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Flashing light bulb problem


Mobius
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Matt, ya don`t need it man :)

 

seriously, if the lightbulb is to be On or Off, it`s simple (I just never thought of it at the time).

 

imagine a square wave, now in THIS situ the ONLY thing that changes is the mark-space ratio, but all that`s kept at /2 time constants :))))

 

so what ever the status is at "switch on" will be the exact same at "switch off" :)))

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Okay, fair enough guys, much more thought went into this than I had originally thought.

 

So to summarise, Matt can give me models for the light bulb being on or off as there is not enough information available i.e. there is only enough information to define its state at some finite time before 2 minutes.

 

YT2095 sees it as a binary progression, so if it is on at 1 minute (odd number) it will be the opposite at 2 minutes (even number) i.e. off, regardless of intervals in between.

 

All good, thanks for the information guys...

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NO!

 

if it`s ON to Start with, ie/ at that point of leading edge on the pulse for ON.

and the whole thing is worked out in a specific time reference (the minute) then at the exact end of TWO minutes (as you started with) then the status will be the same as at Switch On, IF it remains at a constant /2 marker.

the same would occor at 1 minute also. if this minute is divisible by 2 into units, then stop at ANY unit and the status will be the same.

 

it`s not only simple, it`s Provable in real terms :)

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Yes, you've got me bang on.

 

Just because I know something at every point *before* some fixed teim T deosn't necessarily mean that I can deduce what is happening at T.

 

In the ideal situation I wuold have the assumption that things varied continuously (they don't here, it is very discontinuous) or that there was some known extrapolation, but in this case we don't know anything at all (nothing to do with never reaching 2 or some such Xeno type nonsense).

 

There is a continuous variation problem version:

 

sin{1/x} as x tends to zero. it is a pathological counter example to many things (especially in topology)

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huh?

 

this prob doesn`t require maths exactly, just pure logic.

 

we have a 2 state option On/Off. now lets represent this as a square wave.

we see that that for each cycle that the mark-space ratio is reduced by half.

we see that eventualy we`ll need a microscope to see this! but when we use it, this image looks just the same as it did in any other frame.

 

...time goes by and nothing changes except the make-space ratio gets samller and smaller to see...

 

(but since were NOT concerned with pracicalities or other extranious data, this is irrelevant)

 

2 minutes elapsed... the count has stopped exactly, and the lightbulb is captured in the same state as it was when initiated.

 

there`s simply no escaping this fact :)

 

his "problem" is Solved :)

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well, that is obviously nonsense: start the experiment at -2 seconds and have it in the oppposite state at the start of this period as at 0 seconds. now your logic applies perfectly there too so it is in the exact same state as when it was initiated at -2 seconds, only that is now the opposite one.

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eh???

 

the -2 is also irrelevant, we simply shift the set point to -2 and call it 0.

what DOES matter is the periodicity of the mark-space ratio (or pulse width is you like).

since All is /2, what ever occurs in this compression is only a matter of resolution to observe it (but that`s imaterial), if the clock or count is stopped at ANY EVEN division of a minute from the start position, the bulb status will be the same :)

 

I really can`t understand the confusion here?

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"if the clock is stopped at any even division of a minute from the starting position"

 

nope, sorry, that makes no sense. going back to the original formulation you have not specified the wave from time 2 or greater, only for all stages before the time t=2. the coutn has not "stopped exactly" the definition of the wave you gave is not valid at t=2 or later. the self simliarity you use is dimensionless so it implies that the same logic is valid only looking from time t=1 to t=2, and thus concludes that the lamp must be off (if it were on in your orginal reasoning).

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a cycle is point to point in a waveform, so from peak to peak or trough to trough.

we agree upon that yes?

 

now if we start at a peak (ON status for the bulb) and then go to the next peak it will still be on, but Half a phase out and it`s Off.

we should agree upon this also?

 

now if we look at any 3 complet cycles of this waveform, we`ll note that it`s descreasing each cycle by exactly half.

we HAVE to agree upon that.

 

now if you look at those 3 cycles at the beginning, and then look at the another sample of 3 cycles some time later, Both will have exactly the same Ratio decrementation as the original 3 cycles (you`de just need a magnifying glass to see them).

I`m sure you agree with this also, as it`s just the same as with fractions 6/12`th is the same as 1/2 :)

 

now then, we can ignore just how tiny these cycles get (the on/off speed), what Does matter is that they are ALL divided by 2, and so will remain Even throughout.

(remember, we`re Not thinking Real objects or possibilities here).

 

now since All will remain even and at a constant /2, the status at ANY EVEN (base 2) Division of Any arbitrary time unit you wish to employ (the "Minute" in this case) will result in ON status of the bulb.

 

if it`s the same time indexing +half a phase it`ll be OFF status for the bulb.

 

but it isn`t, it`s in minutes, and the decrement count is ONLY in equal /2 counts of this minute.

 

I really can`t explain it any better than this :((

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so, you''re stating that after 2 minutes we've added up an even number of cycles, despite the fact that this is an infinite number of cycles we're talking about and parity makes no sense in this context.

 

let me do it for you in words:

 

you are looking at

 

(on off) (on off) (on off)

 

with each on off cylce being half the length of the previous one and the off period being one half the length of the previous on period, right?

 

however, that observation holds true for

 

on (off on) (off on) (off on)...

 

so what ever the conclusion was for the former the opposite holds for the latter.

 

it is hard to make sense of what you're saying since it is written in some short hand that is meaningless to me (ony in equal /2 counts of this minute, for example)

 

secondly, in your model, no wave peak or trough so described ever falls at the time t=2, only the previous times are defined. or to put it in less couched terms *you never get to time t=2 in your model* (this is merely zeno's paradox)

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the bulb is flashing at an ever decreasing on (or off) cycle, it can Still run for 2 minutes though :)

 

the "shorthand" as you put is is: /2 = divide by 2.

and by this the counts i mean that if you divide a minute up by 2 re-occuring, like 60 30 15 7,5 ... seconds right down to small as possible (the amount is non applicable). that`s what I mean.

 

so for cycle 1 you`ll fit 2 cycles of the next single cycle into the same space (X axis).

and so on down the line for 2 minutes.

 

the OP is not concerned with anything but the Bulb status, and so if the rising edge of the cycle is towards ON, and we can fit 2 cycles into the one preceeding it, it stands to reason that no mater how small the cycles become and you can fit Infinate cycles into the space of the 1`st one, the rising edge will still be towards the ON position/status :)

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The last number in 10/3 would be 3 as all the previous numbers are 3 but as you pointed out earlier, there is NO last number of 10/3. So this is not the same as the flashing light bulb problem.

It is obvious that the sequence provided is inadequate to describe the nature of the bulb at and after the limit of 2 minutes.

I cannot see how you can define your cycles.

 

Physics may be able to explain it with discrete time or maybe a 'tunnelling effect'??? close to the 'barrier' (limit). But maths (as defined here) cannot solve the problem.

 

Regardless of the solution if this does happen your light bulb, stick around for 2 minutes to see what happens then buy a new one ;-)

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the point is that if it ever DID reach 2 the status would be ON.

 

just the same as the last number in 10/3 will always be a 3 :)

 

no that is a completely spurious analogty. as i keep pointing out you logic also implies that the status will have to be off and on simultaneously if the notion of "did reach 2" were actually valid. which it isn't.

 

it is also wrong and down right disingenuous to say that there is a last number in 10/3. "digit" would be better, not that it matters since there is not a last (nonzero) digit in the decimal expansion of 10/3.

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It takes 320 attoseconds for an electron to move from one atom to the next. If you could make a supply switch fast enough and the bulb tough enough, the bulb would be off when the on portion of the mark to space ratio was shorter than 320 attoseconds. So I say at 2 minutes it would be off. :P

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Ah yes RICHARDBATTY, but I think we are well passed practicallity here and talking about the theoretical.

 

In other words, what if the light bulb was not restricted by the limitations of physics i.e. the motion of electrons limited to 10^(-18) seconds.

 

I was going to shorter time spans of 10^(-43) seconds where maybe our theoretical lightbulb may break down....

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the point is that if it ever DID reach 2 the status would be ON.

 

If you consider that the state at 2 minutes will be ON then consider the same situation only limiting to 4 minutes (on for 2, off for 1, on for 0.5...). According to your analagy it should ON at the 4 minute mark, but the poblem with this is that after 2 minutes it is in synch with the origional problem only the opposite state.

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Don't mean to bumb this thread but I have just read about the problem in John D. Barrow's The infinite book.

 

The light bulb is known as Thomson's lamp.

 

Barrow pointed out that the question is unanswerable.

He replaced the 'on' and 'off' with +1 for on and -1 for off.

 

The problem is then reduced to the infinite sum

 

S = 1 - 1 +1 - 1..............

 

this sum does not converge, the answer is either 1 (odd number) or 0 (even number) but interestingly could be a 1/2

 

S=1 - (1 - 1 + 1 - 1............)

 

S=1-S

S=1/2

 

i.e. the bulb is half on!!!

 

The point is that as this is an infinite series there is NO result. It is like asking whether the last number in the natural numbers is odd or even.

 

The book itself is pretty good, I recommend it.....

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I did my utmost to elliminate ANY ref to Physicality' date=' and to keep it as Maths(ish) as I could.

 

the way I see the problem as boiled down to the Raw basics as possible is that which I stated before, it`s division by 2 until it`s no longer divisible (that point being the 2 minute marker).

 

it`s like presenting 10 / 3 =

and asking for a finite answer.

 

there is non. at least Not without specifying a resolution (in decimal places).[/quote']

 

10/3 IS a finite answer.

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