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Mikael

  • 2 years ago

First problem 100 thinking robots are given a challenge : they will communicate only by means of a single light bulb which can be either in the state 1 (light on) or in state 0 (light off). Each second a randomly chosen \[\bf ONE \quad single \] robot can see the bulb (others don't- in that second) and keep it in the previous state or change it to opposite state. Bulb is On at first. Each second one chosen randomly out of 100. The challenge is to know -WHEN every one of them has been at the bulb at least once (or more) times. This info has to reach all of 100 eventually. SEE THE SOLUTION BELOW

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  1. ChmE
    • 2 years ago
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    Sounds like a statistics problem. To me there will always be a chance the same robot doesn't see the bulb, but it will be so small it's considered 0. We have to decide how small is too small. I really can't help any further then relaying my thoughts. sry

  2. ChmE
    • 2 years ago
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    you could set up a limit as x approaches 0 for the last robot. But i don't know how to do that for this problem

  3. Mikael
    • 2 years ago
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    Consider it given that there does occur a visit of each to the bulb. THIS IS NOT THE QUESTION !! The question is to communicate that event AFTER IT HAPPENNED JUST USING THE BULB !

  4. ChmE
    • 2 years ago
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    oh ok. I did read it wrong. Is this a question you need an answer to, or a question for fun to the community? I would have to think about this for a while. still not much help

  5. Mikael
    • 2 years ago
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    For intellectual profit of the community (not fun, god forbid that !)

  6. Mikael
    • 2 years ago
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    Hi there @experimentX

  7. ChmE
    • 2 years ago
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    so you know the answer. Now I'm more interested in this puzzle

  8. experimentX
    • 2 years ago
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    Yo @Mikael what's up!! ... serves as bookmark.

  9. Mikael
    • 2 years ago
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    There is a follow up problem - which HAS practical serious applications and is very deep

  10. ChmE
    • 2 years ago
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    what if the first robot turns the bulb off and every robot you sees the bulb and it is their first time will turn it on. If it is their second time will turn it off. If the bulb is left off for a long time then we can assume every bot has seen it. I don't know how to determine the length of time

  11. Mikael
    • 2 years ago
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    Glad to see you \[ Yoda-Not , \bf @ganeshie8 \]

  12. ganeshie8
    • 2 years ago
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    :) so the state has to be communicated to all the 100 robots, but the bulb has only two states hmm

  13. Mikael
    • 2 years ago
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    The offered method has weakness - a robot doesn't know whether it was a multiple visit by some group that made the light off (if that is his observation) or the complete set of visits.

  14. experimentX
    • 2 years ago
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    lol .. .thanks!!

  15. Mikael
    • 2 years ago
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    Ok let's make it simpler:

  16. Mikael
    • 2 years ago
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    How each robot can be sure with probability of \[p= 1 -10^{-6}\] that all others HAVE visited the bulb - devise a method for that.

  17. Mikael
    • 2 years ago
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    @Algebraic! you are missed here , one needs some critical approach...

  18. Mikael
    • 2 years ago
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    Each robot has also a seconds-counting watch

  19. Mikael
    • 2 years ago
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    Well let's say @ChmE is "warm" in his attempt...

  20. ChmE
    • 2 years ago
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    I'm still thinking I see the problem you mentioned

  21. Mikael
    • 2 years ago
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    Just devise an approach with lower possibility of that

  22. Mikael
    • 2 years ago
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    So ..

  23. ChmE
    • 2 years ago
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    can a robot leave a bolt. so the first time they visit they leave a bolt and there is a robot that collects all the bolts when he has 100 including his own they are done

  24. ChmE
    • 2 years ago
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    and my first method will be complete in 100^101/100! years. lol

  25. Mikael
    • 2 years ago
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    Great beginning of BRAIN -STORMING , NOW that you have "I wish" method - try making "the bolt" only with the ligh on/Off and the conditions given

  26. Mikael
    • 2 years ago
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    I mean - the bulb , in some sense is a third-rate "bolt"

  27. ChmE
    • 2 years ago
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    the first turns it off. and he counts how many times he see it on. The other robots turn it on one of their 2 cycles.

  28. Mikael
    • 2 years ago
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    OK - push forward on the path of SIMPLIFYING !!!

  29. Mikael
    • 2 years ago
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    "the first turns it off"....

  30. Mikael
    • 2 years ago
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    and ignore (for a sec) the actual numbers of seconds

  31. ChmE
    • 2 years ago
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    I will continue to think about it. Gotta catch a bus

  32. Mikael
    • 2 years ago
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    So gentlemen and ladies - who's up to the challenge ?!

  33. KingGeorge
    • 2 years ago
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    I'm going to stay quiet for a while since I've seen the problem before (in slightly different terms).

  34. Mikael
    • 2 years ago
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    \[\bf \text{here is Second Problem - will be posted after this one is done and cleared.}\] \[\bf \color{red}{\text {Now the robots have to somehow}\,\,\\ {\Huge\color{green}{Choose \quad a \quad president}}\\{\text{EXACTLY IN THE SAME SITUATION }} }\]

  35. Mikael
    • 2 years ago
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    No tricks - real new democratic choice of president.

  36. ChmE
    • 2 years ago
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    the 1st robot turns it off and he only ever turns it off. Every other robot turns the light only once. If they have already turned it on they leave it off or if it is on they leave it on. The 1st robot counts how many times he turns it off til 99

  37. ChmE
    • 2 years ago
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    if the robots see the light on but havent touched it then they leave it on til they are given the chance to turn it on

  38. Mikael
    • 2 years ago
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    All right - so here is The second problem. \[ \bf \text{Choosing a specific number by majority of votes.} \\ \text{ Each robot has has own number known to all.}\\ \text{ Using all of the above they have to choose one pf them.}\\ \text{He will be called the president.} \]

  39. ChmE
    • 2 years ago
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    Check my soln to problem 1. I think I've got it

  40. Mikael
    • 2 years ago
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    @ChmE This seems the solution "the 1st robot turns it off and he only ever turns it off. Every other robot turns the light only once. If they have already turned it on they leave it off or if it is on they leave it on. The 1st robot counts how many times he turns it off til 99"

  41. ChmE
    • 2 years ago
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    OMG!!! I swear I didn't cheat. Been thinking about this question on and off all day yesterday

  42. Mikael
    • 2 years ago
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    But you were a bit unclear - you have to STATE that he turns it off during 99 opportunities he is given

  43. ChmE
    • 2 years ago
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    ok

  44. Mikael
    • 2 years ago
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    By the way - he does know - but how do the others know that he DOES know ?

  45. ChmE
    • 2 years ago
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    what do you mean by that

  46. ChmE
    • 2 years ago
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    the robot knows but how to the other robots know he knows?

  47. Mikael
    • 2 years ago
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    First robot counted 99 light-on and reached the conclusion that all have been at the bulb. NOW - how will he communicate that fact to all the others ?

  48. ChmE
    • 2 years ago
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    haha. This isn't fun anymore. I gotta think about this

  49. Mikael
    • 2 years ago
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    This is NOT fun, but i will spare you this time : Only probabilistically they will know. How? by seeing the light in their SEVERAL personal visits off they conclude that the probability that the FIRST - the turning-off guy was right before them is too low.

  50. ChmE
    • 2 years ago
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    ok. thx. Is this a question that was proposed to you in one of your classes?

  51. Mikael
    • 2 years ago
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    Thanks - now let us call here the other people who has been here - so they appreciate our work. @bahrom7893 @sauravshakya @kingGeorge, @ganeshie8 @hartnn @experimentex

  52. Mikael
    • 2 years ago
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    Welcome to our humble abode Mrs @sauravshakya @hartnn and all !

  53. Mikael
    • 2 years ago
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    Much appreciated !

  54. sauravshakya
    • 2 years ago
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    ACTUALLY THIS SIMILAR PROBLEM WAS ALREADY SOLVED BY ME TOO.

  55. sauravshakya
    • 2 years ago
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    http://mathriddles.williams.edu/?p=146

  56. sauravshakya
    • 2 years ago
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    THAT IS THE SIMILAR QUESTION....... only number are different..AND ITS LOGICAL NO SERIES

  57. sauravshakya
    • 2 years ago
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    AS far as I remember.

  58. hartnn
    • 2 years ago
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    ya, i have also seen such problems...

  59. Mikael
    • 2 years ago
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    Dear visitor @sauravshakya - I bet you ten sayings of praise (of your choosing) that the following you have NOT solved before http://openstudy.com/study#/updates/50631053e4b0583d5cd34249

  60. Mikael
    • 2 years ago
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    Because I have met this situation in real life engineering problem.

  61. Mikael
    • 2 years ago
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    Thanks Highlander ....!

  62. Mikael
    • 2 years ago
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    Hey @siddhantsharan

  63. siddhantsharan
    • 2 years ago
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    Hello. :)

  64. Mikael
    • 2 years ago
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    Best of luck. and keep them cool and well nutritioned !

  65. siddhantsharan
    • 2 years ago
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    Haha. Yeahh. Thanks. Nice problem though.

  66. ChmE
    • 2 years ago
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    I was just thinking @Mikael . How is my solution/our solution correct because it relies on the robots communicating prior to seeing the lightbulb? Which by the given conditions cannot happen.

  67. Mikael
    • 2 years ago
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    It is assumed the do communicate and agree beforehand.

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