Is this possible?

[Philosophaster]

I see message boards as a way of introducing people to new ideas and ways of thinking, which is something I rather enjoy doing. They let you put your thoughts out there and then actually interact with the people who encounter them.

In my time on boards, I've had other people introduce me to new ideas as well, refined or reformed my own thoughts on many things, sharpened my debate skills, and even had a bunch of laughs. I definitely consider much of the time I spend on boards to be beneficial one way or another.

Some might say I'm just deluding myself. :-)

[Unidian]

Quite a few would probably say that.

Let them say it. Most of them probably think of themselves as doing "substantial" things like working a job, lol.

The interesting part, though, is that they knock themselves down when they do it. They are participating in the very discussions and activities they say are of no significance or worth. Since they get their sense of making a contribution through their jobs and other (usually regressive) activities, they don't need to see what they do online as being of any importance. That's unfortunate, because it contributes to the devaluation of what could have been a remarkably useful venue for cultural advancement, and keeps people doing things that don't accomplish squat in order to gain self-worth.

[vicdan]

Feeding people in the third world is a total waste of time until the cultural factors that drive third-world hunger and poverty are addressed.

Feeding the voluntarily indolent is a total waste of time until the psychological factors that drive voluntary indolence are addressed.

And how do they get addressed? One discussion at a time, in the "marketplace of ideas." That's the sort of work I think of as very appropriate for those suited to it. It isn't a waste of time to involve oneself in such work, and not all of us view it purely in terms of entertainment.

The sort of cultural work you do is the work of, well, justifying your doing such 'cultural work'.

[Katy]

Certainly the former administrators of the board where this fine thread is found can't be shunning having a bit of fun on a message board?

This thread was originally in the science and math forum, until Kevin dumped all those threads into here, so it did make a bit more sense in that context. But, nonetheless. We're playing games. You play games too.

[Philosophaster]

Sorry, what? Who was "shunning having a bit of fun?"

[Katy]

Sorry, what? Who was "shunning having a bit of fun?"

I wish I could avoid being "the ass" here, but why solving such a problem would be important escapes me. While you have every right to determine the criteria by which one might earn your intellectual respect, I guess I'd have to say that anyone spending time on something like this might lose a bit of mine. These sort of things are mental parlor tricks, no?

[Philosophaster]

Fun is pretty much the reason I come to Genius Forum.

[Unidian]

The hecklers know exactly what I mean. They are dismissing it for known reasons. That's expected.

[Dan Rowden]

Fun is pretty much the reason I come to Genius Forum.

Fuck off, then.

[Laird]

Just curious... does cultural work count for anything with you guys? Is advocating ideas and participating in the sort of dialogue that drives progress bit-by-bit something that anyone here recognizes as any better or more productive than picking one's nose?

Sure, that sort of thing involves a type of learning, but as I wrote earlier, I don't rate the learning that occurs in general on this forum much above the learning that one can get from solving neat problems. Sometimes someone says the odd useful thing, but mostly what I've learnt I've learnt by thinking things through for myself.

The idea that anything that happens on the internet is insignificant by default is an extremely popular one, but I've never understood its basis. Change comes from ideas, and discussion between people is where change begins. Right here, right now, one conversation at a time. Whether it happens around the water cooler, over the telephone, or on an internet forum makes no difference.

Agreed. Probably disagreed on the extent of the effect that the forums that you and I inhabit have though. For the most part the effect is one of providing entertainment and stimulus to relatively intelligent folks, and if they're not relatively intelligent to start with then they're not going to get much pleasure out of it and not going to stay for very long, or in any case not going to appreciate the points that are being made. Actually, scrap "intelligent" - that's too arrogant of me - and substitute "aware", as in "attuned to the norms of the environment".

[Laird]

I'll see if I can come up with a 12-ball solution along these lines, when I have the motivation to think on it...

I found some motivation this morning and it turned out to be pretty straightforward to find a Katy's-way solution - it took about 10 minutes to work this out and validate it:
ABCD vs EFGH
ABFL vs DGHI
AGKI vs FHJL

[note: this solution is flawed - it is for example impossible to know whether C is lighter or E is heavier if the first weighing tips right and the second two balance]

[Trevor Salyzyn]

Uni,

Feeding people in the third world is a total waste of time until the cultural factors that drive third-world hunger and poverty are addressed.

Unless you identify yourself as a humanitarian, feeding people in the third world is a total waste of time, period.

I think someone who doesn't believe in humanity, for whatever reason, might find humanitarian goals unappealing. A Buddhist, for instance, should see no reason to feed starving people.

A Taoist (or Confucian) might only view feeding starving people as valuable in that it brings power. So long as you aren't the ruler of the country, you should not worry about extending yourself beyond your station. The starvation of the Third World is a problem for those in power -- and if they fail to address it, they quickly will lose power. So, if you don't value power, there is another reason not to value consciously striving to end starvation.

Of course, hard-line humanitarians and power-mad nuts will always find something to love about helping the poor and downtrodden. Just as they'll always find a way to vent anger toward humans who don't agree with their perfect ethical system.

[Laird]

Oh, and hey, it's fairly trivial to again extend this - 15 balls this time:
ABCDM vs EFGHN
ABFLN vs DGHIO
AGKIN vs FHJLM

[note that this solution is flawed for the same reason as that of the previous post; and a 15-ball solution is seemingly impossible if the later analyses of this thread are correct]

I wonder what the maximum number of balls is? I can find one upper limit by reasoning like this: there are three weighings and on each weighing there are three possible results (tilts left, balances or tilts right). Three cubed is 27. As far as I can tell this suggests that we can at most differentiate the odd ball out from amongst 27 balls. But are there other limiting factors or is this the supreme limit? There seem to be several different solutions possible for 9 balls; probably also for 12 balls and for 15 balls - but at the limit of 27 balls would there simply be one single solution? Who can answer these questions?

Hmm, parlour games can bewitch... :-P

[Laird]

A Buddhist, for instance, should see no reason to feed starving people.

I don't know what version of Buddhism you're looking at but the version that I know preaches compassion, which includes relieving people of the pain of hunger.

[Trevor Salyzyn]

Relieving people of hunger is a lot closer to pity -- or generosity -- than compassion. To my knowledge, Buddha did not preach pity or generosity (in fact, in the Diamond Sutra, he advises against generosity). Compassion is a difficult virtue that requires genuine sympathy.

[Laird]

Who can answer these questions?

Ha! Turns out I can.

[oops, a little bit overconfident there...]

But are there other limiting factors or is this the supreme limit?

There are no other limiting factors. Twenty seven is the supreme limit.

[wrong: for example, one other limiting factor is that we need not only to pick the odd ball out, but whether it is heavier or lighter]

[A]t the limit of 27 balls would there simply be one single solution?

Yes. And here it is, using all of the letters of the alphabet plus the # symbol to represent the 27th ball:
STUVWXYZ# vs JKLMNOPQR
GHIPQRYZ# vs DEFMNOVWX
CFILORUX# vs BEHKNQTWZ

[note: this solution is very badly flawed for the required purpose but it should work if you already know whether the odd ball out is heavier or lighter]

Challenge: explain how I worked this out! (or pay me money and I'll explain it myself :-P)

[nevermind, the approach that I took was flawed]

[vicdan]

You wrote a constraint-satisfaction program to do it?

i tried to approach it from the information-theoretic side, but I assumed binary system from the get-go, and so I lowballed the highest possible ball count. The trit system would have been the right one.

[Laird]

Relieving people of hunger is a lot closer to pity -- or generosity -- than compassion.

I don't know how you arrive at this understanding. Hunger is suffering. Compassionate people empathise with those who are suffering and relieve it if they can.

[Laird]

You wrote a constraint-satisfaction program to do it?

Dude, I must be the fastest programmer on earth if that's what I did - it was seventeen minutes between my posing of the questions and my answering of them.

[vicdan]

more abstractly, you slice each 9-block into three 3-blocks, and recombine them, leaving one from each initial 9-block in place, and moving the other two from each into two different positions (scale side A, scale side B, off the scale). The same principle applies on the 3rd weighing.

[vicdan]

You wrote a constraint-satisfaction program to do it?

Dude, I must be the fastest programmer on earth if that's what I did - it was seventeen minutes between my posing of the questions and my answering of them.

yeah, good point. See my last post -- you figured out the principle. :)

[Laird]

more abstractly, you slice each 9-block into three 3-blocks, and recombine them, leaving one from each initial 9-block in place, and moving the other two from each into two different positions (scale side A, scale side B, off the scale). The same principle applies on the 3rd weighing.

I'll have to think about what you're saying here because I'm not sure that I understand it nor that I can judge whether it would work, but in the meantime I have to publish a retraction: my solution is flawed because it fails to differentiate between some situations where one ball could be heavier or another ball could be lighter (e.g. if the scales balance on the first two weighings and tip right on the final weighing it is impossible to know whether B is the odd, heavy, ball out or whether C is the odd, light, ball out). I now believe that there *are* other limiting factors so that a 27-ball solution is impossible, but I haven't quite figured out how to work out what the maximum number of balls actually is, nor how to adjust my technique to deal with it. Damn, and I thought that I was so smart and had exhausted all that there was to know about this problem! Back to the drawing board...

[Trevor Salyzyn]

Laird,

I don't know how you arrive at this understanding.

Since any spiritual development requires idleness, I assumed that Buddha's teachings were directed to the idle. Those who build habitats for humanity are not idle. They are all business.

The person who says "the most moral thing for a person to do is help the starving" himself is not the person who feeds the starving. He has a different task.

[vicdan]

more abstractly, you slice each 9-block into three 3-blocks, and recombine them, leaving one from each initial 9-block in place, and moving the other two from each into two different positions (scale side A, scale side B, off the scale). The same principle applies on the 3rd weighing.

I'll have to think about what you're saying here because I'm not sure that I understand it nor that I can judge whether it would work, but in the meantime I have to publish a retraction: my solution is flawed because it fails to differentiate between some situations where one ball could be heavier or another ball could be lighter (e.g. if the scales balance on the first two weighings and tip right on the final weighing it is impossible to know whether B is the odd, heavy, ball out or whether C is the odd, light, ball out). I now believe that there *are* other limiting factors so that a 27-ball solution is impossible, but I haven't quite figured out how to work out what the maximum number of balls actually is, nor how to adjust my technique to deal with it. Damn, and I thought that I was so smart and had exhausted all that there was to know about this problem! Back to the drawing board...

hehe. Could it be that my earlier retraction was premature?

My original figuring was that bi analyzing the amount of information extracted by each weighing, you end up being able to distinguish either one of 16, or one of 22, balls, depending on whether you round down the bit count. I thought I understood your solution, but after some figuring, i am not so sure. The pattern is clear, what's not clear is the highest number of balls it can be applied to. When taken in the abstract, this is indeed a very interesting problem.

[vicdan]

BTW, i think i figured it out... and I think it means your 12-ball solution doesn't work. here's the reason:

if the scales balance out, you have four balls left, and not knowing whether the odd ball is heavier or lighter, you simply cannot figure out which one it is in just two weighings. it's impossible -- because weighing 1 against 1 won't help you if the odd ball is among the other two, and weighing 2 against 2 won't give you enough info to cover all possibilities. Note that all balls known to have normal weight are just filler.

if AB > CD, then the best we can do next is something isomorphic to

AC ? XD

if B or C was it, we are golden. it it was A or D, we are stuck.

We can at best acquire one trit of information per weighing, one three-way choice. However, at the end of any sequence of such trifurcated choices, you will end up with two balls, not sure whether one is heavy or the other is light, and the last weighing will be to figure that out. So, the max number of balls we can discern per n weighings is 3^(n-1) -- i.e. in three steps we can discern 9 balls, in 4 steps 27 balls, etc.

But wait! We are wasting information on the last step! Each weighing is a three-way choice! We can always do one more ball!

Consider: when choosing among 9, if the scales unbalance on the first or second weighing, you end up with 'A is heavy or B is light' choice; but if the scales balance each time, then you are done, because the one ball which never entered the scale is it! But if you add one more ball, then in the case of all weighings balancing we can use the last qweighing, otherwise wasted, to figure out which one of the remaining two balls is the odd one out.

So, as far as i can tell, the most balls we can discern in n weighings is 3^(n-1) + 1. In three weighings, you can do 10 balls; in 4 weighings, 28 balls, etc.