2 = 1

Here's what I think:

Assuming a is 1:

ow...my brain hurts

I just figure it out.

"2=1" is only true in the sense that if you have two of something you have one pair/couple of it.

Pssht you think thats a brain teaser, I'm a math / CS student whose taught graduate level courses in cryptography. I'll blow your mind:

Consider the following:
You have a hypothetical ball pit. You are going to add numbered balls to this pit over the course of exactly 1 hour. Each operation is going to take exactly half as much time as the last operation.

First you are going to add balls numbered 1 to 10. At the same time you do this, you are going to remove ball number 1. This takes you 1/2 hour. Next you add balls 11 to 20, then remove ball 2. This takes you 1/4 hour. Next add balls 21 to 30, remove ball 3, it takes you 1/8 hour. 31-40, remove 4, 1/16 hour. 41-50, remove 5, 1/32 hour... etc

For the nth operation, you add ball 10n - 9 to 10n and remove ball number n. It takes you half as much time as the previous operation (or 1/(2n) of an hour).

How many balls are in the ball pit after 1 hour?

---

An oldie but a goodie:
In order to move anywhere, we must fist move half way. How is motion possible?

---

Famous and Important:
Is the set of all sets which are not members of themselves, a member of itself?

An easier yet slightly less accurate formulation: There is a barber who shaves all men who do not shave themselves. Does the barber shave himself?

---

I've got more, but most of them take some explaining. Some of my favorites are the Commitment Problem aka "Coin flipping over the phone" (yes it's possible and your online games do it all the time), Zero-Knowledge proofs (prove it's true without showing anything other than "it's true"), and the Discrete Logarithm (public key cryptography).

Oh yea, and fractal dimension (there are objects which exist "in between" dimensions, for example the Sierpinski Triangle lives in dimension 1.584962...). There was a nice lecture given at MIT about this I'll have to see if I can dig it up.

I'm guessing 10n - n?
Sorry but i don't know the method to do this one.

Ow... My brain hurts...

Eh, I guess I could have explained it better. Think of it this way:

You are adding 10 balls and removing 1, faster and faster, until at the 1 hour mark, you are adding and removing them infinitely fast. You are always adding more balls than you remove (9 more to be exact).

So intuitively, people would say "oh, of course, the ball pit is infinitely full", but the fact of the matter is the time of removal for any ball is known. Because each time, you remove 1 ball right? But not just any ball, you remove the nth ball. So you can ask me, when did you remove ball number 87,145,699,728,023,344? And I could simply say, "After 1 - (1 / (287,145,699,728,023,344) ) hours.

That is, the ball pit is both infinitely full and empty at the same time. Problem of the Supertask.

I have a basic idea of what you explained, but considering it always takes half the time of the previous operation, you would never reach the 1 hour mark, so your question is null in any case.

This is pointless but if you were to graph it, you would get an Asymptote I'd be willing to bet.
Finally being in high maths in high school is paying off :P

Which brings us to the crux of the matter. So, the second paradox I mentioned, about you always having to go "half way" first, is also called Zeno's Paradox. We "solve it" (make it go away) through the notion that motion is taken to some limit.

You mentioned an asymptote, and indeed the sum of the series 1/(2^n) does approach 1. But the limit is one. That is, if we ask ourselves, what does this formula equal "at" infinity (I quote "at" because that's not really correct, you cannot take the image of a continuous function at a non-number, what we are really asking is what is the ultimate behavior of this function), it is 1. Go ask your math teach, next time he shows you an asymptote at 5 or something, "what is the limit of this function towards infinity (or negative infinity)" and he will tell you 5.

Now this is getting into hair splitting territory with the 0.99999\bar = 1 trolls. DO NOT THINK LIKE THAT. There are two distinct concepts here. Numbers. And Limits. When you take calculus you will get a lot more of that.

So anyways, the idea is that we can eventually "get there" because we take the tiny little itty bits of infinitely small non-motion and integrate them to get motion from A to B. When we are at B, we have "gotten there." We have effectively "done something" an infinite number of times in a finite time span.

What this does, is it makes motion a Supertask. And it allows us to formulate the problem of the Supertask, to say "well if that is possible, then so is this." The example is just a conceptual model for a much deeper generalization which basically says, if motion is possible, then so is this. If this is impossible, then motion isn't.

From the start I knew you were wrong. Without even having to look at the first step...
A and B are already given a separate value....then you treat them as if they were again variables..

twofish - I spent some time writing answers to your questions and by then you had already answered everything. lul
Either way, loved these. I then went on to read the wiki page for Supertask. Really interesting stuff.

However, I didn't quite understand the problem of Achilles and the tortoise. Here's what's written on wikipedia.

oh snap D:

1. An infinite number. While the number of balls you subtract is also infinite, it is growing more slowly than the number of balls added. While you can calculate the time of removal for any individual ball, you can also calculate the number of balls in the pit at the time of any given step and this is always growing without bound.

2. Same thing, limits/Riemann sum.

3. I don't know. Clearly if the answer is yes it is also no and if it is no it is also yes. Can you resolve this paradox in a clever way without bringing in something else?

I hate math...

Reminds me of this...

http://www.shoujoai.com/forum/topic_show.pl?tid=33136;pg=1

1 + 2.) Here you are missing the point of "an infinite number of steps in a finite time." This is a pretty common misinterpretation of the problem as well, where people assume "I can always say there were more balls added than removed at any step, even saying that I can calculate when any ball is removed, I can also show that there are more balls in the pit at that step." The main concept at fault here is infinity. Where one assumes that "there are always more balls being added" at this step. Unfortunately, this implies that our operations do not actually end, which contradicts what we know (or have widely assumed) about Zeno's Paradox and Supertasks. You need to think pretty hard about your concept of infinity, in a very abstract way, to understand what is going on here. In effect, after 1 hour "infinity has happened" (I know that sounds silly but bare with me). That is, there is no "this step." You cannot say "well you can show that ball 450,323,943 was removed and there should be 10*450,323,942 balls in the pit at that step" BUT in a later step we can show that all those balls have been removed. We can do this ad infinitum. Constantly adding one then showing there are more in the pit, then jumping forward to where they have all been removed, yet still more added.

But it's a Supertask. It's done. It has happened. There is no next step or previous step anymore.

3.) Is called Russel's Paradox and it does not have a solution. It is, for many mathematicians, one of the most disturbing antimonies in all of their field. We "ignore it" through The Axiom of Choice in Zermelo–Fraenkel Set Theory (often abbreviated ZFC).

It pains me that some of the deepest things like that and Godel's Incompleteness theorem just get totally ignored by many mathematicians.

So I looked up the problem and it looks like there is no consensus "true" answer. If you look at the total number of balls in the pit, then you get an infinite number. But, as you say, you can also show that for any given ball, it isn't in the pit, so it is empty. And people have argued both sides. Yet others think the answer is undefined or the problem is impossible for varying reasons. So I guess now 'm not really sure what your point is, other than a call for thoroughness or that mathematics is inconsistent.

[font=Verdana][color=green]I swear I've seen this posted before...but yes. Everyone has pretty much covered why this is wrong.