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Mikael Group Title

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

  • one year ago
  • one year ago

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  1. ChmE Group Title
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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

    • one year ago
  2. ChmE Group Title
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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

    • one year ago
  3. Mikael Group Title
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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 !

    • one year ago
  4. ChmE Group Title
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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

    • one year ago
  5. Mikael Group Title
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    For intellectual profit of the community (not fun, god forbid that !)

    • one year ago
  6. Mikael Group Title
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    Hi there @experimentX

    • one year ago
  7. ChmE Group Title
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    so you know the answer. Now I'm more interested in this puzzle

    • one year ago
  8. experimentX Group Title
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    Yo @Mikael what's up!! ... serves as bookmark.

    • one year ago
  9. Mikael Group Title
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    There is a follow up problem - which HAS practical serious applications and is very deep

    • one year ago
  10. ChmE Group Title
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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

    • one year ago
  11. Mikael Group Title
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    Glad to see you \[ Yoda-Not , \bf @ganeshie8 \]

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

    • one year ago
  13. Mikael Group Title
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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.

    • one year ago
  14. experimentX Group Title
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    lol .. .thanks!!

    • one year ago
  15. Mikael Group Title
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    Ok let's make it simpler:

    • one year ago
  16. Mikael Group Title
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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.

    • one year ago
  17. Mikael Group Title
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    @Algebraic! you are missed here , one needs some critical approach...

    • one year ago
  18. Mikael Group Title
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    Each robot has also a seconds-counting watch

    • one year ago
  19. Mikael Group Title
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    Well let's say @ChmE is "warm" in his attempt...

    • one year ago
  20. ChmE Group Title
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    I'm still thinking I see the problem you mentioned

    • one year ago
  21. Mikael Group Title
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    Just devise an approach with lower possibility of that

    • one year ago
  22. Mikael Group Title
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    So ..

    • one year ago
  23. ChmE Group Title
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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

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

    • one year ago
  25. Mikael Group Title
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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

    • one year ago
  26. Mikael Group Title
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    I mean - the bulb , in some sense is a third-rate "bolt"

    • one year ago
  27. ChmE Group Title
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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.

    • one year ago
  28. Mikael Group Title
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    OK - push forward on the path of SIMPLIFYING !!!

    • one year ago
  29. Mikael Group Title
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    "the first turns it off"....

    • one year ago
  30. Mikael Group Title
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    and ignore (for a sec) the actual numbers of seconds

    • one year ago
  31. ChmE Group Title
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    I will continue to think about it. Gotta catch a bus

    • one year ago
  32. Mikael Group Title
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    So gentlemen and ladies - who's up to the challenge ?!

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

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

    • one year ago
  35. Mikael Group Title
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    No tricks - real new democratic choice of president.

    • one year ago
  36. ChmE Group Title
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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

    • one year ago
  37. ChmE Group Title
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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

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

    • one year ago
  39. ChmE Group Title
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    Check my soln to problem 1. I think I've got it

    • one year ago
  40. Mikael Group Title
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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"

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

    • one year ago
  42. Mikael Group Title
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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

    • one year ago
  43. ChmE Group Title
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    ok

    • one year ago
  44. Mikael Group Title
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    By the way - he does know - but how do the others know that he DOES know ?

    • one year ago
  45. ChmE Group Title
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    what do you mean by that

    • one year ago
  46. ChmE Group Title
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    the robot knows but how to the other robots know he knows?

    • one year ago
  47. Mikael Group Title
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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 ?

    • one year ago
  48. ChmE Group Title
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    haha. This isn't fun anymore. I gotta think about this

    • one year ago
  49. Mikael Group Title
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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.

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

    • one year ago
  51. Mikael Group Title
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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

    • one year ago
  52. Mikael Group Title
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    Welcome to our humble abode Mrs @sauravshakya @hartnn and all !

    • one year ago
  53. Mikael Group Title
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    Much appreciated !

    • one year ago
  54. sauravshakya Group Title
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    ACTUALLY THIS SIMILAR PROBLEM WAS ALREADY SOLVED BY ME TOO.

    • one year ago
  55. sauravshakya Group Title
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    http://mathriddles.williams.edu/?p=146

    • one year ago
  56. sauravshakya Group Title
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    THAT IS THE SIMILAR QUESTION....... only number are different..AND ITS LOGICAL NO SERIES

    • one year ago
  57. sauravshakya Group Title
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    AS far as I remember.

    • one year ago
  58. hartnn Group Title
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    ya, i have also seen such problems...

    • one year ago
  59. Mikael Group Title
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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

    • one year ago
  60. Mikael Group Title
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    Because I have met this situation in real life engineering problem.

    • one year ago
  61. Mikael Group Title
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    Thanks Highlander ....!

    • one year ago
  62. Mikael Group Title
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    Hey @siddhantsharan

    • one year ago
  63. siddhantsharan Group Title
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    Hello. :)

    • one year ago
  64. Mikael Group Title
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    Best of luck. and keep them cool and well nutritioned !

    • one year ago
  65. siddhantsharan Group Title
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    Haha. Yeahh. Thanks. Nice problem though.

    • one year ago
  66. ChmE Group Title
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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.

    • one year ago
  67. Mikael Group Title
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    It is assumed the do communicate and agree beforehand.

    • one year ago
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