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Question

None of the following experiments can be carried out in practice.

But that doesn't stop us from thinking about them.

In one way or another each of them involves the infinite series

1/2 + 1/4 + 1/8 + ... + = 1.

Each asks for a response. Which responses, if any, can be given?

  1. A lamp is switched on and remains on for 1/2 minute.
    Then it's switched off for 1/4 minute.
    Then on for 1/8 minute, and so on.
    After 1 minute, of course, the switching stops.
    Describe the condition [on or off] of the lamp.
    .
  2. A supercomputer is programmed to calculate a print the value of pi, one digit at at time.
    The first digit prints after 1/2 minute. Each succeeding digit prints in 1/2 the time, and,
    [to keep the supply of printer paper finite], in 1/2 the space of the previous digit.
    After 1 minute, of course, the printing stops.
    Can we at last determine pi's final digit?
    .
  3. A marble rests in tray A.
    After 1/4 minute a machine transfers the marble to tray B and rests for another
    1/4 minute while a different machine transfers the marble back to tray A.
    During the next 1/8 minute the first machine puts the marble back to tray B
    and rests 1/8 minute while the second machine returns it to tray A.
    After 1 minute the halved-time transferring process stops.
    Describe the location [tray A or B] of the marble.
    .
  4. A constant-speed runner completes 1/2 of his journey in 1/4 minute; then he rests for 1/4 minute.
    In the next 1/8 minute he completes 1/4 of his journey, resting thereafter for 1/8 minute. And so on.
    Describe the position of the runner after 1 minute.
    .
  5. A certain toy ball returns to 1/2 its original height 1/2 minute after being dropped.
    It falls from that point and returns to 1/4 the original height 1/4 minute later,
    and to 1/8 the height 1/8 minute after that. And so on.
    Describe the motion of the ball 1 minute after the initial drop.
    .
  6. The ball in question 5 is initially red.
    It changes color, alternating between red and blue, each time it strikes the ground.
    Describe its color after 1 minute.
    .
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Question: For 1, 3, 5 + 6 ... Would there be an amount of time for the processes in between (i.e. switching the lamp, transferring the ball, time it takes a ball to slightly flatten on contact then spring back) to occur? Or am I being too practical for these thought experiments?

Edited by Gusbob
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Would it be slightly pedantic of me to say that 1/2+1/4+1/8 etc woud never actually equal 1, so none of the experiments will be completed?

Pedantic or not, it would be incorrect. ;)

But let's leave the discussion of infinite series for another time.

For the present purpose, make the assumption that it does in fact equal 1.

If it helps your approach, change "after 1 minute" to "after 1.0000000000001 minutes".

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though for many of the questions you cannot garentee an answer you can come up with a "likely" answer.

1. light on would be = 1/2 +1/8 +1/32... = 2/3

light off would be 1/4 +1/16 +1/64... = 1/3

so its most likely to be on.

2. using the same logic, you can calculate that the most likely last digit is 3.

3. since the second machine never rests, i's say the ball's definitely in tray A.

4. his journey would be completed.

5. it would be at rest. (relative to the ground.)

6. using the same logic as before, its most likely red.

Edited by phillip1882
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for 1,3,4,and 5:

whatever the position each one is in after a 1/2 minute, wouldn't it be the opposite after a minute?

so for:

1. the lamp will be off

3. marble in tray A, since after 1/2 minute the marble is in tray B

4. the runner will be running across the finish line

5. the ball will be on the ground

for number 2...aren't the digits of pi infinate? I'll say no you couldn't determine the last digit after 1 minute

for number 6...the ball starts out red...after 1/2 minute it is blue, since it hit the ground in that time, so I'll say red as it hits the ground after a minute.

all that may be probably is wrong...if any of it is right I can try to explain why I thought all that..if wrong, then no need to... :P

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1, 3 and 6:

In terms of switching states on-off / a-b / red-blue it happens infinite times, so you can not determinate the state.

If you say the switch is in some particular state / position at time x<1min and 1min - x is allmost 0, and there is one last switch (so it will have its final state), this switch will happen in time (1min - x)/2. So there is still time to make one more switch because there is (1min - x)/2 until the end of 1min and it takes only (1min - x)/4 to make another switch... and so on, an on, and on...

Basically, when you decide on final state, there is still time for one more change of state. 1min is infinity for this sequence and there are infinite number of switches in 1 min. I'm not sure if my explanation is clear in english :(

2. The question is, will we know the final digit of pi. Well, as pi has infinite number of digits, and there are infinite number of prints (the last one will be in a infinite small time - 1min/inf., almost 0 but not 0), after 1min we should see the whole pi printed out, because after 1min infinity is reached... mathematically speaking.

If you print 1 digit per x time, it would take you forever to print pi, but in this sequenc, 1 min "is forever".

4 and 5 - basically apply logic from num. 2

so... ok?

Edited by klmn
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though for many of the questions you cannot garentee an answer you can come up with a "likely" answer.

1. light on would be = 1/2 +1/8 +1/32... = 2/3

light off would be 1/4 +1/16 +1/64... = 1/3

so its most likely to be on.

There appears to be a flaw in this mathematical derivation superimposed on something as philisophical as this question.

Here's why: consider the experiement conducted for 2 minutes where for the first 1 minute the bulb is kept off, then then on for 1/2 minute and so on... in this case, the bulb would be off for 4/3 minutes out of 2. So the probability that will be on would be 1/3!

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For 1)

After some time, the bulb will have to be switched on/off at a speed greater than that of light (3x108 m/s)! So, the bulb would appear to be on at the end of 1 minute.

For 2)

The print of the last few digits would have to be far smaller than the size of an electron and no present super-duper magnifying microscope would be able to read it perfectly! So, I say no, you wont be able to see the last digit.

For 3)

The marble would be in transition. Neither on plate A nor B

This is same as adding 1-1+1-1... and the "accepted" value of this series is 1/2

For 4)

He will appear to be running (same logic as in Q1)

For 5)

It will appear to be at rest (same logic as in Q1)

For 6)

It will apear to be a combination of red and blue colours... that means purple (same logic as in Q3)

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an intriuging set of questions.

i find the runner one the most interesting as it might be mathematically calculatable. I shall attempt that later but for now i think as fort the lightbulb one, either its the same as the question is sine of infinity positive or negative? It can't be answered. OR because the sequence converges to 0, the time between the last switch off and switch on is 0 and therefore its both on and off simultaneously.

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For 1)

After some time, the bulb will have to be switched on/off at a speed greater than that of light (3x108 m/s)! So, the bulb would appear to be on at the end of 1 minute.

For 2)

The print of the last few digits would have to be far smaller than the size of an electron and no present super-duper magnifying microscope would be able to read it perfectly! So, I say no, you wont be able to see the last digit.

For 3)

The marble would be in transition. Neither on plate A nor B

This is same as adding 1-1+1-1... and the "accepted" value of this series is 1/2

For 4)

He will appear to be running (same logic as in Q1)

For 5)

It will appear to be at rest (same logic as in Q1)

For 6)

It will apear to be a combination of red and blue colours... that means purple (same logic as in Q3)

1. I think there is a problem with your conclusion... You are correct for time < 1min or lets say almost 1min

2. It should be a math problem... so I dissagree

3. I don't have any thoughts on this one... yet.

4. He would be exactly at the end of his journey - "finish line"

5. I agree

6. Same as 1, in would appear as "purple" just before one minute mark, so while time expired < 1min

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an intriuging set of questions.

i find the runner one the most interesting as it might be mathematically calculatable. I shall attempt that later but for now i think as fort the lightbulb one, either its the same as the question is sine of infinity positive or negative? It can't be answered. OR because the sequence converges to 0, the time between the last switch off and switch on is 0 and therefore its both on and off simultaneously.

the time between the last switch on and off is never 0 - it's infinitly close to 0, but it's not. You should look at it mathematically, and not in terms of real world

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the time between the last switch on and off is never 0 - it's infinitly close to 0, but it's not. You should look at it mathematically, and not in terms of real world

I think it is mathematical. for a series to converge, it's sequence must converge to 0

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here appears to be a flaw in this mathematical derivation superimposed on something as philisophical as this question.

Here's why: consider the experiment conducted for 2 minutes where for the first 1 minute the bulb is kept off, then then on for 1/2 minute and so on... in this case, the bulb would be off for 4/3 minutes out of 2. So the probability that will be on would be 1/3!

deegee, well of course it would be more probable to be off, if you keep it off for the first minute.

Edited by phillip1882
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deegee, well of course it would be more probable to be off, if you keep it off for the first minute.

That is exactly the flaw I was talking about. The end result must be the same whether you keep it off for the first minute and then start the whole process or if you just start the whole process given in the problem.

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Would say the light would appear on due to the persistence of vision and the maximum frame rate perception of the human eye. For most this would be around 1/30 of a second so this would happen after not too many switches. After so many switches of course the bulb would burn out, in which case the lamp would be off. Hmmmm.... And this scenerio of course can only occur if the lamp is not in the middle of a forrest.

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Pretty much everyone concludes 4 and 5 are answerable.

The runner has constant speed and completes his course in 1 minute, even with the pauses.

This is a red herring version of Zeno's paradox - it's no paradox at all.

The ball quits bouncing and lies at rest.

This experiment most closely illustrates that the infinite series = 1.

These are fundamentally the same:

They all are equivalent to asking whether the final integer is odd or even.

Since there is no last integer, the answer is indeterminate.

But two other positions were taken: let's look at them.

  1. One state [or the other] obtains, with a certain degree of probability.
    .
  2. Both states [with perhaps different weights] obtain.

Consider the questions that arise:

  1. At the end of 1 minute the state has stopped changing.
    In the absence of any stochastic processes, repeated experiments will obtain the same result.
    Given a sample space of completely identical outcomes, describe what probability means.
    .
  2. During the switching process [which we take as instantaneous] one state or the other exists.
    After 1 minute, the state stops changing.
    Describe what a mixture of states means after 1 minute, when it never existed prior to that.
  • There's a lot more going in question 2: how can you read something that has 0 size, etc.

    While both positions [yes and no you can't find pi's final digit] were taken,

    the reasoning was harder to make clear.

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    During the switching process [which we take as instantaneous] one state or the other exists.

    Taking the switching process as instantaneous makes the idea of being in one state tenuous at best.

    Consider:

    As time approaches 1 minute the switching instants approach eachother. At some infinitesimal instant in time before 1 minute the change will occur in both directions at once. True "race condition" anyone?

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    Taking the switching process as instantaneous makes the idea of being in one state tenuous at best.

    Consider:

    As time approaches 1 minute the switching instants approach eachother. At some infinitesimal instant in time before 1 minute the change will occur in both directions at once. True "race condition" anyone?

    Not really.

    Specify any time t<1; the OP will describe the state with unconditional clarity.

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    There's a lot more going in question 2: how can you read something that has 0 size, etc.

    While both positions [yes and no you can't find pi's final digit] were taken,

    the reasoning was harder to make clear.

    Here's my thought...

    First we have to acknowledge that the final digit will not have the size of 0. It wil have the size of a point, so almost 0, but not 0. So if you have a supercomputer that can calculate at given speed, and a printer that can print with that speed AND size, then we could assume that we have the equipment to read what the printer wrote... In our world of puzzles and theory, we have a supercomputer, and a super-printer, so why not super-microscope or even super-vision :)

    So I still stand by my initial claim, that the final digit of pi would be reached and it could even be red.

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    i havent read all the responses but theoretically

    numero dos'o

    if the computer could reach an infinite speed yes no matter if there is one or not

    either the computer will not have the speed to finish. or it will print it out.

    for example say pi has 200 digits the computer would need to be able to print and think of a digit in (1/2)^200s

    if it cant do this it would never reach that digit even though this pi is supposedly finite

    as the digits increase by one the computers needed speed doubles. so even if pi has infinite digits. if the computers speed is 2^infinite it wins. This is how the incredibly annoying (yet useful) concept of infinite works

    so the question is really if a computer's speed could be infinite would the world explode

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