Question

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

Answer 1

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

Answer 2

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

Answer 3

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 !

Answer 4

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

Answer 5

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

Answer 6

Hi there @experimentX

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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.

Answer 14

lol .. .thanks!!

Answer 15

Ok let's make it simpler:

Answer 16

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.

Answer 17

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

Answer 18

Each robot has also a seconds-counting watch

Answer 19

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

Answer 20

I'm still thinking I see the problem you mentioned

Answer 21

Just devise an approach with lower possibility of that

Answer 22

So ..

Answer 23

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

Answer 24

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

Answer 25

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

Answer 26

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

Answer 27

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.

Answer 28

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

Answer 29

"the first turns it off"....

Answer 30

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

Answer 31

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

Answer 32

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

Answer 33

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

Answer 34

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

Answer 35

No tricks - real new democratic choice of president.

Answer 36

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

Answer 37

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

Answer 38

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

Answer 39

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

Answer 40

@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"

Answer 41

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

Answer 42

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

Answer 43

ok

Answer 44

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

Answer 45

what do you mean by that

Answer 46

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

Answer 47

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 ?

Answer 48

haha. This isn't fun anymore. I gotta think about this

Answer 49

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.

Answer 50

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

Answer 51

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

Answer 52

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

Answer 53

Much appreciated !

Answer 54

ACTUALLY THIS SIMILAR PROBLEM WAS ALREADY SOLVED BY ME TOO.

Answer 55

http://mathriddles.williams.edu/?p=146

Answer 56

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

Answer 57

AS far as I remember.

Answer 58

ya, i have also seen such problems...

Answer 59

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

Answer 60

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

Answer 61

Thanks Highlander ....!

Answer 62

Hey @siddhantsharan

Answer 63

Hello. :)

Answer 64

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

Answer 65

Haha. Yeahh. Thanks. Nice problem though.

Answer 66

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.

Answer 67

It is assumed the do communicate and agree beforehand.