1. Anyone care to have a stab at a maths puzzle?

The puzzle is:

An infinite number of coins must be dropped in a well before lunchtime. An infinite number of steps are then performed, such that at each step coins 10 coins are added and 1 then removed from the well. The question then is, how many coins are left in the well when the task is finished?

To complete an infinite number of steps, it is assumed that the well is empty at 1 minute to lunchtime, and that the following steps are performed

The first step is performed at 30 seconds before lunchtime.

The second step is performed at 15 seconds before lunchtime.

Each subsequent step is performed in half the time of the previous step, i.e., step n is performed at 2−n minutes before lunchtime.

This guarantees that a countably infinite number of steps is performed by lunchtime. Since each subsequent step takes half as much time as the previous step, an infinite number of steps is performed by the time one minute has passed.

At each step, ten coins are dropped in the well, and one coin is removed from the well. The question is then: How many coins are in the well at lunchtime?

2.

3. I need to check something. You add 10 coins and remove one? In other words, you add 9?

4. I also don't get why you would add 10 and then remove 1. Why don't we just add 9. In which case this puzzle is just the sum from 1 to infinity of 9. Demanding that a countably infinite amount of steps are taken in a finite amount of time means that the series diverges and you get an infinite amount of coins in the well...

I don't see how the time intervals come into play in any way other than to guarantee that an infinite amount of steps are taken.

5. After reading a few Wiki pages on sets, I have an idea. I'll take a stab at it (though I'll likely miss terribly. don't judge me please). Answer follows in white...

The well ends up empty. In the end, there are no coins. This is because overall, you're adding a set and subtracting a set with the same size. With an infinite number of steps, it doesn't matter if you're adding more than you subtract in each individual step, because the sets will end up with the same cardinality.

If I was completely wrong, don't blame me. This is all far above me, and I suspect the answer is some surprising, analytic set theory work.

6. I'll work out the answer, but can I have lunch first?

7. Ok here's my guess:

It takes 60/2 seconds to finish the first; the second 60/(2*2); the third 60/(2*3)......... the 'n' 60/(2*n).
Let us assume that after 'n' steps it's finally lunch time!

Thus

You just have to figure out 'n' and multiply that by 9. I'll post the final answer later. Got to eat

8. Again I learned a lesson: Never do math on the computer!

I went to my desk and wrote down the equations and everything, here's the answer:

Infinite number of coins ... no actually ... I got no idea!

How I came up with this:

I check my answer and found my previous post incorrect. The actual equation looks like this:

It's like 1+0.1+0.01+0.001... = 2

No matter how great n is, we'll never get 60, even when n is infinite I believe. So ya what is the answer!

9. Originally Posted by Wise Man
Again I learned a lesson: Never do math on the computer!

I went to my desk and wrote down the equations and everything, here's the answer:

Infinite number of coins ... no actually ... I got no idea!

How I came up with this:

I check my answer and found my previous post incorrect. The actual equation looks like this:

It's like 1+0.1+0.01+0.001... = 2

No matter how great n is, we'll never get 60, even when n is infinite I believe. So ya what is the answer!
well done! Wise Man the answer is indeed indefinable! to complete the infinite number of steps in the ever shortening time you can never reach lunchtime, so you can never say how many coins are in the well.

10. Well done everybody who had a look at the maths puzzle and especially well done those us of you brave enough to have ago at solving it.
Wise Man solved it with the correct answer that the number of coins is indefinable.

11. Chrisgorlitz that's one trick question! No wonder you earned a 'Forum Masters Degree '.

12. Well, the most obvious answer is an infinite amount of coins, but you have to think beyond that and it is also useful to present to those who like the concept of infinity, I for one am not one of them.

I would have been interested to see the workings though of anyone getting either 0 or 9, could have been interesting.

14. Originally Posted by halorealm
The answer is, as Wise Man worked out, you can never get to lunchtime because you can never get to the end of the task. Therefore the number of coins in the well at lunchtime is unknowable.

15. But your time step increments do in fact converge, I fail to see why you wouldn't be able to make it to lunch. Unless you are putting a real life constraint on the task because obviously a person in real life couldn't do an infinite number of steps infinitely fast but I didn't think that restriction was apart of the puzzle. We proved a theorem in my calculus class that you can get a divergent series to add up to any number you wanted if you allowed for rearrangement of the terms, but it seems here you specified the order. I still think it should be infinite.

16. You have me most interested in the idea that the steps could converge, please explain how?

17. Well I mean provided that person could be putting coins in the well infinitely fast to meet the requirements that he puts 9 coins in at each step, lunch is going to be in 30 seconds regardless. It isn't like that 30 seconds is getting pushed further away, the guy just starts putting coins in the well at a trillion miles an hour. Unless you are saying that the person couldn't be moving that fast to be doing the steps, which obviously he can't in real life.

18. Yes I guess he would have to be moving pretty fast.

19. So I think I have a bit of a grip on this problem now. A good way to think about this problem is to number the balls. First we put in balls #1-10 and then take out ball #1. Next step put in balls #11-20 and take out ball #2. Continue putting in #21-30 and take out #3. Continue this ad infinity. Now both sets (The set of 1-10, 2-10, ect. and the set 1, 2, 3,...) are countably infinite and a bijection can be formed between them, so we can ensure that both sets are covered entirely by this process. This would lead you to believe that no balls are left since in the infinite amount of steps, we actually take out every ball.

This is exactly why divergent series are super non intuitive.

20. Originally Posted by Chrisgorlitz
You have me most interested in the idea that the steps could converge, please explain how?
A wise man I knew once said...

The actual equation looks like this:

Notice the last term in the series, 60/(2^n). A better way to formulate this is .

30 = 30
30+15 = 45
30+15+7.5 = 52.5
... + 3.75 = 56.25

Jumped to the hundredth term and you're already approximated at 59.99999999999999999999999... So it's easy to see how that even though there are infinite steps of time, the time intervals decrease geometrically resulting in a finite result. This is much like the dichotomy paradox.

21. Well if you're going to be that pedantic then you're all wrong. The instant of 'midday' is only meaningfully described to the precision of 1 Planck time, which is about seconds. So you are complete after steps. In which case you have just 9*149=1,341 coins in the well.

22. Suppose that the coins of the infinite supply of coins were numbered, and that at step 1 coins 1 through 10 are dropped into the well, and coin number 1 is then removed. At step 2, coins 11 through 20 are dropped, and coin 2 is then removed. This means that by lunchtime, every coin labeled n that is dropped into the well is eventually removed in a subsequent step (namely, at step n). Hence, the well is empty at lunchtime. It is the juxtaposition of this argument that the well is empty at lunchtime. However I subscribe to the idea that lunchtime cannot be reached as this is a 'supertask' with no end. Thus showing the difficulties with an infinite 'supertask'.

23. Originally Posted by Chrisgorlitz
Hence, the well is empty at lunchtime.
That does not follow at all. In the finite case that is obviously false, and in the infinite case it is also false since the series diverges to +infinity (keep in mind that the time interval between adding and then removing a coin becomes ever larger). And if you want to rely entirely on intuition here then you'll have to throw out infinity altogether.

24. In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

(
cryptic)

25. If you take Zeno's paradoxes as having any real significance then can I interest you in some magic beans?

26. "to complete an infinite number of tasks, which Zeno maintains is an impossibility."

27. "Today there is still a debate on the question of whether or not Zeno's paradoxes have been resolved."

How'd you like them beans.

28. What if we disregard obvious physical restrictions? Disregard the physicality and just look at the math being applied.

29. Zeno's paradox has one incorrect assumption: Distance can be infinity divided. Physicists discovered that when arrived at a distance so small, it becomes meaning less.

Here's Zeno's logic: Because a finite distance can be infinitely divided, thus the distance is infinite.

Clearly wrong. BTW, Can I have a magic bean?

30. Originally Posted by Wise Man
Zeno's paradox has one incorrect assumption: Distance can be infinity divided. Physicists discovered that when arrived at a distance so small, it becomes meaning less.

Here's Zeno's logic: Because a finite distance can be infinitely divided, thus the distance is infinite.

Clearly wrong. BTW, Can I have a magic bean?

Ok so if we apply this to the puzzle and take time instead of distance and say that when time gets so small it becomes meaningless it would then be impossible to complete the infinite number of steps so the task cannot be completed and thus we cannot know the number of coins in the well, in which case your answer is still correct.

31. I wonder also if physicists might apply that logic the other way when trying to work out wether the universe is infinite and decide that when it reaches a suitabley large size it becomes meaningless.

32. Originally Posted by Chrisgorlitz
"Today there is still a debate on the question of whether or not Zeno's paradoxes have been resolved."
Only amongst the philosophers. The fact of reality is that Achilles would beat the tortoise, so it's not a problem. And in light of analysis, mathematicians don't care for it either.

The debate around Zeno's paradoxes reminds me of when people claim that the flight of bees violates the laws of physics. Putting aside the laughable origin of that claim, I want to grab them and say 'Look! It's flying! ...'

33. There's this phenomenon called the 'Quantum Zeno effect'. Where a particle that decays quickly, when observed continuously, will never decay!

Link: Quantum Zeno effect - Wikipedia, the free encyclopedia

However I do not know why the name 'Zeno effect' was coined. What does this have to do with Zeno paradox?

34. I like nano's (post #20) idea about Planck scale. It solved the infinite problem. This thing has happened before in physic... its called "Ultraviolet catastrophe" which was solved in the same way.

In REAL LIFE there's CONSTRAINT that PREVENT INFINITY

35. Originally Posted by msafwan
I like nano's (post #20) idea about Planck scale. It solved the infinite problem. This thing has happened before in physic... its called "Ultraviolet catastrophe" which was solved in the same way.

In REAL LIFE there's CONSTRAINT that PREVENT INFINITY
I would tend to agree with that, also what TheObserver pointed out about speed because obviously light speed couldn't be exceeded so this would also prove a natural limit.

Well conjecture about beans magic or otherwise, the relevance of Zeno and the actual answer aside, it has been fun going through this puzzle with you guys. Who knew it would be so involved.

36. Originally Posted by halorealm
Bingo! The assertion, I can't remember who made it, that it is zero is appealing but wrong

To see this (forget all about "physical constraints" and other irrelevant nonsense - this a math forum after all). First notice that the number of halves-of-halves-of-halves-of-....... in the interval 0 to 15 min is infinite.

So we are throwing in 10 coins an infinite number of times. Now we know that . Likewise we know that removing a single coin an infinite number of times removes an infinite number of coins, so let us assume that the number of coins remaining is zero, that is .

But suppose, by an unnoticed slip of the hand, not 10 but 11 coins are tossed in at least once. So what do we have? which, if we assume associativity is (since which implies that .

And if the tosser's slip of the hand should occur twice we will have and so on and on and......

Subtracting infinity from itself is not defined, so that tempting as it might seem,

37. Saying is like saying . Arithmetic operations are not defined on infinity.

38. Originally Posted by Guitarist
Originally Posted by halorealm
Bingo! The assertion, I can't remember who made it, that it is zero is appealing but wrong

To see this (forget all about "physical constraints" and other irrelevant nonsense - this a math forum after all). First notice that the number of halves-of-halves-of-halves-of-....... in the interval 0 to 15 min is infinite.

So we are throwing in 10 coins an infinite number of times. Now we know that . Likewise we know that removing a single coin an infinite number of times removes an infinite number of coins,
Sure: But if you remove each other coin then theres as many left as was taken out.
Originally Posted by Guitarist
so let us assume that the number of coins remaining is zero, that is .
That depends (as we saw above) on how the coins were selected.

Originally Posted by Guitarist
But suppose, by an unnoticed slip of the hand, not 10 but 11 coins are tossed in at least once. So what do we have? which, if we assume associativity is (since which implies that .

And if the tosser's slip of the hand should occur twice we will have and so on and on and......

Subtracting infinity from itself is not defined, so that tempting as it might seem,
Are you reaching a conclusion without stepping out of the assumptions?
If so the conclusion is as valid as the assumptions...and the first was not valid. So the conclusion is not valid.

39. Originally Posted by nano
Saying is like saying . Arithmetic operations are not defined on infinity.
Actually, even though infinity's not considered a number, it's not absolutely inapplicable with operations... as far as I know. I think one's allowed to treat it "in the extended reals", and that's how we calculate nontrivial limits at infinity. Of course, we can't relate this to ordinary elementary algebra since we get things like 1=2 (or generally, every real number equals every other real number), but since it's implied in the context of limits we're not actually proving so.

40. Originally Posted by nano
Saying is like saying . Arithmetic operations are not defined on infinity.
Hi! Not making it rain on your parade nano, but didnt Cantor do just that? Define an arithmetic of Transfinite Numbers?

41. The number of coins that we end up with depends on the order in which the coins are removed from the well. As I said earlier, the coins can be added and removed in such a way that no coins are be left in the well at lunchtime. However, if coin number 10 were removed from the well at step 1, coin number 20 at step 2, and so forth, then it is clear that there will be an infinite number of coins left in the well at lunch.

42. Originally Posted by Chrisgorlitz
The number of coins that we end up with depends on the order in which the coins are removed from the well. As I said earlier, the coins can be added and removed in such a way that no coins are be left in the well at lunchtime. However, if coin number 10 were removed from the well at step 1, coin number 20 at step 2, and so forth, then it is clear that there will be an infinite number of coins left in the well at lunch.
If I am not mistaken then you are correct. The way was: 10 in, one out. That leaves coins in the well each time, and that definitely leaves us at lunch with infinitely many coins both inside and outside the well.

43. Originally Posted by sigurdW
Originally Posted by nano
Saying is like saying . Arithmetic operations are not defined on infinity.
Hi! Not making it rain on your parade nano, but didnt Cantor do just that? Define an arithmetic of Transfinite Numbers?
Of course we can define new types of numbers which are characterised as being both 'infinite' and arithmetically operable (transfinite numbers being one of numerous examples) but that is not the infinity that we're speaking of in this context. So I stand by my statement.

44. Hi Nano

Doesn't the extended real number system deal with this exact type of infinity though. An unbounded, increasing sequence converges to and at least some of the normal arithmetic operations can be extended to this number system.

45. Originally Posted by river_rat
Hi Nano

Doesn't the extended real number system deal with this exact type of infinity though. An unbounded, increasing sequence converges to and at least some of the normal arithmetic operations can be extended to this number system.
Not exactly. The only arithmetic operations defined on the infinity in the extended reals (which, again, is a patch) are between the infinite (for which and are the same thing) and finite elements. Something such as , which was my main gripe, is undefined even on the extended reals.

46. Originally Posted by nano
Something such as , which was my main gripe, is undefined even on the extended reals.
Why is this a "gripe"? Did you not like my proof that is undefined? Surely you recognize a proof by contradiction when you see one?

Anyway the extended reals were mentioned. Let's do this......

Suppose I take a segment of the real line and join the two ends together. This we recognize as a circle; grand folk call it the 1-sphere . Now since this is essentially the "same" as our line segment , we might expect to be able to find a one-to-one correspondence between them. This correspondence - technically a "homeomorphism", since these are topological spaces - will be given by an invertible function from one to the other (this is called a bijection, btw).

Of course no such bijection exists; the map sends the two ends of to same point of - our "join". And since well-behaved functions are not allowed to send a single point in the domain to distinct points in the co-domain, there can be no inverse .

Now there is a theorem (due I think to Dedekind and Pierce) that states that an infinite set is always isomorphic to a proper subset of itself. Since and since homeomorphism is an equivalence relation, by transitivity we might expect to be able to do something like the following.

Place our circle anywhere on the real line, and project each point of "down" to a point in in such a way that none of these projections intersect each other or . (Let's recall that the standard topology on mandates that the real line is the open interval (or rather union of open intervals) i.e. infinities are NOT included)

All goes swimmingly until we get to the North pole, where now our projections run parallel to and seem never to meet it.

But we have the perfect construction specifically made for just this purpose, called the "projective real line". This is formed by adjoining what's called the "point at infinity" to to form . So now our point at the North pole projects to this point at infinity, and a homeomorphism can be recovered, but at some cost; I cannot make this construction of 1-dimensional spaces without embedding them in, say a 2-dimensional space. And, if I am willing to do this, I have to say that parallel lines meet!

Worse still, since the two projections - "left" and "right" - from the North pole map to the same point in our projective line, this implies that, in , the "end-points" of the projective real line may be considered as the same point - that is

Which is no more, or less, than to assert that the projective real line has the same topology (up to homeomorphism) as the real circle.

And if we drop the topological constraints, we may say that is the "extended reals". But note that whereas is a field (both additive and multiplicative inverses exist), by the above, is not a field, since there exists no

This line of thought can be extended to the plane and its projective "partner" , but here the geometry is a little more interesting; it's called "projective geometry", where "points" are "lines" and "lines" are "planes"

Gosh, I am a long-winded bastard. My excuse is that it is hammering rain here in Northern France - this post is courtesy of McD's whose food looks like shite, but whose coffee is excellent

47. The coin removals shows that both:

infinity - infinity = 0

and:

infinity - infinity = infinity

are possible, just by ajusting the order coin removal, which seems strange.

48. Originally Posted by Guitarist
Originally Posted by nano
Something such as , which was my main gripe, is undefined even on the extended reals.
Why is this a "gripe"? Did you not like my proof that is undefined? Surely you recognize a proof by contradiction when you see one?

Anyway the extended reals were mentioned. Let's do this......

Suppose I take a segment of the real line and join the two ends together. This we recognize as a circle; grand folk call it the 1-sphere . Now since this is essentially the "same" as our line segment , we might expect to be able to find a one-to-one correspondence between them. This correspondence - technically a "homeomorphism", since these are topological spaces - will be given by an invertible function from one to the other (this is called a bijection, btw).

Of course no such bijection exists; the map sends the two ends of to same point of - our "join". And since well-behaved functions are not allowed to send a single point in the domain to distinct points in the co-domain, there can be no inverse .

Now there is a theorem (due I think to Dedekind and Pierce) that states that an infinite set is always isomorphic to a proper subset of itself. Since and since homeomorphism is an equivalence relation, by transitivity we might expect to be able to do something like the following.

Place our circle anywhere on the real line, and project each point of "down" to a point in in such a way that none of these projections intersect each other or . (Let's recall that the standard topology on mandates that the real line is the open interval (or rather union of open intervals) i.e. infinities are NOT included)

All goes swimmingly until we get to the North pole, where now our projections run parallel to and seem never to meet it.

But we have the perfect construction specifically made for just this purpose, called the "projective real line". This is formed by adjoining what's called the "point at infinity" to to form . So now our point at the North pole projects to this point at infinity, and a homeomorphism can be recovered, but at some cost; I cannot make this construction of 1-dimensional spaces without embedding them in, say a 2-dimensional space. And, if I am willing to do this, I have to say that parallel lines meet!

Worse still, since the two projections - "left" and "right" - from the North pole map to the same point in our projective line, this implies that, in , the "end-points" of the projective real line may be considered as the same point - that is

Which is no more, or less, than to assert that the projective real line has the same topology (up to homeomorphism) as the real circle.

And if we drop the topological constraints, we may say that is the "extended reals". But note that whereas is a field (both additive and multiplicative inverses exist), by the above, is not a field, since there exists no

This line of thought can be extended to the plane and its projective "partner" , but here the geometry is a little more interesting; it's called "projective geometry", where "points" are "lines" and "lines" are "planes"

Gosh, I am a long-winded bastard. My excuse is that it is hammering rain here in Northern France - this post is courtesy of McD's whose food looks like shite, but whose coffee is excellent
Nah! This is rather a bit too short.

49. I frequently read the old threads in the math section of the forum. If I know Guitarist's style, it's that one should always expect something long and detailed whenever number theory or set theory are involved. :P

50. Originally Posted by halorealm
I frequently read the old threads in the math section of the forum. If I know Guitarist's style, it's that one should always expect something long and detailed whenever number theory or set theory are involved. :P
I was not sarcastic! I like readable set theorethical analysises.
Some theoreticians use notations I cant follow... Guitarist tries to be readable.

51. Originally Posted by sigurdW
Originally Posted by halorealm
I frequently read the old threads in the math section of the forum. If I know Guitarist's style, it's that one should always expect something long and detailed whenever number theory or set theory are involved. :P
I was not sarcastic!
I doubt that

I like readable set theorethical analysises.
Some theoreticians use notations I cant follow... Guitarist tries to be readable.
True. I think the mod does do a pretty good job of presenting advanced ideas in a readable manner, but that doesn't change the fact that his posts are long. If I'm a mathematician, how will I usually work problems out and then present my work? With proofs, which are usually quite long. Since it seems he takes this approach, that's why I said we should expect it to be that way.

By the way, what in the world is "Set Theoretical Analysis"? I haven no clue as to what that might deal with (well, besides obviously sets).

52. Originally Posted by halorealm
Originally Posted by sigurdW
Originally Posted by halorealm
I frequently read the old threads in the math section of the forum. If I know Guitarist's style, it's that one should always expect something long and detailed whenever number theory or set theory are involved. :P
I was not sarcastic!
I doubt that

I like readable set theorethical analysises.
Some theoreticians use notations I cant follow... Guitarist tries to be readable.
True. I think the mod does do a pretty good job of presenting advanced ideas in a readable manner, but that doesn't change the fact that his posts are long. If I'm a mathematician, how will I usually work problems out and then present my work? With proofs, which are usually quite long. Since it seems he takes this approach, that's why I said we should expect it to be that way.

By the way, what in the world is "Set Theoretical Analysis"? I haven no clue as to what that might deal with (well, besides obviously sets).
Lol! Means any analysis involving sets and such. Like Cantors concept of the Absolute. Something nobody discusses. Discussing Hegel on absolute being seems to attract more attention. Charlatans gets interpretators... Deep thinkers stay buried.

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