I believe d(n) is just shorthand for "the number of digits in n."

I wonder about the fact that n^2 and pi(n) are not dependent on the base, but d(n) is.

## Search found 401 matches

- Sat Sep 02, 2017 10:06 am UTC
- Forum: Logic Puzzles
- Topic: Arithmetic puzzle
- Replies:
**5** - Views:
**1868**

- Sun Apr 02, 2017 9:42 pm UTC
- Forum: Logic Puzzles
- Topic: Two secrets
- Replies:
**20** - Views:
**3630**

### Re: Two secrets

Oh, that was silly of me.

- Sun Apr 02, 2017 8:55 am UTC
- Forum: Logic Puzzles
- Topic: MisterGC's PUZZLES
- Replies:
**17** - Views:
**4440**

### Re: MisterGC's PUZZLES

**Spoiler:**

- Sun Apr 02, 2017 5:45 am UTC
- Forum: Logic Puzzles
- Topic: Two secrets
- Replies:
**20** - Views:
**3630**

### Re: Two secrets

In truth, N = 5 should require only 3 questions. It's true that adversarially the secret-holder will always answer "no" to a 2-element question if possible (otherwise, guessing those 2 elements will guarantee at least one is in the set), but if you strategically ask about, let's say, {1, ...

- Thu Mar 23, 2017 3:23 am UTC
- Forum: Logic Puzzles
- Topic: MisterGC's PUZZLES
- Replies:
**17** - Views:
**4440**

### Re: MisterGC's PUZZLES

**Spoiler:**

- Wed Jan 04, 2017 10:08 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

The primary reason I can't fully get behind your argument is that a nonempty jug leads to (many) logical contradictions. I'm not sure on what basis we can say anything is objectively incorrect if that does not qualify. What is "objective validity"? Where in objective reality can you find p...

- Wed Jan 04, 2017 4:23 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Again, Wildcard, do you hold the same views about the roots of negative numbers? I can literally replace two lines in your original post and everything else holds: The question poses a situation that is impossible in the physical world we live in. Therefore, the rules of the physical universe may no...

- Fri Dec 30, 2016 6:05 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Wildcard: Are we allowed to make objective statements about the square roots of negative numbers, or is that too imaginary for you?

- Fri Dec 23, 2016 9:14 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Kryptonaut, let's try this. Instead of removing balls at each step, we'll install timers on each ball such that when the timer runs out, the ball immediately vanishes and ceases to exist. We set the timer so that for each ball n, it will evaporate and disappear at 20/2 n minutes before midnight. So ...

- Fri Dec 23, 2016 2:28 am UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

It's amazing this has gone on for so many pages. I think it's worth reiterating how simple the solution really is. Yes.... there is another thread this reminds me of. I'll spare you which one it is, but it's the first one I posted in. :) Jose For me it reminds me of Blue Eyes. The bulk of that thre...

- Thu Dec 22, 2016 8:50 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

You can describe a general case where the balls are un-numbered and everyone (as far as I know) agrees you end up with an infinite number of them. No possibility whatsoever of any other outcome. Except for the half dozen users in this thread, the ones who are actually having a conversation with you...

- Wed Dec 21, 2016 9:00 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Your problem is your continued insistence that somehow balls like ω and ω+1 and 2ω just magically show up. All the balls we started with had natural numbers on them. So which natural number was originally on the now-lowest ball in the jug, which you claim is now numbered at least ω? When did it acq...

- Wed Dec 21, 2016 6:12 am UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

What is the fundamental difference between (at midnight) there suddenly being balls that were never added and a state that was never reached? I think this is a really good question, because in a way it highlights why it can be so difficult to adjust our intuitions to reflect the mathematics of the ...

- Tue Dec 20, 2016 8:57 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Kryptonaut, I would still like for you to address the point that I made. Before midnight, there is only finite-numbered steps. At finite numbered step k, we only interact with the following eleven balls: {b k , b 10k-9 , b 10k-8 , b 10k-7 , b 10k-6 , b 10k-5 , b 10k-4 , b 10k-3 , b 10k-2 , b 10k-1 ,...

- Sat Dec 17, 2016 11:16 am UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

then we could say... We could say anything, really. What matters is what we can justify . So when you say this, what you're actually saying is "then I am going to say..." unless you actually justify it. it's just that the actual cardinal numbers are unknowable. What we have is a situation...

- Tue Dec 13, 2016 2:54 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Defining B n ={n+1,n+2,...10n} L sup = ∩ n≥1 ∪ j≥n B j Every ∪ j will contain all members of B ∞ , therefore the intersection of all ∪ j also contains all members of B ∞ There is no such thing as a ∪ j . L inf = ∪ n≥1 ∩ j≥n B j ∩ j=∞ B j = B ∞ , therefore the union of all ∩ j≥n B j = B ∞ There is n...

- Mon Dec 12, 2016 5:00 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Kryptonaut's Conjecture: For any sequence of sets {An}n=1→infinity, |limn→infinity An| = limn→infinity |An|. First off, I have not claimed anything for "any sequence of sets", so for you to then go on and argue about a different sequence of sets is not proving anything about the case in p...

- Sat Dec 10, 2016 9:09 am UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

However, the simplest version that displays the "interesting behaviour" is N . If it's not there to begin with, it can't be there in the end, and if that gives xir trouble, the more subtle case of unreachable numbers will not be any easier. Jose Don't get me wrong; I'm all about reducing ...

- Sat Dec 10, 2016 4:57 am UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Since at the limit, (1/2) n and (9/10) n are both equal to 1... You mean 0 Yes, thanks for that. @kryptonaut: "1,2... and so on" is used to refer to the set of natural numbers greater than 0. If people want to specify some set that contains infinitely big numbers(such as the superreals), ...

- Fri Dec 09, 2016 4:59 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Well, it leads to a paradox in the '9n=0 for some finite n>0' sense. My set-based proof from many pages ago resolves this paradox. The intuition is that since the cardinality of the set of balls in the jug is increasing without bound - and is in fact 9n for all finite n - that the cardinality of th...

- Thu Dec 08, 2016 3:50 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

We have differing opinions on whether a fully instantiated infinite set of numbers includes infinite numbers or not. So whatever I answer here will only lead to us going round in circles. And my opinion is consistent with set theory and abstract algebra, and yours is not. So the circle we've been i...

- Wed Dec 07, 2016 1:22 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Regarding set partitioning: Take an infinite set S ={1,1,1,1,....} Append another 1. Or a hundred of them, or an infinite number. You have the same set. Do that backwards, you have partitioned S into an infinite set and another set. Take an infinite set N ={1,2,3,...} Append an infinite value. Or a...

- Tue Dec 06, 2016 2:06 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

↶ You want me to explain what partitioning means? I think I will anyway, for clarity. Two sets A and B are "partitions" of a set S if: ⋅ The intersection of A and B is empty (ie A and B are disjoint, do not overlap) ⋅ The union of A and B is S (note: not "maps 1:1...

- Tue Dec 06, 2016 12:07 am UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

One such complication: We can split the supertask into two supertasks No we can't. For the same reason that you can't say 10+infinity-infinity=10. The whole paradox revolves around the fact that two supertasks run in parallel but at different rates, which fact is lost if you run them sequentially. ...

- Mon Dec 05, 2016 3:53 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Point (A) should be clarified to read "For any finite number n...". Whilst it is true that only balls with finite numbers exist at any time before midnight, there are an infinite number of balls with infinite numbers (in the sense that you can never count that high) present at the end of ...

- Sun Dec 04, 2016 9:18 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

It was argued that since balls have special memories of their past digits and Hilbert's Hotel has an incompetent assistant manager, the limits should actually be different. You sound unconvinced. 1. In the original, remove-the-lowest-ball game, there exists at some point a ball with the label "...

- Sat Dec 03, 2016 5:27 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

It was argued that since balls have special memories of their past digits and Hilbert's Hotel has an incompetent assistant manager, the limits should actually be different. You sound unconvinced. 1. In the original, remove-the-lowest-ball game, there exists at some point a ball with the label "...

- Fri Dec 02, 2016 6:22 pm UTC
- Forum: Logic Puzzles
- Topic: Stock Redistribution
- Replies:
**10** - Views:
**2549**

### Re: Stock Redistribution

ThemePark wrote:Okay, I'll bite.Spoiler:

My solution gives

**Spoiler:**

I think one of us has misunderstood the question.

EDIT: I suppose I could include my distribution:

**Spoiler:**

- Thu Dec 01, 2016 11:56 am UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

I'll say just this, then I'm gone. I thought you were gone already. But in any case, you once again did not acknowledge anything I said in my post. We're not on a live debate stage. You have an opportunity to sit back and take your time reflecting on and responding to something. Yet you don't. If y...

- Wed Nov 30, 2016 8:54 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

I have yet to see those other reasons. Several people spent the better part of five pages asking for them. I think if there were any, they would have been provided. I gave up trying to teach kryptonaut mathematics, but I think that for posterity the following deserves comment (emphasis mine) As ano...

- Mon Nov 28, 2016 5:34 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

kryptonaut wrote:If a is infinite then the angle is zero. I'm sorry if I didn't acknowledge your discussion

With that it becomes clear that you aren't interested in "exploring" anything. Take care.

- Mon Nov 28, 2016 4:33 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

ucim wrote:Second, omega is an ordinal. It refers (as you say) to the position in an ordered set. Omega minus one refers to the position of the element before it.

In fairness to kryptonaut, there is no element before an omega-th element for "omega minus one" to refer to.

- Mon Nov 28, 2016 2:30 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

How would you like to resolve this? The value 'driving' the puzzle is a n , which is analogous to n in the jugs game. θ is the thing that you change at each step, from which all other values in the puzzle are defined. So why is a n the value that drives the puzzle? If we think of the puzzle as a di...

- Sun Nov 27, 2016 10:18 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

1) I agree, θ and φ converge to zero 2) The intersection points are at infinity if that's how you like to describe it, or further away than any distance you can compare them with (although I would argue that the φ intersection is still in some sense 'even further' than the θ) Let's focus on these a...

- Sun Nov 27, 2016 10:31 am UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Consider the following scenario: What we are going to do is take two parallel lines and connect them with two line segments. Then, we will rotate those two line segments at different rates, such that the angles between the line segments and one of the parallel lines becomes zero. We will find that, ...

- Sat Nov 26, 2016 8:39 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Everything is set up so that the limits work that way - the discards are {1,2,3...n} followed by the keepers {n+1,n+2,...10n} which converges to N followed by another set isomorphic to N but higher than any number in N . If the set of keepers converges to the null set, then, magically, the whole th...

- Sat Nov 26, 2016 1:07 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

No they don't, see reasoning above. Your argument that they do is based on an assumption that the union of discards and keepers can be mapped to N , which it cannot. They have to map to the balls actually added to the jug in order for K to be in the jug. You have yet to show how the balls added by ...

- Sat Nov 26, 2016 9:06 am UTC
- Forum: Logic Puzzles
- Topic: Hidden Cards
- Replies:
**5** - Views:
**2289**

### Re: Hidden Cards

Essentially, the information immediately available to each person is "who is the next person in my cycle?" So whatever guess someone makes is them guessing who the third person in their cycle would be. Obviously, any 1-cycle is immediately eliminated. After that, regardless of the distrib...

- Fri Nov 25, 2016 8:02 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

Kryptonaut: The "keepers" in the original game converges to the null set. Then everything that you say is consistent with the jug being empty. A major relevant difference between a game in which you sequentially remove every naturally numbered ball (because there is a naturally numbered st...

- Thu Nov 24, 2016 8:44 pm UTC
- Forum: Logic Puzzles
- Topic: Infinite Balls and Jugs [solution]
- Replies:
**611** - Views:
**68753**

### Re: Infinite Balls and Jugs [solution]

The same goes for the balls in jugs. If you want to change the game to include balls and steps with numbers greater than any natural number, you can do that, but it's a different game. The game as it's defined leads to the jug being empty, because a ball with properties that allows it to remain in ...