A lovely simple but surprising mathematical problem this week (via Timothy M) …
Imagine you have a piece of string long enough to stretch around the earth (40,074 km or 4,007,400,000 cm). Then you take an extra meter of string and add it to the string around the Earth. Now you spread this extra string around the Earth, supporting it somehow, so that the string forms a circle off the ground.
How high off the ground would the string be?
As ever, please feel free to say if you think you have solved it, and how long it took, but please do not post your answer. Solution on Monday. Oh, and for an extra 10 points, what is the surprising aspect about the answer?
I’ve used this example lots of times over the years as a counter-example to using an “instinctive guess”
I think we did this in maths in 6th form. Which was some time ago for me…
Yup, think I got that one…
Intuition tells me the height increase would be negligible; probably much less than the thickness of the string.
Well known puzzle involving very simple maths, but when applied to the real world at this scale has a result that is still hard to actually comprehend.
yes i know this one – a stunning answer. Although telling the problem the other way around is even more stunning, I think (e.g. how much extra string is required if you suspend the string 1m above the ground…)
And the surprising thing? well if I said then it would spoil the surprise… shall I say it starts with ‘L’…?
Lemon squeezy.
Umm, that seemed straight-forward. Maybe I”m missing something ?
That’s what I’ve asked me as well :-/
I think I got it. Embarrassingly long time to remember which question I needed to ask myself, but when I finally got that it was only a couple of minutes.
I know this so doesn’t took so long ^^
I think I got it right. The math isn’t that hard. But I am quite surprised about the result. My guts were telling something different.
So I went with intuitive answer but it didn’t sit well with me, so I did some mental math, then checked it with pan and paper…
Ooh. That’s quite a lot more than my mind was telling me. Had to use google calculator as my iPhone was having none of it lol
With my little bit of paper and a pen, it took me a couple of minutes to give me an answer. I was a bit surprised by the answer I got so checked my maths. As long as I haven’t been a complete dimwit, then the answer still stands. Astonishing really!! But definitely fun to know (though I think I’ll check the answer on Monday, just to be on the safe side and before I wow people down the pub with this golden nugget!!)
lovely
Yes, I know this one, and I know the surprising thing also.
I calculated an answer immediately which I thought was pretty reasonable. But these comments are throwing me off, am I missing something?
Here goes my friday..
OK I took another approach and now I am indeed getting a pretty surprising result! I’m betting this is it
My instinctive answer and my math tell me the same thing which, if the comments are any indication, are both wrong. I’ll just have to wait until Monday….
It is a little surprising…. my instinct was wrong on this one. (I’ve now adjusted my instinct…)
It sounds like a good multiple choice question for an exam. Give a wide range of possible answers and see what people answer.
If one takes a fractal approach the result is more like an instinctive answer ……. mind you the string would have to be of infinite length minus 1 meter ……
A fractal approach? How are fractals involved in stretching a line around a circle? I think your over-complication is unnecessary, misleading, and simply wrong. I dont think fractals necessarily have to have infinite borders, anyway, do they? I dont know if youre joking or not but “infinity minus 1″ should have been a dead give away, presuming of course youre even remotely rational. Twisting a question into absurdity is an easy escape for someone who is too lazy or stupid to do any real computations.
Nice one
Indeed, if the puzzle would have been turned the other way round as suggested by Andrew MW, it would have been even more stunning
And a nice kicker too
Most unexpected
Nice one! Took me about 10 minutes… but the answer is quite interesting – as always.
Now release the Earth this instant!
Ah, the surprising aspect might be that you always get those 10 extra points no matter what…!
Pretty easy this time…
I know this one, so it only takes a few seconds to solve, but it’s completely counter intuitive
Wow, I was way off with my guess. Nice!
When I first came across this one the question was the other way round. If you want the string to stand XXX above the ground, how much longer do you have to make it? It’s a great example of the uselessness of intuition in maths.
I wonder why the answer is surprising?
Maby it is natural to compare the differce in hight with your own hight but the extra meter with the circumference of the earth?
The difference in hight may feel larger then expected but compared to the radus of the earth it’s really small….
What do you think?
Yeah v easy, a couple of mins – just some simple circumference/radius of a circle maths – that is if you work with the assumption that the earth is a sphere for this exercise.
The answer does seem surprisingly large, much larger than I would have guessed. In fact it’s almost hard to believe how large it is until you start to appreciate it is about the difference in that measurement compared to the radius of the earth.
But real surprising part of this is that you can work it out without knowing the circumference, radius or diameter of the earth!
Huh. Of course. So each meter of string would yield the same height? (that’s not giving too much away, is it?
Yeah I was very surprised when I spent the extra minute to work out the formula and saw those terms cancel out. The implications are just wierd.
I took 1 sec to make a guess that was completely wrong and 1 min to compute a value, which was – surprisingly – much higher.
I reckon I have it but reading the comments makes me feel dubious about the whole thing. Shall be looking forward to Monday.
I shouldnt have read the comments as now I am convinced its wrong! Oh well! lol
got it in about 5 mins – simple radial geometry. Suprising thing about answer is it’s the same no matter what starting radius you use.
And now go to the North pole, pick up that rope and pull it up so that it becomes taut; would you need a stepladder to do this?
Killed me with that one. But there is a slight probability that – assuming 10 ft high stepladders – you’d need 1.194.404 stepladders to do it. Or an approximately 3.640 km-high ladder…?
Look here for the answer (and to brush up on your Dutch):
http://dutiaw37.twi.tudelft.nl/~kp/stukjes-pythagoras/jg44/2004-11/
Haha!, I’d never get there, I forgot most of those things! Thanks though, very interesting follow-up question. So apparently you’d need to fold a 0.1 thick sheet of paper 21 times to get there.
What I know of the Dutch is that you have great cheese – and Fristi, of course.
LOL thanks for the Dutch site as I was starting to wonder if I was right with all these comments! but I had it totally correct!
and as a cheesehead myself I had no problem with reading it
haha, ran the site through babelfish, all makes perfect sense now
How many buses named Alfred do we need to put in line in order to complete the perimeter of Earth? And what if two of the buses are named Frank, and one has no doors at all?
Hint: you might consider wise to build a Large Bus Collider (LBC).
I like the cut of your jib, Faltese Malcon. Personally, I think this string is a threat created by the petri dish bacteria, and only Alfred can save us.
Thanks – likewise! And about the plot involving planet-saving public transportation, paper-folding bacteria and a lethal string embracing the earth… I agree. The dish must be destroyed and Al is the 4-wheel entity to do it. The string in love with the Earth is obviously the plot’s fall guy. Now it’s just a matter of waiting for tomorrow’s grand finale.
The grand finale – or just the beginning?
Solved the inverse puzzle before, so i did this very quickly
Remembered where I saw it before. It’s in Lewis Wolpert’s ‘The Unnatural Nature of Science’
wouldn’t the answer depend on how of the two strings you use to tie them together?
wouldn’t the answer depend on how much of the two strings you use to tie them together?
A more difficult question is: instead of supporting the string so it forms a circle off the ground, suppose you hold it at one point and lift it until all the slack is gone. How high off the ground would the highest part of the string then be? (Assume the string does not stretch.)
121.5 meters to Nick’s question? I’ll check later after reaching home. Nice twist, Nick.
That’s about what I get, Joao Pedro. I find that more counterintuitive than Richard’s original question!
Indeed, Nick. I think so too. Going from an originally taut string, adding only one meter, and arriving at 121 meters of slack is quite impressive.
I think both are counter intuitive. The funny thing is to be such, with so such simple math (well, not yours, had to resort to numeric methods to solve it, geezh).
I thought of a twist of my own, but for a moment, I thought it already presented by Hart… now I think his problem is the same as yours (only he assumed the string was around earth in a certain way… I misunderstood his problem completely). My twist is this one: suppose the string was around the equator and already stretched to form a circle above earth (with the 1 meter additional). Now, we pick the poles supporting the string (very close ones to form a circular string), and walk with them to south, raising them continuously to continue to form the circle. How much I had to walk until the poles have to be extended to have an height of 1 meter?
(the twist on Wiseman puzzle would be to specify a smaller height to attend, I’ll not say how much)
PS.: Nice puzzle pages of yours, Nick!
I didn’t really understand the question until i read some of the comments. Then I started calculating. Hope I did that right…
Like a sharp mind above already commented, the answer might seem abnormaly big, but in comparison to the sizes we’re dealing with it is quite small.
Simple problem, nothing surprising about the answer, unless you don’t know what “linear” means.
Got it. The second part is interesting, but after I thought about it for awhile, it really shouldn’t be surprising.
Ah, this just reminds me how bad my math is, awesome puzzle though.
On behalf of your American audience… Don’t you think we’ve had enough π for one week? Sheesh!
I’ve been an arts student for too long to do the maths, but I’m expecting it’s much more than ‘a few centimetres’, which is my instinctive answer!
Good puzzle, very counter-intuitive, even if the answer is immediate. The nice thing about it is that the answer would be the same, starting here, in Jupiter, or around a marble. It’s also an half way to explain the advantages of big dinosaurs over small ones.
It’s much more fun if you add three possible solutions, so people get tempted to guess.
Or, even better. Put it like that:
“You have a globe on your desk, the circumference is 50 cm. If you put a string around it and add 1m to the lenght of the string, there’s is x cm of air between globe and string.
What would happen if you did the same to the earth?”
Without possible answers, people will have nothing to compare it to and “just” do their math and end up with an solution.
ok I’ve actually gone and done the maths and I have to say it’s not as surprising as I thought.
However, I now know what’s surprising about it, which appeared in the calculation. Dunno if I can say it here? Let’s say that the same applies to other round objects, for instance?
Another interesting twist on this one is if you have a taut string stretched along a football field (say 100m), and add 1m of length to the string, how high could you lift up the string at the half way line?
Told the guys this one in the pub at lunchtime, and no-one believed me before doing the maths!
These get weirder.
it does not matter whether it is the earth or the sun or jupiter or moon … the extra height wud be the same …
it took me hardly a second to solve … not that i do not feel sleazy that i sound like boasting … anyway this is hardly surpriising as i am doing PhD in maths …
i like Tony’s twist in the problem !
EDIT :: somebody above commented that it would be the same if it were jupiter or a marble ..
actually this is wrong … for a marble it wouldn’t be the same … as long as the circumference is very large compared to one meter the answer is same … not for the marble though …
PS : i notice your restriction on not writing the solution. however i had alreday written it and no way to undo that … sorry about it.
“actually this is wrong … for a marble it wouldn’t be the same ”
And you said you were doing a PhD in math?
That’s the kind of comment I would expect from a PhD in Physics, not Mathematics.
And even I would be pressed to find how, a different metrics from the Euclidean, maybe 
The answer is always the same from all over the world … (and things … !)
Nice quiz. I think I got the solution. It was easy for me to draw the problem in a piece of paper to resolve it. It’s surprising!
If you take the answer you get and invert it (1/x), the number you get ought to look somewhat familiar – and a clue that something’s not quite as one would expect.
PS: to Niraj, about what happens if the sphere is very small: have you tried a really small example – say, the size of an electron? I think you should ..!
I hope I got it right….. but no, I checked my answer and it wasn’t what I had been expecting.
This problem is becoming interesting at every minute. My comment to Niraj shouldn’t be read as a joke on him: Mathematicians usually reduce problems to the ideal world of math, Physics people are usually lured to the physical interferences to it. A typical difference in the approaches would be the first ones to think zero radius strings while the second ones would question if the radius of the string would have influence. This is an interesting problem. Imagine a string with a certain radius. To be around a marble, the outer perimeter of the string must be bigger than the inner perimeter. Now, add the one meter length perfect symmetrical string around is length to the first one… question, after spread the string around the marble, are it going to be stretched or compressed (or became thinner or grosser) in any way because we are giving the same additional length to both the outer perimeter and inner perimeter of the string?
What do you think?
Well, after much messy math, I got an answer, but I don’t find it particularly surpirising… I’m pretty sure I go this one wrong. Damn.
I know this one, so it only takes a few seconds to solve, but its completely counter intuitive
Assuming that Earth is a perfect sphere then it’s a matter of applying the formula for the perimeter of a sphere/circle. I thought it would be much less than the value i obtained!
Here is a version from the book “The Chicken From Minsk“, which is not a bad source of puzzles, compiled by Yuri B. Chernyak and Robert M. Rose.
A fibre-optic cable completely encircles the Earth, by chance running through a new, privately owned chichen farm on the outskirts of Minsk. The chickens refuse to walk or fly over the cable and will only pass under it. Clearly, the cable must be raised off the ground by 1 foot if the chickens are to survive.
For technical and bureaucratic reasons, the cable must then be raised 1 foot higher everywhere, around the entire circumference of the Earth. The farmer, exercising his newly won individual rights (no more USSR!) refuses to permit the cable to cross his land unless it is raised. The bureaucrat in charge of the project is a holdover from the old days, He maliciously agrees to raise the cable only if the farmer agrees to pay for all of the additional cable, at a rate of $US1 per foot. The farmer agrees, provided that the government will pay for the additional supporting structures. How much will the chicken farmer pay?
Got the original answer immediately. Love this version. Stick it to the bureaucrat!
I’m terrible at math, but I know the theory. It took me just a minute to do the work. As many have said, the answer is really surprising to me. Can’t wait to check my answer on Monday.
Good one, one minute, surprising answer when put in a real world context, though in the abstract the answer I think is more intuitive. I also like the football field example from Tony and Nick’s twist as well.
wow, that was the first one i figured out in like 3 seconds
i couldnt even figure out the other one
Got the answer using simple circle formula, subtracting smaller radius from larger. The surprising thing is how counter-intuitive the result is – it seems too large for the small amount of string added and the size of the earth.
This is too easy. You need to know that to pass primary school.
In less than a minute.
It is interesting in that the earth’s circumference does not affect this value.
I have an answer that I worked out in about 1 minute… it is indeed surprising, but I’m not at all convinced that it is correct. I’ll keep thinking about it until the answer is posted… I could cheat and look it up elsewhere on the web, but what’s the fun in that?
I think I sort of have it? I’m not sure.
Is the earth-catching string something that only Alfred the number 14 Moonbus can save us from?
I think I’ve got it!
I found the answer less than a minute thanks to the many digist of Excell
The answer is surprising.
I think I’ve got it!
I found the answer in less than a minute using Excel
The answer is surprising.
wait, what? I just tried this with actually doing some mathematics. I don’t know what’s more surprising, the answer orthefact that I did some maths.
[...] It’s the Friday puzzle! A lovely simple but surprising mathematical problem this week (via Timothy M) … Imagine you have a piece of [...] [...]
Anybody who read this article will be feel usefull, I will wait your more articles next time. thanks mate.