end and the light MUST be on one state or the other; one of the answers is correct. None is an incorrect response.
Actually 2= 1 + 1/2 + 1/4 +... It's a convergent series.
If you were to attempt to graph this at t=2min it would be undifined. As simple as that. KISS Keep It Simple Stupid
This is a thought experiment known as Thomson's lamp paradox . Ignoring the physical impossibility of this scenario, there are several mathematical arguments that can be made to support the conclusion that the final state of the light bulb cannot be determined. As an example, if we were to assign a value of 0 to the 'off' state and a value of 1 to the 'on' state of the bulb, then the sequence of states would be 0 , 1 , 0 , 1 , 0 , 1 , . . . . . Since this sequence is infinite and does not converge, the "final" state of the bulb is indeterminate.