On Friday I set this puzzle…

Imagine the Earth is perfectly circular and has a circumference of 24,000 miles. If you had a piece of rope stretched around it, touching the floor, that rope would also be 24,000 miles long. If you were to add 3 feet to that rope and then make it hover above the ground until it’s a perfect circle again, approximately how far off the ground would it get?

If you have not tried to solve it, have a go now. For everyone else the answer is after the break….

The rope would raise 6 inches off the ground if you added 3 feet to it! Can anyone explain why and what is surprising about this answer?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called **PUZZLED** and is available for the **Kindle **(UK here and USA here) and on the **iBookstore** (UK here in the USA here). You can try 101 of the puzzles for free here.

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I didn’t and don’t have a clue about how to do this one.

I have not now posted for over two weeks. Not since the name calling and other unpleasantness reached for me an unacceptable level.

Yet I see in the past two weeks more and more posts being unpleasant and rude towards me. It is now clear where the problem lies. It lies with the people who are using this blog to vent their splene rather than make civilised discussion.

As to the puzzle at hand. There is an easy visualisation. Imagine the new three foot length rolled into its own circle. It would have a radius of about six inches.

Now imagine the big circle being slightly elastic and streched so the new little circle can slip between it and the Earth. The little circle would be holding the big circle 6″ off the ground. And that is the answer needed.

Puzzles like this do not need complicated maths or similteneous equations. Simple visualisation and intuition will result in the right answer at an instinctive level.

Because even before we had language and maths and the need to write everything down we could dodge thrown rocks and hide from tigers and make fire. We have these innate abilities. Do not let others undermine you because they have “clever” learning methods.

I have yet to get one of these puzzles wrong yet all I do is quietly meditate for a few moments until my mind is attuned with the solution.

In para 4 you say ‘that is the answer needed’. But why is that the answer?

I missed you, Barry. Thought you wouldn’t be coming today. Good to see you. You always make me smile

Barry, I’m surprised you didn’t point out the variability of the possible answers to this puzzle given the obvious elasticity of any rope, and the fact that we weren’t told how tightly the rope was stretched, nor the effects of the various temperatures and pressures that the rope would endure as it made its way through the varying weather conditions. I think your analysis is slipping. 😉

Surely the thickness of the rope needs to be taken into account too? Would 1 cm thick rope be higher off the ground than say a 4 cm thick rope?

You must experience time differently from everyone else, Barry. Your previous comment was last Wednesday, 16 July at 5:01 PM which, by my calculations was less than 5 days before you made this one, not “over two weeks”.

You’re confusing radius and diameter again. If you let a circle with a 6 inch radius slip between the big circle and the Earth, it will lift the big circle 12 inches off the ground.

Your last sentence has already been shown by many people to be false.

@Barry

I don’t know how you can easily visualize that a 3 foot rope rolled into a ball would have a radius of 6 inches. I have to do some math to determine that radius. When I do use math I find the actual answer to be 5.7 inches. Maybe you can simply visualize that but not everyone can. Don’t disparage people because they can’t visualize what you can.

Then of course if you did slip this circle of 6 inch radius between the earth and the rope, it would rise by twice the radius. So it appears you are wrong and your meditation and visualization didn’t quite help you out here.

I’ll have to resort to my math, which wasn’t all that complex and I didn’t need simultaneous equations either.

I think you’ll find it’s *similtaneous* equations (according to Bazzer anyway).

I see what Barry means : if you consider the extended rope forming a circle, but touching the Earth in one point, then at the other end of the diametre, the rope is above the Earth, with a distance equal to the difference of the two diametres (D=d(rope)-d(Earth)).

This difference of diametres is twice the distance that the rope would hover above Earth if it hovered uniformly, and, since the diametre of a circle is proportional to its circumference, it is also the diametre of the circle formed by the additional length of rope. So if the additional circle has a 12″ diametre, then the rope would hover 6″ above the Earth.

(Sorry if my explanation is a little confuse, I’m not a native english speaker. And I don’t even know if I could explain this clearly in my own language ^^)

That’s “spleen”, Barry.

You may be an instinctive problem solving genius, but your spelling is terrible.

Its “you’re” not “your”

Phil, if *you’re* referring to the comment immediately above *yours* then I’m afraid *you’re* mistaken.

and Phil – your “its” should be “it’s”. (“its” is possessive, “it’s” is short for it is)

But perhaps you meant that and realised only after clicking ‘Post Comment’.

Lack of an edit option (unless you’re Richard and change the question 😉 ) can lead to unwanted embarrassment but does ensure all our ramblings remain unchanged for posterity – which perhaps has some benefits

@Sydious

Of the many who come to sneer one understands. If only the “mathematically literate” who are stuck in their ways could be as open to intuitive insight.

Even those who point out they can solve this puzzle by similteneous equations (one for each rope) have much to learn from a more holistic approach. I am happy to teach those who wish to learn.

The one who sneers yet again fails to understand. If only this mathematically illiterate person who is stuck in his ways could be open to introspection.

No-one has claimed to solve this puzzle by similteneous (or simultaneous) equations, that is another fabrication on your part. I am happy to teach you but you have made it clear that you have no wish to learn.

… and the bait is taken.

So Barry’s just taking the piss is he? Thought so.

Troll / not troll, he’s still being a dick. There’s no justification for him to behave like that. Who does he think he is?

@Self Appointed Guardian

Thank you.

It is humbling to be called an instinctive problem solving genius.

I believe my talents are humble and available to all.

Yet one way to measure a human mind is to measure the contributions it makes to society. If my humble contributions be recognised as genius that is all I need to feel welcomed and humbled.

Some weeks ago @Steve called me a godlike being. That too was high praise. Perhaps too high. Yet it was his authentic response and I welcome that.

Yet any one of us can learn my skills. Just follow what I do and do the same. It is my pleasure to help you all.

You’re brilliant, Barry, a genuinely talented performance artist.

You’re better than I ever was.

All hail to the demi-god that is Barry Goddard

The circumference of a circle (i.e. the length of the rope) is related to the radius of that circle (i.e. the distance from the *centre* of the Earth) via the formula C=πd. So, if you increase C by 3 feet, πd also increases by 3 feet. Divide by 2π to convert that to the radius and (because π is about 3) you’re left with about half a foot.

It’s surprising because intuitively one wouldn’t expect the tiny addition of just 3 feet to an overall length of 24,000 miles to result in any noticeable increase in how far off the ground the rope is. The truth, though, is that the original circumference and radius are both irrelevant. If it were 1 inch to start with instead of 24,000 miles the radius would still increase by about 6 inches, just as it would if the initial length of the rope were 1,000 light years.

Very clever. Thank you for the explanation, Mark.

Thanks for the explanation, quite fascinating when you think about it

Thanks Mark. That’s a good explanation, as are Simon’s and Ken’s below.

Circumference of a circle is 2 x [Pi] x r

adding 3 ft to the circumference of any circle always adds 3ft / (2xPi) = 5.7inches to the radius.

Can anyone explain why and what is surprising about this answer?The ratio of the circumference of a circle (the length of the rope) to the radius (the distance that the rope is from the Earth’s centre) is 2 × (pi) (or

exactly2 × 22/7 for DAve). This is near enough to 6 for an approximation.Therefore, if you increase the length of the rope (circumference) by 3 feet, its distance from the Earth’s centre (radius) will increase by approx 1/6 of that, so about 6 inches.

I didn’t find anything surprising about the answer but I imagine that some people would assume that adding 3 feet to 24,000 miles would not make a noticeable difference. Measured relative to its distance from the centre of the Earth, it didn’t make much difference, but when viewed by us tiny things way out on the edge, it does seem like quite a lot.

Beaten. Thought mine was going to be the first.

careful with the word “exactly”. 22/7 is just an approximation. http://en.wikipedia.org/wiki/Pi#Continued_fractions

I’ll try to remember to add a 😉 next time I make a joke.

Thank you Simon that’s a very clear explanation.

Aimée

What Mark said

Why does Richard think it’s a surprising result? There’s a linear relationship between the circumference and diameter of a circle. The results exactly what you’d expect.

Only if you’re a giant smuggy smartypants.

Is that your baby-talk term for somebody numerate?

I ended up with 5.99 inches. But I was pretty sloppy with my sig figs throughout on account of doing it on the back of a Taco Bell receipt.

C=2πr

C’=2πr’

C’=2πr+a

2πr’=2πr+a

r’-r=a/2π

Metric system, please?

Ok, so you add 10 feet to the rope. It will then be ~10/6 ft = 1 ft 8 in above the ground. 😉

Imagine the Earth is perfectly circular and has a circumference of 40000 km. If you had a piece of rope stretched around it, touching the floor, that rope would also be 40000 km long. If you were to add 1 meter to that rope and then make it hover above the ground until it’s a perfect circle again, approximately how far off the ground would it get?

The rope would raise 16 cm off the ground if you added 1m to it!

Given the variables:

Initial Circumference (IC), Final Circumference (FC), Added Circumference (AC)

Initial Radius (IR), Final Radius (FR), Added Radius (AR)

We are looking for AR (Added Radius).

IC + AC = FC

FC – IC = AC

IC / pi / 2 = IR

FC / pi / 2 = FR

AC / pi / 2 = AR

AC = 1

AR = 1 / pi / 2 = 0.15915494309189533576888376337251

I’m thick at maths so I don’t understand any of this – but it does surprise me that such a comparatively tiny increase would add 6 inches to the distance from the surface. On a very obvious and immediately intuitive level this seems very weird! And now, Richard, please let’s have a few non-maths puzzles for those of us whose brains don’t really work well with this sort of stuff ! Had a lot of maths ones lately – let’s have some nice interesting logic puzzles! 🙂

The counter-intuitiveness is because you’re not actually comparing like with like. You’re adding 3 feet to 24,000 miles and the effect is ½ foot added to about 4,000 miles (distance to Earth’s centre).

My main question is what fraction of respondents hadn’t seen this puzzle before. I suspect it is mentioned in most high school physics classes or wherever they teach c=2 pi r.

I got an elevation of 0,477464829 feet, based on the assumption that a mile is 5280 feet and that Pi is 3,14159265359. But nobody really knows what a mile is, right? 😦

50 miles is what my car tells me I drive every day.

Weekends too? I want a talking car.

You’re problem is the assumed value of Pi. The unit of a mile is a complete irrelevance to the question as the circumference of the Earth is wholly immaterial to the answer which would have been the same whether the object in question have been Jupiter or a dinner plate.

A reasonable case might be made for needing to know how long a foot is in terms of inches. However, even that isn’t strictly necessary as the result could be expressed in factions of a foot.

@Jones. It’s “Your” and not “You’re”. It’s “fractions” and not “factions”. Apart from that, your four sentences of aloof self-importance are more or less presentable.

And “had been Jupiter” instead of “have been Jupiter”.

Maths: 11/10

Speling: 7/10

Sure, but I wasn’t sure if a mile in the UK is 5280 feet, as the US-driven Google told (metaphorically, Tim!) me when I looked it up.

@Christian Geiselmann – I see where you are coming from now. Apologies. Too busy chatting with my car this morning.

I’ve changed my mind, I want a flying car. It’s 2014! – where is my jetcar?

And my rocket pants?

A mile isn’t a mile isn’t a mile. For examples, a nautical mile isn’t 5280 feet, but 6000 feet. When a place kicker misses a field goal “by a mile” how far from the nearest upright was it? How about a country mile? North Michigan Avenue in Chicago is often called the Miracle Mile. Is it a mile long? Yes, Barry is rubbing off on me but he’s the most interesting person here so I don’t mind catching a bit of that.

You need to watch the episode of horrible histories where they talk about the London mile and the Oxford mile etc. There used to be so many more miles… I prefer kilometres myself. The name tells you exactly how may metres it has in it.

Indeed an advantage of kilometres. Funnily, a “mile” (from mille passuum) should mean something similar (1000 paces). But obviously, people who then made up their various “miles” were not that enthusiastic about the strict concept of 1000 = 1000…

I know. Unfortunately Richard’s site does not allow post-editing when I spotted the typo. Also, I now see I hit the wrong reply box and the poster was using the convention of a comma for the decimal point.

In any case, his post was sarcastic. It deserves a sarcastic response.

Your reply did not read sarcastic; it read self-important. Big difference.

“I now see I hit the wrong reply box”

Twice! 🙂

I read it as being (mainly) sarcastic about our culture being wedded to an outdated system of measurement, but never mind. Also you present a false dichotomy; sarcasm is not mutually exclusive with being self-important. Indeed they can be bedfellows. Quite cozy ones.

On the problems, its always interesting to see some of the lateral approaches, but those posts based purely on making points about irrelevances are surely fair game, As, I should add, are those who think it almost admirable to understand what is surely among the simplest of geometric and algebraic principles by throwing around classroom insults.

Nb. I do hate getting punctuation and grammar wrong, not to mention the typos. It comes from a troubled childhood, dreadful physical coordination and being trained to spell phonetically rather than by rote. Unfortunately English is written phonetically. Such mistakes cause me almost physical pain.

Hello Steve,

It is a little bit off the main topic here, but I would be interested in learning what you mean by “Unfortunately English is written phonetically”? In my understanding, “phonetically written” would mean that one phoneme is represented by one (and ideally only one and the same) grapheme. We find this in languages such as Serbian (etc.) where the written language was constructed, relatively late in history, purposefully in order to match the phonetic repertoire of the spoken language. In English, I would say, things are different, due to the very long development of both written and spoken language in parallel… But perhaps I have mistaken your intention? (This is a serious question, not sarcasm or so… just ot prevent misunderstandings…)

@Jones. I take it your post of 7.53 pm was intended for me. I am not even sure where to begin with this latest dollop of nonsense. I have presented no false dichotomy (although fair play to you for actually spelling that word correctly in the alphabet-soup mess of the rest of your post). But to call English a phonetic language is untrue, as thirty seconds of the lightest of light research would have confirmed to you before making this ludicrous claim.

Is it just me or should it be spherical and not circular?

It only needs to be circular around the diameter where the rope is laid so it could be an oblate spheroid or even a torus. Surprised no-one yet has questioned whether “floor” = “ground”.

But if it’s circular and you make the rope hover above it, the circle formed by the rope does not change in size. If it’s spherical (or the rope moves away from the centre of the circle), then you avoid that problem.

A sphere is a ball. A rope around the earth, would be a ring, or circle.

Yes, it’s you.

It’s a lot easier to calculate if you imagine the rope is being extended by approximately PI feet (or metres or whatever). Adding PI to the circumference gives an increase of PI in the diameter, or half-PI in the radius.

The reason it’s surprising is that:

1. It applies to any circle, whether one the size of the earth, or one the size of a coke can.

2. You feel like, in this case, a 6 inch gap all the way around the earth is a lot of space to get from just 3 feet, and that’s down to the fact that our perception of how big the circle is, is related to its area, which grows based on the square of the total of the radius, not just the amount you add on.

The surprising thing about the answer is that an insignificant change in the circumference results in an apparently very significant change in the height of the rope. Six inches is quite visible; one might expect it to be microscopic. But that’s a myopic perception. Considering that the original “height” of the earth’s surface is about 4,000 miles from its center, six inches is just as insignificant as the 3-foot change in the 24,000 mile circumference.

Bottom line: because the ratio of circumference to radius is fixed (roughly, 6:1), adding 3 feet to the circumference always adds about 6 inches to the radius — no matter how huge or tiny the circle is.

This was the most convincing contribution here, from my perspective. Thank you, Ken.

The answer is actually pretty simple

Here is Math explanation

R – radius of the Earth

C – circumference

pi – is the magical pi number 3.14

2 * pi * R = C

With the new 3 feet, it will look like this:

2 * pi * (R+x) = C + 3

2 * pi * R + 2 * pi * x = C + 3

C + 2 * pi * x = C + 3

x = 3/(2*pi) = 0.4777 feet or 5.73 inches. Roughly 6 inches

The exact measurement is 18/pi inches.

3 feet/2*3.14

note that parens are required around the 2 pi, because the expression is resolved innermost expression(s) first then left-to-right.

Actually the answer is every value from 0 to 12. Because there is nothing in the question that says it must be exactly centered. It can be touching at one point and maximally distance at the other side.

And anyway . . . is a fraction of a foot a toe????

What is surprising about the answer (6 inches) is that it depends on pi being 3 rather than 3.1415926etc

The question asks “…approximately how far off the ground would it get?”. Calculating the exact value using pi is a fairly straightforward maths problem but I wouldn’t call it a puzzle. Using 3 as an approximation of pi and working out the answer mentally is the trick that Richard expects us to work out IMHO. Given that interpretation the “official” answer should be “about 6 inches” not “6 inches”.

@Tim: Der, I missed the “approximately”, so when I got 3 / (2 pi) I thought I missed something.

Ratio C:d = pi:1 ≈ 3:1

Increasing C by 3 units (furlongs anyone?) extends d by 1 or r by 0.5.

Is where I was coming from…..

Surely the measurement system has nothing to do with the problem. (A “problem” by the way I thought everyone had encountered at school.) But can it be a coincidence that the best explanation is by GeoDetective who uses the metric system?

I got this wrong because I took the extra 3 feet to be added to the diameter, not the radius. I am still not sure why I am wrong.

The rope is around the circumference, not the diameter – or radius, for that matter.

Right, but I think of the formula for circumference to be pi*diameter. The right answer is 3 / 2 / Pi. I do not understand why it isn’t 3 / Pi.

That has to do with radius vs. diameter. The extra diameter is 3/pi. And that extra diameter is the total of that on both sides of the moon. Devide by 2, to get one side.

You need the radius, not the diameter.

Holy cow! I am humbled (and humiliated)! This entire bunch of explanations, replies, and retorts astounds me. You are all so much fun to read. I’m coming here again. Thanks for the fun from beginning to end.

In the interest of accuracy, 22/7 as a proxy for Pi, is IMHO totally unacceptable. I suggest that we work with the following approximation (off the top of my head)

pi=3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196 44288 10975 66593 34461 28475 64823 37867 83165 27120 19091 45648 56692 34603 48610 45432 66482 13393 60726 02491 41273 72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 78925 90360 01133 05305 48820 46652 13841 46951 94151 16094 33057 27036 57595 91953 09218 61173 81932 61179 31051 18548 07446 23799 62749 56735 18857 52724 89122 79381 83011 94912 98336 73362 44065 66430 86021 39494 63952 24737 19070 21798 60943 70277 05392 17176 29317 67523 84674 81846 76694 05132 00056 81271 45263 56082 77857 71342 75778 96091 73637 17872 14684 40901 22495 34301 46549 58537 10507 92279 68925 89235 42019 95611 21290 21960 86403 44181 59813 62977 47713 09960 51870 72113 49999 99837 29780 49951 05973 17328 16096 31859 50244 59455 34690 83026 42522 30825 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69998 92256 95968 81592 05600 10165 52563 7567

You misplaced one of the 7’s.

That should be close enough for practical purposes 🙂

It is never late to contribute some exactitude to a scholarly discussion.

PS. I think this prooves that there is no God: If God existed, would he or she have created circles that have such a ridicolous ratio of diameter and circumference? Hard to imagine…

Au contraire: What if God has left us a secret message in the decimal expansion? Suppose, say, after trillions of trillions of digits in the expansion, there’s a sequence of all 1’s and 0’s (and no other digit) that results in a message spelled out when the digits are arranged in a 1000×1000 grid? That would be irrefutable evidence that God exists.

That’s fantasy; but seriously, I think it would be impossible for God to have pi be any other value that what it is. It would be just as impossible as making 2 + 2 equal anything besides 4. It’s the very nature of the numbers–not something God did or did not do.

If!

Ken, not only does every possible “message” gets spelled out in a sequence of 0s and 1s when arranged in a 1000×1000 grid, every such message happens an infinite number of times.

that’s because pi is irrational.

As for the question if God Almighty could or could not create an universe where pi is exactly 3 (or anything else more tidy than the odd pi of our universe), I would say: shouldn’t be impossible even if you think in terms of physics: multidimensional space, curved dimensions, etc. gadgeds of string theory and the like. But if there is a God, he, she or it should be almigthy, and if h/s/i is almighty, what would hinder h/s/i to create such an universe?

Of course, again: if !.

Uh… here is a solution to the God-and-his-odd-Pi problem:

God’s first universe of course had Pi = 3. But then came the fall of man, and ever since we are living in a basically disarranged universe, with even a Pi that never ends. So, if you like, you may take the deplorably lenghty Pi as a proove that God exists… 🙂

Last contribution to this topic for today: Ken, I am afraid, even if you find in the long, long array of numbers that are behind the comma in Pi a section with 1 million consecutive 1s and 0s which form, arranged in a square 1000 x 1000, a message that you can read in your preferred language (by the way: which character encoding would be used?), this would prove nothing: under the assumption that Pi does never end, there MUST be, anywhere, far, far away in the queue of Pi a section where such a consecutive array of 0s and 1s occurs. Given the infinite length of the string…

I even feel a little bid sad about the loss of a very nice proof for God’s existence… (not that I bothered, but I liked the argument.)

quote […there MUST be, anywhere, far, far away in the queue of Pi a section where such a consecutive array of 0s and 1s occurs. Given the infinite length of the string…]

what is surprising is that, if the expansion of pi really is infinite, such a consecutive array of 0s and 1s will occur an infinite number of times !

Um… or could God possibly have outruled such occurences? Then, THIS would be another proof of God’s existence 🙂

I think pi’s overrated.

Discuss.

“I am the Anti-Pi

I am an anarchist

Don’t know what I want

But I know how to get it

I wanna destroy passerby”

Which god are you referring to? There are so many.

Well, maybe not the last..:)

It’s not true that such any sequence exists, just because the decimal expansion is infinite, even with the constraint that the expansion be non-repeating. Consider the following number with a non-repeating, infinite decimal expansion thus:

.202002000200002000002…

where the number of zeros increases by one each time. Nowhere will you find ANY sequence of 1’s and 0’s, let alone the one we’re imagining.

It’s a common misconception that an infinite set of things must contain all possible things that COULD belong to the set. The set of real numbers between 0 and 1 is infinite, but that set does not contain the real number, 2. Similarly, just because the decimal expansion of pi is infinite, it doesn’t PROVE that any given finite sequence must exist therein.

Now, however, having said that, I admit that it is highly likely that any finite sequence WILL exist somewhere in pi’s decimal expansion, assuming there’s no unknown bias in the nature of pi that would prevent it.

Good argument, well presented.

Ooh, I missed this. Yes, Ken is definitely correct when he says that the decimal (or any other base) expansion of pi doesn’t necessarily contain every possible sequence of digits. It’s an unanswered question although the generally accepted answer is “probably”.

It’s actually a mistake to refer to pi’s digits as “random” since pi has a fixed value and is not truly random. The sequence of digits are unpredictable but they are not random, although they do have the appearance of randomness (as far as they have been checked). However, we don’t even know if the digit ‘1’ appears exactly one tenth of the time in the infinite series or whether it appears an infinite number of times, both of which would be the case if the sequence were truly random.

With regard to an embedded secret message being proof of an intelligent creator, if the digits of pi do truly contain every possible sequence of digits, then they will contain an infinite number of messages, messages that contradict each other, messages that are self-contradictory, messages that are verifiably false, and those same messages with typos, spelling mistakes, poor grammar… hang on a minute – I think the World Wide Web proves the existence of an intelligent creator!

Well… then… couldn’t it be that the intelligent creator had been come into existence at random?

Messages that contradict each other or are false, with typos, spelling mistakes, and poor grammar? Sounds more like Goddard than God.

This has been a fascinating discussion. Thanks for posting.

Permit me to say a few words here ………

22/7 differs from π by about 0.04%.

355/113 differs from π by only about 0.000008%.

Good puzzle this – maddening but great when you solve it.

Barry:

Your intuitive answer works for a single additional loop. It also works for two loops half as long positioned 180 degrees apart. But it doesn’t work for three loops one third the size 120 degrees apart or any higher number (though it does work for n loops all in a line). That would suggest your “intuitive” answer is a coincidence, rather than a provable, generalisable, mathematical answer.

july 25 puzzle: “10 bets you will almost win”? doesn’t that mean you will lose?

Typo in my last post: “…such any sequence…” should read “…any such sequence…”

Is it me…… or are none of you bothered that a rope would be hovering above the earths surface with no visable means of support? ……… would health and safety have to put a ‘caution trip hazard’ sign every ten feet along its length? and how many signs would it take to stop someone tripping over and putting a claim in?

There is no number of signs that would stop every idiot from tripping over the rope.

I have a problem with measuring the length of that rope (or its elevation above ground, respectively). Supposed, as kosmologists tell us, that the universe expands continuously, and with it all things in it, obviously Earth’s diameter (or circumference, respectively) constantly grows; so, how can we then possibly define the length of such a rope at all?

The expansion of the universe (actually “space”) doesn’t mean that all things in it also expand. That is how we actually know that space is expanding – because the distances between things (generally galaxies) is greater now than it was in the past. If absolutely everything expanded, the distances would appear to be the same

To clarify – if all things expanded, this would include all of our rulers, measuring tapes, micrometers, etc, and our sense of how long anything was, so that it would be impossible to detect any change.

So the suggestion is: space expands, but things in it do not?

But wouldn’t this mean that space and things are separate?

I rather would believe space is everywhere, also “in” things (so to say), and if space expands, things should be part of that process. No?

How would quantum mechanics still work if everything was expanding? What about Planck’s constant and the speed of light?

Metric system, anyone?

Well… perhaps my supposition was wrong: Does the universe expand, but things in it remain the same size?

it seems that this answer is correct if Pi is considered to equal 3 (rounded down)