## Mikael 3 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

1. ChmE

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

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

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

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

For intellectual profit of the community (not fun, god forbid that !)

6. Mikael

Hi there @experimentX

7. ChmE

so you know the answer. Now I'm more interested in this puzzle

8. experimentX

Yo @Mikael what's up!! ... serves as bookmark.

9. Mikael

There is a follow up problem - which HAS practical serious applications and is very deep

10. ChmE

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

Glad to see you $Yoda-Not , \bf @ganeshie8$

12. ganeshie8

:) so the state has to be communicated to all the 100 robots, but the bulb has only two states hmm

13. Mikael

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

lol .. .thanks!!

15. Mikael

Ok let's make it simpler:

16. Mikael

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

@Algebraic! you are missed here , one needs some critical approach...

18. Mikael

Each robot has also a seconds-counting watch

19. Mikael

Well let's say @ChmE is "warm" in his attempt...

20. ChmE

I'm still thinking I see the problem you mentioned

21. Mikael

Just devise an approach with lower possibility of that

22. Mikael

So ..

23. ChmE

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

and my first method will be complete in 100^101/100! years. lol

25. Mikael

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

I mean - the bulb , in some sense is a third-rate "bolt"

27. ChmE

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

OK - push forward on the path of SIMPLIFYING !!!

29. Mikael

"the first turns it off"....

30. Mikael

and ignore (for a sec) the actual numbers of seconds

31. ChmE

I will continue to think about it. Gotta catch a bus

32. Mikael

So gentlemen and ladies - who's up to the challenge ?!

33. KingGeorge

I'm going to stay quiet for a while since I've seen the problem before (in slightly different terms).

34. Mikael

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

35. Mikael

No tricks - real new democratic choice of president.

36. ChmE

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

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

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

Check my soln to problem 1. I think I've got it

40. Mikael

@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

OMG!!! I swear I didn't cheat. Been thinking about this question on and off all day yesterday

42. Mikael

But you were a bit unclear - you have to STATE that he turns it off during 99 opportunities he is given

43. ChmE

ok

44. Mikael

By the way - he does know - but how do the others know that he DOES know ?

45. ChmE

what do you mean by that

46. ChmE

the robot knows but how to the other robots know he knows?

47. Mikael

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

49. Mikael

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

ok. thx. Is this a question that was proposed to you in one of your classes?

51. Mikael

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

Welcome to our humble abode Mrs @sauravshakya @hartnn and all !

53. Mikael

Much appreciated !

54. sauravshakya

ACTUALLY THIS SIMILAR PROBLEM WAS ALREADY SOLVED BY ME TOO.

55. sauravshakya
56. sauravshakya

THAT IS THE SIMILAR QUESTION....... only number are different..AND ITS LOGICAL NO SERIES

57. sauravshakya

AS far as I remember.

58. hartnn

ya, i have also seen such problems...

59. Mikael

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

Because I have met this situation in real life engineering problem.

61. Mikael

Thanks Highlander ....!

62. Mikael

Hey @siddhantsharan

63. siddhantsharan

Hello. :)

64. Mikael

Best of luck. and keep them cool and well nutritioned !

65. siddhantsharan

Haha. Yeahh. Thanks. Nice problem though.

66. ChmE

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

It is assumed the do communicate and agree beforehand.