Is this possible?

[Katy]

i ran across this on a website earlier this morning and I can't solve it, so I'm hoping someone else will be able to

If you have six dots

X X X
O O O

Is it possible to connect each O to each X (9 lines) without any of the lines crossing one another?

[Dan Rowden]

Do the lines have to be on a flat 2 dimensional plane or can they be curved in 3 dimensional space?

[Laird]

Hi Katy,

Interesting problem - I can see a solution if I'm allowed to make two assumptions - firstly that each line of each cross is the same length as the diameter of each circle and secondly that the angle at which the cross lines intersect is set to 60 degrees (and 120 degrees for the larger, complementary angle).

The solution then goes like this: arrange the circles into a triangle, with each circle touching each other circle once. Place one cross inside each circle and orient each cross so that each line of the cross touches the same point that the circles meet at. If you have trouble visualising this then let me know and I'll draw a picture and post it.

Laird

[Katy]

lol i looked it up a couple days after posting. It appears Dan got the right answer. Laird's doesn't work because they have to be in two rows.

[Laird]

I have a problem for you in return. It's one of my favourites. If you haven't solved this one before, and if you manage to solve it without assistance within 48 hours, then you will have my intellectual respect (whether you particularly care about that is another matter...):

You are given 9 balls and a pair of balancing scales. All of the balls are of equal weight except for one of them, but you are not told which one this is, nor whether it is heavier or lighter than the rest. Your task is to discover which ball is a different weight and whether it is heavier or lighter. You may use the scales three times to weigh any combination of balls against any other combination. You may of course base your choice of balls for a weighing on the results of a previous weighing. The solution is a description of the process of weighing that you would go about to find the odd ball and whether it's heavier or lighter than the rest.

[Dan Rowden]

Can't you just judge it by hand like most guys ordinarily would?

[Laird]

Dan, if you want to handle your balls, then be my guest, just don't expect me to weigh them.

[Laird]

Katy's 9-ball solution and my own different solution being lost in The Great Crash of several months ago, and her subsequent challenge to discover a 12-ball solution likewise lost, this post might seem a little out of context. Suffice it to say that such lost posts did in fact occur. I put aside Katy's 12-ball extension problem when I failed to solve it immediately and I only returned to it recently, whereupon a solution struck me whilst I was in the bath a couple of days ago. It is a fairly simple extension of my 9-ball solution. No doubt Katy has a similar extension to her own, different, 9-ball solution, which I would love to see published here.

Labelling the 12 balls from A to L:

Step 1 (first weighing). ABCD vs EFGH
Result 1a. The scales balance => goto 2.
Result 1b. The scales are unbalanced => goto 4.

Step 2 (second weighing; from 1a). IJ vs KA
Result 2a. The scales balance => L is the odd ball out - weigh it against A to work out whether it is heavier or lighter.
Result 2b. The scales are unbalanced => goto 3.

Step 3 (third weighing; from 2b). JK vs AB
Result 3a. The scales balance => I is the culprit and we know whether heavier or lighter from the direction of the scales in step 2.
Result 3b. The scales reverse direction from step 2 => K is the culprit.
Result 3c. The scales remain unbalanced in the same direction as step 2 => J is the culprit.

Step 4 (second weighing; from 1b). AIJK vs EBCD
Result 4a. The scales balance => the culprit is one of FGH. In the third and final weighing, weigh F vs G to determine which of the three it is (we already know whether it is heavier or lighter due to the direction of the scales in step 1; if F vs G is balanced then the culprit is H).
Result 4b. The scales reverse direction from step 1 => the culprit is one of BCD. In the third and final weighing, weigh B vs C to determine which of the three it is (we already know whether it is heavier or lighter due to the direction of the scales; if B vs C is balanced then the culprit is D).
Result 4c. The scales remain unbalanced in the same direction as step 1 => the culprit is one of A or E. Weigh A vs C to determine which of the two it is (if A vs C is balanced then we know that it is E and whether E is heavier or lighter due to the direction of the scales in step 4).

Now Katy, let's have your elegant three-line solution. :-P

[Unidian]

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?

[Unidian]

I guess it probably sounds like I'm saying it's "beneath me" or whatever. That's not the case, though. Rather, I think it's beneath anyone. The size of one's IQ is not the relevant factor here - priorities are. Shouldn't we use whatever mental energy and skill we might have in thinking about how to make a world where people don't get screwed over so often? I find that I just don't have any time to figure out where the train that left Chicago will meet the one leaving New York. Motivation and skill are limited, and there's so much to do.

[|read|]

Without looking at your solution, here's mine after an hour. That was fun.

Weigh 3 balls against 3 balls and leave 3 out (measurement 1). If the scale balances, the odd ball is among the 3 you left out. Weigh 1 of those against another 1 and leave 1 out (measurement 2). If the scale balances, the odd ball is the 1 you left out. Weigh it against any other ball to see if it's lighter or heavier (measurement 3). If measurement 2 doesn't balance, you know the odd ball is 1 of the 2 you weighed. Weigh the 1 that went down against a different ball than you did in measurement 2 - if it goes down, it's the heavy ball, and if the scale balances, the 1 that when up in measurement 2 is the light ball (measurement 3). If measurement 1 doesn't balance, the odd ball is among the 6 you weighed. Now comes the neat bit. Say balls A, B, and C went down, balls D, E, and F went up, and balls G, H, and I sat out. You know either one of ABC is heavy or one of DEF is light, and GHI are all normal. So weigh ADG against BEH (measurement 2). If it balances, either C is heavy or F is light. Weigh C against any ball except F (measurement 3). If C goes down, it’s heavy, and if not, F is light. If measurement 2 doesn’t balance: If ADG goes down, either A is heavy or E is light. Weigh A against any ball except E (measurement 3). If A goes down, it’s heavy, and if not, E is light. If BEH goes down, either B is heavy or D is light. Weigh B against any ball except D (measurement 3). If B goes down, it’s heavy, and if not, D is light.

[|read|]

As far as whether this particular problem is important, it's not, it's a parlor trick. But doing it indicates 2 things. First, it indicates you have some logical problem solving ability. This is necessary to solve a host of more useful logical problems. Second and perhaps more importantly, simply attempting the problem shows you want to solve logical problems and have the desire to learn how. The second can lead to the first. And of course, some people will undermine the importance of solving this problem, the importance of logical problem solving in general, and even the desire to learn how, because they lack these things and they don't want to feel they lack anything important.

To accomplish any goal, the ability to do so is just as essential as the motivation. How to make the world a fairer place is complex logical problem indeed. It's easy to have good intentions, but you also have to figure out how to implement them effectively if you want to make a difference. All metrics of intelligence are incomplete and flawed, but many also give some indication of useful abilities. Better to rise to the challenge, albeit with a grain of salt, than spend your mental energy figuring out why you don't have to live up to metrics of intelligence. Given the long periods most of us spend in relative leisure, introspection, and other non-world-changing activities, I know my time priorities could benefit from a few hours to sharpen my mind.

[hsandman]

Code: Select all

 .(X)(X)
   (X)

The only way I see is 3 stright lines? Could add 6 more, but they are redundant, but that would fit the 9 lines specified.

[Katy]

They serve the same purpose as sudoku puzzles and crosswords. Things to do to pass time that aren't "watch TV" or "sit on internet" There's nothing wrong with them anymore than there's something wrong with watching a movie.

It's called amusement.

[Imadrongo]

Dots in 3-space? O's and X's in 3-space?

If Dan really got the right answer it is pretty dumb.

[vicdan]

|read|, dude, long time no see! Hi.

Hahaha at Uni's post! Yeah, there are so many so much more important things to be found by digging in one's navel lint!

I have solved the initial post on the spot, in about 30 seconds, complete with proof of impossibility and all -- and I have a witness to prove it. :D

[|read|]

Victor D! Good to see you. Thought you had dropped off the face of the internet. How the hell are you?

Try Laird's ball problem, it's way more fun.

[Unidian]

And of course, some people will undermine the importance of solving this problem, the importance of logical problem solving in general, and even the desire to learn how, because they lack these things and they don't want to feel they lack anything important.

Yeah, I knew this would be suggested. Believe what you want - although I think anyone of reasonable intelligence can solve these things with some effort. I used to solve them all the time, starting in the "gifted" class in elementary school, in which such time-wasting activities were pursued frequently. I guess there was some excuse for it then, though, as training. But adults doing this sort of thing? It smacks of "IQ society" meetings, which are intolerable ego sessions. And yeah, I was invited and declined, so it's not sour grapes.

And the importance of solving logical problems is huge, of course. But these problems? I don't think so. I guess it could be a pastime for people who have already solved all the logical problems that actually matter, such as rooting out all the contradictions and fallacies in popular religious and social beliefs, etc. If there's nothing else to do, I suppose it couldn't hurt to do this sort of thing. But how many people are actually in that situation?

[vicdan]

Victor D! Good to see you. Thought you had dropped off the face of the internet. How the hell are you?

I generally post at YAPB these days. At the moment, I also stand in utter (but not inactive) amazement of the intellectual cesspittidude that is this forum. Your arrival here is good news.

You?

Try Laird's ball problem, it's way more fun.

I already knew the solution from before. it's an excellent challenge though. it's also a useful tool to understand the nature of information better.

[Philosophaster]

And of course, some people will undermine the importance of solving this problem, the importance of logical problem solving in general, and even the desire to learn how, because they lack these things and they don't want to feel they lack anything important.

I'm generally not interested in logical puzzles like that, although lately I've taken more of a liking to them for whatever reason.

Hard science and math people tend to have higher IQs than everybody else. Their work deals in precision, leaves little room for excuses, and their academic training is tougher as a general rule. So I respect them for their intelligence and hard work, and sometimes I even get a little insecure about my "intellectual status" (or whatever you want to call it) with regard to them, because I don't really know a lot about science; I'm mostly a humanities sort of guy:

People who make their livings in “soft” sciences and the arts are not entirely at ease in the company of chemists and physicists and other “hard” scientists. In such company, the psychologists and sociologists and the professors of English feel like touch-football enthusiasts who have wandered by mistake into the locker room of the Pittsburgh Steelers.
- Richard Mitchell, The Graves of Academe

:-P

Of course, a few humanities domains do have pretty low "BSability" (as we used to call it on the debate team) -- philology for example, and certain areas within philosophy.

[Unidian]

"BS" isn't a very good standard, anyway. Someone might read Nagarjuna (for example) and think it is rubbish, while for another, it might be gold. There's always obvious BS like "pyramid power," etc - but once factual nonsense and fantasy are ruled out, it's much less clear how we can establish a consistent "BS standard."

Science is great, but I wouldn't let them intimidate you. Too many of them are after that - and it's really quite baseless. The humanities have just as much importance in the whole experience, and the ones who insist otherwise are biased egotists, just as the ones who say science is worthless are.

[Laird]

I wish I could avoid being "the ass" here, but why solving such a problem would be important escapes me.

Yeah, bit of a downer Nat but point taken. If I were to be true to the way that I believe that I *should* be living my life I'd probably be working for an aid organisation in some third world country, or otherwise attempting to redress some of the poverty/technology/justice imbalance in the world. Having said that, there's a time and a place for recreation too. Also, the ability to solve problems like this is one of the more significant and noticeable ways in which humans are more skillful than other animals which explains my respect for those who are good at it. The ability to solve problems generally is what has led humans into the relatively abundant life that we (at least in western countries) have nowadays, and "parlour tricks" like this can IMO be considered to be a training tool for general problem-solving ability that might have some actual relevance for alleviating suffering or promoting comfort in the "real" world. But Victor's point is spot-on - I don't class the posing of this problem on this forum in much different light than any of the other threads - for the most part I view participation in this forum as stimulating, challenging and pleasurable but not as something that has any more real-world relevance than any other type of learning such as that involved in problems of this type.

Without looking at your solution, here's mine after an hour.

Dude, that's some serious horsepower you've got going there. From memory it took me a day or two of non-continuous (I was working full time) thought and pottering to come to a solution the first time that I solved it. Respect. Oh, and welcome to GF. I enjoyed reading you on FP.

That was fun.

Now for even more fun, try Katy's extension: same problem, but with 12 balls this time (that's what my solution above is actually for, so don't peek yet!). By the way, check out Katy's 9-ball solution - it's bloody elegant mate! From memory (the post where she described it in own words was lost in the crash) it was something like this - correct me if I'm wrong, K:
ABC vs DEF
ABC vs GHI
ADG vs BFH

Succinct, huh? She tells me that her 12-ball solution was an extension of this one, but that she's forgotten it. 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... anyway, that's an additional challenge for anyone who's interested.

[Unidian]

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?

I suspect not, in most cases.

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. 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.

Some here realize full well how dismissive they are being when they refer to such work as unimportant or non-existent. They don't care because they have no respect for it. But I think others are genuinely unaware of what they are implying when they act like the only thing one can do on internet forums is waste one's time or play around for entertainment.

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.

[Imadrongo]

Anyone who can't solve the ball problem in 48 hours belongs in a mental ward.

[Unidian]

Why is that, Neil? Need some company?