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by Xias
Sun Apr 02, 2017 9:42 pm UTC
Forum: Logic Puzzles
Topic: Two secrets
Replies: 20
Views: 2952

Re: Two secrets

Oh, that was silly of me.
by Xias
Sun Apr 02, 2017 8:55 am UTC
Forum: Logic Puzzles
Topic: MisterGC's PUZZLES
Replies: 17
Views: 3822

Re: MisterGC's PUZZLES

Spoiler:
the power of
accurate
observation
is commonly
called
cynicism by
those who
have not got
it.
by Xias
Sun Apr 02, 2017 5:45 am UTC
Forum: Logic Puzzles
Topic: Two secrets
Replies: 20
Views: 2952

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, ...
by Xias
Thu Mar 23, 2017 3:23 am UTC
Forum: Logic Puzzles
Topic: MisterGC's PUZZLES
Replies: 17
Views: 3822

Re: MisterGC's PUZZLES

Spoiler:
"I guess that's why they call it the American Dream because you have to be asleep to believe it"

And now I've got "I guess that's why they call it the blues" by Elton John stuck in my head :)
by Xias
Wed Jan 04, 2017 10:08 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Wed Jan 04, 2017 4:23 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Fri Dec 30, 2016 6:05 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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?
by Xias
Fri Dec 23, 2016 9:14 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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 ...
by Xias
Fri Dec 23, 2016 2:28 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Thu Dec 22, 2016 8:50 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Wed Dec 21, 2016 9:00 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Wed Dec 21, 2016 6:12 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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 ...
by Xias
Tue Dec 20, 2016 8:57 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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 ,...
by Xias
Sat Dec 17, 2016 11:16 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Tue Dec 13, 2016 2:54 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Mon Dec 12, 2016 5:00 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Sat Dec 10, 2016 9:09 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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 ...
by Xias
Sat Dec 10, 2016 4:57 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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), ...
by Xias
Fri Dec 09, 2016 4:59 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Thu Dec 08, 2016 3:50 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Wed Dec 07, 2016 1:22 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Tue Dec 06, 2016 2:06 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Tue Dec 06, 2016 12:07 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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. ...
by Xias
Mon Dec 05, 2016 3:53 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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 ...
by Xias
Sun Dec 04, 2016 9:18 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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 "...
by Xias
Sat Dec 03, 2016 5:27 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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 "...
by Xias
Fri Dec 02, 2016 6:22 pm UTC
Forum: Logic Puzzles
Topic: Stock Redistribution
Replies: 10
Views: 2197

Re: Stock Redistribution

ThemePark wrote:Okay, I'll bite.

Spoiler:
The answer is 6 hours and it sounds like Settlers of Catan to me


My solution gives

Spoiler:
21 hours, 55 minutes, 58 seconds.


I think one of us has misunderstood the question.

EDIT: I suppose I could include my distribution:

Spoiler:
208202 Wood
271812 Clay
208202 Iron
0 Wheat
by Xias
Thu Dec 01, 2016 11:56 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Wed Nov 30, 2016 8:54 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Mon Nov 28, 2016 5:34 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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.
by Xias
Mon Nov 28, 2016 4:33 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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.
by Xias
Mon Nov 28, 2016 2:30 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Sun Nov 27, 2016 10:18 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Sun Nov 27, 2016 10:31 am UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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, ...
by Xias
Sat Nov 26, 2016 8:39 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Sat Nov 26, 2016 1:07 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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 ...
by Xias
Sat Nov 26, 2016 9:06 am UTC
Forum: Logic Puzzles
Topic: Hidden Cards
Replies: 5
Views: 2001

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...
by Xias
Fri Nov 25, 2016 8:02 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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...
by Xias
Thu Nov 24, 2016 8:44 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

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 ...
by Xias
Thu Nov 24, 2016 3:36 pm UTC
Forum: Logic Puzzles
Topic: Infinite Balls and Jugs [solution]
Replies: 611
Views: 62586

Re: Infinite Balls and Jugs [solution]

@xias, omega+1 has the same cardinality as N , that is it is countably infinite, but yes, there is no order preserving bijection between omega+1 and omega. But yes, HH is usually defined to be isomorphic to omega Ah, I think I was conflating the two ideas. Obviously they're not the same thing or yo...

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