Dottown Deaths: Newsletter Challenge (November 2019)

If the entire population had blue dots on their foreheads and could see everybody else had a blue dot, then the red dot must be them, so everybody would die...If there was a mix of red and blue dots then nothing happens, if everybody has a red dot nothing happens....

There is one person with a red dot. That one person sees all blue dots, he then knows his dot is red, and he dies.

Everyone else sees the one person with a red dot, but still doesn't know the color of their dot.

After a little more thought...

There is one person with a red dot. That one person sees all blue dots, he then knows his dot is red, and he dies.

Everyone else sees the one person with a red dot, but still doesn't know the color of their own dot.

The next day, the blue dot people notice the red dot person is gone, and since he figured out his dot was red, they know their dots must all be blue. The following day, no one shows up.

No, the stranger said 'at least one person with a red dot'....but he might have been lying...so anybody seeing that everyone else had blue dots on their foreheads would have to assume that it was him or her with the red dot...they all would assume they were the one with the red dot...

"If anyone knows the color of his or her own dot, that person will die that night. Everyone else will notice this at the gathering at noon the next day (and not before)."

The operative word is "knows".

If the stranger is lying and all are blue, but each assumes he is red, then he doesn't really know his color, he has made a false deduction, and he doesn't die.

I suppose you could argue that at this point each could deduce that since he is still alive, his previous belief was false, his true color was blue, and he will die the following night. You have to split hairs between "know" and "believe". In that case, everybody would die the second night.

However, I still contend that "know" implies a logical deduction based on a true statement, not based on a lie by some homicidal stranger.

If there were more than one red dot person, then everybody would have seen both red and blue. The stranger would have provided no new information. Nothing happens.

If there is only one red dot person, then that person would have seen only blue and would know his dot was red from the stranger's statement.

Everybody knows everybody and sees everybody at the town meeting. All the blue dot people would notice that the red dot person was gone. They would realize that the red dot person deduced he was the red dot because he could see no more red dots, and they would then know that their dots were blue. I'm afraid the following day, the town would be empty of survivors.

Yes that could be so, in any case the introduction of new information destroys the village....so either way it's a ghost town....

Either way , have to love Bodie

All believed the mysterious stranger; there is at least one person in this town with a red dot.

What was the stranger’s method, has he made an error, why did he use the words, “at least”?

Options, believing the stranger:-

Zero red dots, The village people, would be aware that no one else in the village has been seen to have a red dot, therefore, they could be that person, with the red dot. With this thought, they would all die that night.

One red dot, If there is only one red dot in the village, then every one would know who that person was, they would ignore that person. But the use of the wording, at least, suggests one or more people have a red dot, and that person could be them? And the person with the red dot, not seeing another person with a red dot, is in the same position as the others. With this thought, they would all die that night.

More than one red dot, If there are more than one person with a red dot, then every one would know, and as the stranger did not specify actually how many, then every one would have no reason to think it refered to them. Life would continue on as normal

If two or more people have red dots? Nobody dies . If one person has a red dot. they will die and then nobody else will.

Prior to the noonday meeting, the mayor and the priest acquire a sticky back red dot to apply over the other's dot in privacy. Still, neither of them know if they themselves have a red dot under this applied dot. Everyone at noon sees at least two red dots and nobody dies that night. This solution intrigues the other residents and from then on a lottery for which two will wear the red dots the next day.

The stranger is the only one who dies...He was sacrificed to the dot god for breaking the rules about speaking of the dots...

One way to reduce the burgeoning population .

Whether the stranger lied or the problem giver (He said that there are blue and red dotted persons ), every person would have to leave the world, like the usual Homo Sapiens .

Re Rixter comment #6. Knows as opposed to believes.

I think it hinges on 'at least' one red one.

I think any combination of multiple red and blue will cause doubt until it narrow down to only two persons, A and B, with red dots - even then no problem until the stranger rides into town.

Either A or B will see a red dot on the other person - and therefore cannot be certain of their colour.

Except next day, as they are both alive, one will deduce that the other must have seen a red dot.

Exit A and B the next day - followed by the blues in quick succession.

As I said earlier in my answer #15, Rixter #6 draws attention to 'belief' versus 'know'.

The decision logic to 'calculate' your colour by sound reasoning (to 'know' without doubt) - then to die as a consequence - is only valid if the original conditions (the rules) are true - that must be believed. Then belief in the truth of the stranger - then belief in the new rule.

So logically if boils down to 'belief'. And if your powers of reasoning are at fault - and you get it wrong - you die because you believe you are right.

Back to my previous reply #15, with only two of us left, both red - and the new rule, 'at least 1 red one. Then I will know I am red if I see blue - and die.

The other person likewise will know they are red if they see blue - and die.

Only those seeing red will turn up the next day - and soon realise they are both red.

However, I cheat and do not turn up. The other person will (wrongly) deduce they are blue - and believing it - will die.

I am the sole survivor for another day (at least 1 red one must be me).

.......sod the stranger!

I believe the stranger added no new information - at the noon meeting everyone has seen a possible mixture of red and blue dots - assuming they are not all color blind. So being told at least one has a red dot is not new information - they still don't know there own dot.

I think the status quo continues.

Well, I thought of that too but I didn't find time until now to write it down. What I can add is that, in a way, the stranger's statement caused a kind of timer start. Although the initial statement was "there is at least one...", the fact that the next day nobody died was equivalent to "there are at least two..." and so on. Until the day when it became equivalent to "there are at least N..." and the red dotted ones realized that (as they see only N-1) they have a red dot.

On the other hand, if they are all extremely intelligent, they should realize that the best is not to use the stranger's information at all because using it can have only two consequences:

- they find out the color of their dot - and die

- they cannot find out the color of their dot - and trying to use the information is pointless.

If they all decide to do so, they all survive. But: "...there is no way to avoid this.", so...

If there are at least 3 with a red dot, then no one will die, since everyone will be satisfied that they have observed "at least 1" person with a red dot.

If 3 people have a red dot, those 3 people all witness 2 people with red dots - so they know that the village either has 2 people (the ones they observe) or 3 people (in the event that they have a red dot themselves). Their own dot color cannot be devised, so there is no way that any of those 3 people would die. They can rest satisfied that the Stranger's words are true, and that they still cannot know their own dot color.

Everyone else witnesses 3 people with red dots, and such an observation is congruent with the Stranger's words. They have no way of knowing if their own dot is a 4th red dot or not - and none of the people with red dots will die, so everyone lives.

If the number of people with a red dot is at least 2 higher than whatever number the Stranger mentions, then everyone will live.

To switch it up... If the stranger says there are at least TWO red dots, then:

* If there are 2 or 3 red dots, everyone will die very soon; but...

* If there are 4 or more red dots, everyone will live, since everyone will witness at least 3 dots and be satisfied in the truth of the Stranger's words, and rest in still not being able to know their own dot color.

The only way one could realize the color of their dot is if there was only one red dot. If all others are blue. You die.

Very simple.

I agree. If all I see is blue - then I must be red.

And if there is a random (heads / tails) selection of dots - then it is equivalent to stating there is at least one head with a large sample of tosses - So if everyone sees a selection both red and blue dots stating the obvious has no change on the outcome of your "random" dot.

Nothing happens. If the stranger was telling the truth or not the people only know what they can see. They are intelligent enough to know if they figure out what their dot color is they know they will die.

So one can only assume what color their dot is. This is not knowing what it is. Assuming what your color is is not going to get you killed. You can assume wrong and that will get you killed. If you assume it is red and you don't die that night you know it is blue and you will die the next night.

No one really cares what their color is otherwise the whole town would have died off long before a stranger came to proclaim that at least one person has a red dot.

Here are some hints from a well-known solution, to keep you thinking.

Let R be the total number of persons with red dots, B the total number of persons with blue dots, and P the total number of persons, so P = R + B.

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Hint 1. All persons with the same color dot have exactly the same information, so if they die, they will all die the same night.

Each person with a red dot will see exactly the same number of persons with red dots and blue dots that every other person with a red dot sees.

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Hint 2. If anyone dies, then all persons in Dottown will die on the same night, or on consecutive nights.

If all ascertain their dot colors on the same day, they will all die that night. If, say, all those with red dots die one night, those still living (if any) will notice the next day at noon that there are no red dots left, thus they must have blue dots, and will therefore die that night.

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Hint 3. Consider one resident of Dottown, called JQP. Let R_v (for "Red dots visible") be the number of red dots that JQP sees; let B_v be the number of blue dots that JQP sees.

Then R_v + B_v + 1 = P.

Consider the following statements:

1) JQP_has_red (JQP has a red dot)

2) R = R_v + 1 (there's one red dot that JQP doesn't see)

3) B = B_v (JQP sees all the blue dots)

4) JQP_has_blue (JQP has a blue dot)

5) R = R_v (JQP sees all the red dots)

6) B = B_v + 1 (there's one blue dot that JQP doesn't see)

Then Statements 1, 2, and 3 are equivalent;

Statements 4, 5, and 6 are equivalent;

Statements {1, 2, 3} are the logical opposite of {4, 5, 6}.

Thus, either (a) 1, 2, and 3 hold, with 4, 5, and 6 false, or

(b) 4, 5, and 6 hold, with 1, 2, and 3 false.

So, if we know the true/false value of any of the six statements, we know the value of all of the statements. If JQP knows the value of any of the six statements, JQP will die that night.

Bonus hint: The stranger said that there is at least one person with a red dot. What happens if the stranger was lying? That is, everyone has a blue dot, no one has a red dot, but everyone believes the stranger anyway. What happens?

If it is possible that the stranger could have lied, then we can proceed like this:

If there are more than one red dots, then all can see at least one, so they know that the stranger tells the truth. Business as usual; all die at some point, as explained in one previous post.

Now, if there is only one red dot, the one that has it, has no idea whether the stranger tells the truth or not. Everybody else though does know he told the truth. They see that the red-dotted one is not dying, therefore he sees no red dots (and thus got puzzled), so they all know they have blue dots and immediately die. The red-dotted one sees that nobody shows up the next day, so he realizes that it was his red dot that caused their demise. He dies too.

If there is no red dot, then surely the stranger has lied, but nobody knows it as yet. Each and every one wonders whether he is the only one with a red dot or just everybody has a blue dot. But if one had a red dot, then all the rest should not show up the next day. But all turn up. So they realize they all have blue dots. They all die.

Hint 1 is flawed.

"Hint 1. All persons with the same color dot have exactly the same information, so if they die, they will all die the same night.

Each person with a red dot will see exactly the same number of persons with red dots and blue dots that every other person with a red dot sees."

No one knows their own dot color. So they do not all "have the same information" until all opposite dots perish. Until then dot color is an assumption, even with the stranger's comment.

"Until then dot color is an assumption, even with the stranger's comment."

I agree.

That has been my point. "N" the number of dots of any one color is always unknown. Any one person will see a total of B blue dots and R red dots - but their own is still unknown. With a mixture of dots (unless there are only one or two of one color) then the observation of at least one red dot is what everyone has seen every day.

Every resident with a red dot would die that evening. The following afternoon, the

remaining residents would see that everyone with a red dot was gone. They would

then realize that they must have a blue dot. That evening they to would die. The

poison seed planted by the stranger led to the demise of Dottown.

All but a very few died that night because they knew their color.

Doesn't it depend on what is said at the "formal proclamation at noon each day"?

One thing is unclear.

"The dots are always either red or blue."

As in they change color.?. like blood in a vein? If not the whole problem is a problem.

Nope, they don't change color. As in, each person has either a static red or blue dot.

-new answer

If 2 or more people have red dots on their forehead?

Nobody will die.

If only one person has a red dot. they will die that night.

The next night everyone else will die.

It's simple.

OK thanks for that. GA

But now I'm going back to the idea that the dots change color. Otherwise the town law would never let visitors in.

If mirrors would be suicidal?

..so would a visitor making an observation.

That's my observation.

Riddle flawed..

It's simple

Don't forget what is written at the end: "Note: this is a purely mathematical puzzle. There is no trick hidden in the narrative".

With your logic, would it not imply that the original state was unstable? - knowing your dot color you die (already a statement of truth) - then someone stating what everyone should already know- that there are a selection of dots - changes nothing.

The ban on mirrors etc. implies they already know the death problem and have taken active steps to prevent themselves knowing.

It appears to me your logic implies in the original state people were not cognitive of the fact they were dotted.

Alex, GW

I was assuming each day the dots may or may not change.

I don't understand the whole "point of pride" if all you have is a deadly unchanging dot.

Why gather and see everyone's dot at noon etc if the dot is always the same?

Makes no sense.. the math isn't supported by the narrative.

Doh!

"Why gather and see everyone's dot at noon etc even if the dot is always the same?

Makes no sense.. the math isn't supported by the narrative"

The narrative says there are red and blue dots, not maybe all one color. So if everyone is blue except one (never mind the visitor) then the die off is eminent anyways.

If the visitor had said there are 2 red dots (or one red dot or N red dots) then extinction is mandated. I see N-1 red dots therefore I am red -- statement of at least one red dot the N is unknown.

.That is my question - even with static dots! They gather at noon - see everyone's dots - don't know their own, so a statement by someone that says someone has a red dot - I cannot see changes anything. There is no new data.

Perhaps I am being obtuse.

What if they decide to tattoo a green dot over their color dots to become an invincible army of mercenaries!!!

I guess there are no water glasses or water in dot town to see their reflection causing all to perish.

The answer is now in the original post.

I wonder what the role of the stranger really is, in case there are more than one red dot (i.e. N>1). He could be completely absent from the narrative, as his statement is irrelevant. In this case would all the populace die after some days?

I know, this is not strictly mathematical approach, as induction needs a first step, which is N=1. But apart strict mathematical reasoning, what would happen in the minds of the populace in such a scenario? Wouldn't they assume by themselves that the N=1 step already is true so that they can apply induction anyway?

Although I provided one of the answers, I still cannot grasp that a piece of no-information given by an outsider could trigger a change in a system.

Flawed riddle.

It's simple

I hear you.

What you have wrong is that the stranger does bring new information to the system. They bring the information that on day zero a minimum number of one set exists. The immutable system then itterates into oblivion.

Your "gut" feeling smells a paradox but cannot identify it. I think I just stated the crux of the paradox. A truly immutable system cannot change. Therefore, the stranger's words should be ignored. A mutable system and particularly a mutable system with any degree of self preservation should respond to the new information to prevent self destruction.

Disclaimer: I'm only the Challenge Question Guy and don't have the ability to delve as deeply into these puzzles as CR4 users. This puzzle is evidently decades old and frequently bounces around the classrooms of engineering schools.

I interpreted the stranger as essentially breaking Dottown's tense equilibrium between its residents. The residents stay silent about the dots -- they're intelligent enough to know that knowing any non-trivial information about dot colors is going to cause some degree of death. The stranger, being unfamiliar with Dottown's equilibrium and its importance, nonchalantly mentions some detail about dot color and ends up upsetting the balance.

This iteration of the puzzle, its hints and solution came from a former MIT student. An alternate version of the puzzle, from Dartmouth professor Peter Winkler, goes like this:

Each resident of Dot-town carries a red or blue dot on his (or her) forehead, but if he ever figures out what color it is he kills himself. Each day the residents gather; one day a stranger comes and tells them something—anything—non-trivial about the number of blue dots. Prove that eventually every resident kills himself.

Comment: “Non-trivial” means here that there is some number of blue dots for which the statement would not have been true. Thus we have a frighteningly general version of classical problems involving knowledge about knowledge.

Solution:

The proof (my own) of this very general version uses backwards induction on the number of disallowed numbers of blue dots. In the base case there are n−1 such numbers, where n is the population of Dot-town.

In that case everyone can immediately deduce his own dot-color, so the town is wiped out immediately.

Say there are n residents and the stranger announces that the number of blue dots does not belong to the set K, where K is some non-empty subset of the numbers from 0 to n. Suppose the actual number of blue dots is j, so that the red-dotted residents see j blue dots and the others see only j−1. If j−1 is in K then the blue-dotted residents will all kill themselves the first night.

The remaining residents (if any) will deduce that their spots are red, and dispense with themselves the following night. If j + 1 is in K then the red-dotted residents will all kill themselves the first night, and the remaining residents will follow suit the next night after deducing that their spots are blue.

If no one kills himself the first night, everyone can deduce that neither j−1 nor j+1 is in K, i.e. the actual number of blue dots is not within one of any number in K. This increases the number of forbidden numbers of blue dots, and the induction hypothesis can now be applied. Note that the proof shows that n nights suffice; further, that the full n nights are needed only when K = {0} or {n} and either all dots are the same color or n ≤ 2.

"at least one red dot" ...

If ALL of the Dot People can see at least 2 red dots (meaning, there are at least 3 people with red dots) - then no one will die, because it will be impossible for ANY of them to deduce their own dot color. Everyone will conclude that the Stranger added no new information. In this example, everyone has ALWAYS been able to see at least one red dot.

If there are 3 with red dots, then each of them has always witnessed 2 red dots on others, while all the blue-dotters have always witnessed 3 red dots. Thus, no one is intrigued by the statement that there is at least 1 red dot. No matter what one's own dot color is, everyone already knows there is at least 1 red dot. With no new actionable data, there is no successive waves of new knowledge and deaths.

It only ends in catastrophe if there are only 1 or 2 red dots - otherwise, everyone is satisfied if they can already see at least 2 red dots.

I agree with the math up to 3 red dots. After that, not so much. Let’s say 50 of 100 have red. They all say, “no shit Sherlock.” There is no daily reassessment of self doubt. No one dies. Explain how this scenario is wrong. {note; editor really difficult on iPhone}

Maybe I am missing something in all the Dottown answers, but I wish to add these thoughts even at the risk of revealing myself to be of lesser intelligence.

If all the residents of Dottown are of high intelligence and will figure out their dot color if possible, then they should have already counted the number of red and blue dots each day at noon gathering. Assuming that there are both red and blue dots among the populace, then the residents would already know that there is at least one red dot, but they also would know there was at least one blue dot (visitor offers no new information). Agreeing with the N-1 math and all residents of that color dot dying that night, then the countdown of N-1 days would already be ongoing prior to the visitor proclaiming there to be at least one red dot. I hypothesize that the countdown would be for the lowest color dot though, not neccesarily the red dot (if there were less blue dots). And if that is all true, then the populace of Dottown would only still be alive because the Blue or Red (lesser quantity) population of Dottown was large enough to put them still somewhere within the N-1 countdown. If outside of that countdown, then the visotor made his proclamation in a ghosttown.

Maybe some flaws in my logic, so please be kind in your critique.