The first prisoner says Black if the number of black hats in front of him is odd, and White if the number is even. Every succeeding prisoner notes the number of black hats in front of him and the number of times he’s heard ‘Black’ from someone behind him, and says Black if the number is odd and white if even. The first guy has a fifty fifty chance of survival, but everyone else is fine. This won’t work for an infinite line of prisoners,though.
All the declarations are done simultaneously. A prisoner doesn’t get to know anything about the declarations of the other prisoners (except as predicted by whatever strategy they work out beforehand).
Oops. I was thinking of a different puzzle scenario, I guess.
And there are an infinite number of hats in front of him. How is going to decide odd or even?
There are an infinite number of prisoners, but no individual prisoner has an infinite number of hats in front of him.
(I nitpick because it is also related to the 0.99999… issue. There is an infinite number of nines, but there every nine has a finite position – there is no “infinity-th” position like some would suggest.)
To clarify, prisoner n can see the hats of all prisoners m > n.
Which, to be explicit, means every prisoner does have an infinite number of hats in front of them, but only a finite number behind them. Zakalwe’s concern was well-founded.
It’s worth noting that this prisoners problem is quite analogous to the Banach-Tarski paradox: we might reason that each particular prisoner has no information about their own hat, and thus, no matter what strategy they use, has probability 0 (on the outlandish “Hat color is a real between 0 and 1” version) of getting their hat correct. And therefore, the average number of hats which are gotten correct must be 0 + 0 + 0 + … = 0; by this reasoning, we cannot have a strategy which even has a positive probability that at least one prisoner will get their hat correct, much less guarantees all but finitely many will.
The reconciliation of this argument with the winning strategy provided by the Axiom of Choice (still a challenge for anyone in this thread to try discovering) is in the realization that no one ever guaranteed us we could find a way to assign a “probability” to every property of prisoner-hat-configurations in such a way as to satisfy all the rules we expect for such probabilities. If we restrict attention to just allowing ourselves to consider some and not necessarily all properties of prisoner-hat-configurations as having well-defined probabilities, the other properties being forbidden from probabilistic reasoning, we know how to pull it off, but in general, in contexts satisfying the Axiom of Choice, it will not be possible to consider every property of hat-configurations as having a coherent probability, precisely because of the above argument in combination with the argument that a winning strategy exists.
That is to say, if “The N-th prisoner gets their hat correct” is not considered to actually have a well-defined probability under some particular strategy, then the argument of the first paragraph above fails to run to its stated conclusion, and room is opened for that strategy to indeed guarantee all but finitely many prisoners go free.
In the same way, no one ever guaranteed us we could assign a notion of “volume” to every arbitrary subset of points in R^3 such that we get all the properties we expect (e.g., volume doesn’t change under translation, volume of a whole is the sum of the volume of its parts, etc.), and the Banach-Tarski paradox shows we actually need to consider certain Choice-definable sets of points (used intermediarily in the sphere-doubling process) as lacking well-defined volume. It’s all exactly analogous.
Normally I grok you indie. But I’m gonna have to read that a couple of times with my brain on.
J.
I gave in and googled. The solution on wikipedia does my head in.
The description of equivalence classes given intuitively (to me) seems to lead to at least a countable infinity of such classes (which makes everything that follows a little difficult to believe). Am I missing something?
Perhaps this deserves a seperate thread.
He can’t decide odd or even. That’s why I said my solution wouldn’t work for an infinite line.
I am not missing your point at all, sir. I understand what you and professors want me to believe just fine, actually! 
The number system would work fine without trying to represent certain objects as infinite decimals.
And this system has it flaws that I acknowledge, but many people turn a blind eye to.
1 = lim 0.999… and not “0.999…” but you leave out the limit part.
I clearly show his error of number shifting, and how it is circular. I find this very relevant.
No, but building up the reals, ie, *constructing *them, outputs the real number. The objects used (the sequences), are not the numbers themselves. This is one of the problems I have. I take a sequence, find its limit of partial sums, and arrive at a finite number. The sequence is not this number, nor is the series !!
That is analogous to saying flour+sugar+egg+milk = cake
The ingredients are used to make the final product, but they are not “the same”.
No. The series is not defined at N=0. What if you wanted to add multiple “zeros” to the beginning as terms? What about infinite 0’s ??
Also, n=0 would give “9” as the first term, not 0.
I must have missed them.
Also, I don’t feel obligated to answer every new question when my posts have not been refuted properly either.
See above. how about we discuss 1 item at a time ?
I find it amazing that people forget the “remainder” in the 1/3 = 0.(3).
dividend/divisor = quotient + remainder
in the solution 0.(3), they are writing the quotient repeatedly, but do not account for the remainder whatsoever:
1/3 = Σ(n=1 to N) 3/10ⁿ + 1/ (3 x 10^N)
See my post in response to MsKaren.
What is x - y ? That is what you are asking !!
How can you solve a finite number minus an infinite series ? It will never end ! It is a meaningless question unless you say specifically:
1 - lim 0.999…
What would be the point of numbers if they all have multiple values ?!?! :eek:
Thanks for identifying one of the downfalls !
1 = lim 0.999… yes
1 = 0.999… no
If you want to write “lim 0.999…” where ever you see the number 1, I know what you mean. However, 0.999… is meaningless, which is why a limit is used in the first place… to attempt to give it meaning. however, 0.999… on its own is still meaningless without “limit”.
So if you want to make it a convention that “most mathematicians will know what is meant by 1 = 0.999…” where 0.999… is really lim 0.999…, then that is fine, but it will be always implied that 0.999… is lim 0.999… ie, we just don’t write “lim” …
What end ??
I am fine with using limit 0.999… = 1
however, this does not automatically make 0.999… = 1 without the limit. We can’t just drop it .
Not at all.
See, you’re still not getting it.
“Belief” has nothing to do with it.
That’s the great (and often confounding) thing about math. Once you set your definitions, what follows requires only rigorously following through the logic.
It’s possible to get the logic wrong but that’s not happening here.
What’s happening is you don’t like the definitions but you still want to keep the conclusions.
Doesn’t work that way. If you change the definitions, you change the conclusions.
Correction: Such a number system could be designed.
But “the” number system would not (as I stated above).
If you claim otherwise, proof? There’s plenty of proofs for what you are still trying to deny (even if you don’t accept/understand them).
No remainder exists. You claim otherwise, but you’re wrong.
This is a prototypical example of what I am trying to explain. You have a particular notion of how things work that is at odds with how they have been defined to work.
This is a common misconception. Just because you have an infinite series doesn’t mean you require infinite time or infinite computation to solve.
For example, here’s an extreme example of where this works just fine.
x = sum (i = 0 to inf) g*
where g* = 2 if i = 0 and g* = 0 else.
The difference 5 - x = 3. No problems there.
Don’t like that example, ok, how about:
1 - [ sum(i = 0 to inf) 1/2^i]
This is equivalent to sum (i = 1 to inf) 1/2^i, since 1/2^0 = 1. The difference of the number (a “finite number” is a repetition and not necessary - something I think is revealing about how you’re thinking about things) and an infinite series is also quite easy to compute in this case without appealing to an infinite number of steps.
Who (other than you and some other folks who are very wrong) said they have multiple values?
Now, you’re being circular. If you begin by assuming two different decimal representations must represent different values, of course you will conclude they have different values. You are assuming your conclusion.
Real numbers have one value. But they can have multiple decimal representations.
A decimal representation is NOT the same thing as the value of the number. It is, as seen from the words themselves, a “representation” of that value. There are numerous things which have multiple representations. 1/2 = 2/4 is a pretty simple one.
In the case of decimal representations, you can start by simply not knowing if two different decimal representations can have the same value or not. It turns out that for the real numbers, it’s quite possible for a single number to have no unique decimal representation.
Again, not having a unique decimal representation doesn’t mean a number suddenly has multiple values. That is, unless you assume they must, but that produces an inconsistent number line (if you want to keep all other properties of the real numbers intact).
Appealing to Euler and ignoring Cantor means you don’t have to consider how mathematicians actually think about the real numbers. It’s quite relevant and ignoring it means you’re dealing with a strawman representation of what mathematicians claim.
There will in fact be an uncountable infinity of equivalence classes. But what’s the problem with that?
I’m worried I’m going to sound like Cognitively Tired, but that isn’t possible! It isn’t saying “have some way to name/reference each class” but "memorise a representative of each of an infinite number of classes.
I can’t remember what I had for breakfast, but worse no conceivable gizmotronic-mega-brain could memorise that list.
I’m missing something, aren’t I?
Just that this is a math problem and not a real-life scenario. I don’t expect you to have the optical resolution to distinguish the colors of infinitely many hats in your visual field either. But we are to presume these prisoners better than us, completely fluent in the handling of infinite quantities of data.
Fair enough. But as the prisoners become more and more impressive the set-up is sounding less and less impressive:
“There are an infinite number of omni-capable prisoner bots capable of memorising and comparing an infinite number of infinite series…”
I think I’d just give them the (infinite number of) keys and beg them not to tase me.
For the people who drive-by and keep wondering where the claim
…999999999 = -1
is coming from (I believe someone previously explained it in a slightly different way). This result can be obtained by using Borel regulariztion.
In similar manner
…8888888888 = -80/9
…1111111111 = -1/9
and
…999999999990 = -10
In other words, it’s a Reductio ad Absurdem demonstrating that the Axiom of Choice must be false. :eek: It’s similar to refuting the “obvious” fact that deterministic games are determined.
So again your problem is with the notation. As I indicated in my post, I agree that 0.999… is sloppy notation, but if 0.999… is going to have any well defined meaning it is going to have to equal 1. It can’t have a mean a value that is less than 1, and is also greater than 0.999…9 for all finitely long lists of 9’s, because no such value exists in standard number theory. Most mathematicians understand that “…” implies a limit in the same way they understand that Sigma sum notation with an upper limit of infinity represents the limits of the partial sums.