There are 2 flagpoles that are each 100 foot high.  A rope that is 150 feet long is strung between the tops of the flagpoles.  At its lowest point the rope sags 25 feet about the ground (see schematic diagram below).  How far apart are the flagpoles?

As ever, please do NOT post your answers, but do say if you think you have solved the puzzle and how long it took.  Solution on Monday.

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.

1. Job David says:

Nailed it. Simply geometry and a little algebra

2. Niraj says:

10 seconds

3. Anonymous says:

I think I got it fairly quickly.

4. Criticalbounce says:

One minute. First tried to make it harder than it actually is.

A more interesting question would be “what’s the relationship which describes the distance between the poles and the height of the rope above the ground?”

6. z says:

I think I have it… in symbolic, non-reduced form…

1. ivan says:

I think you haven’t got it.

7. Mickey D says:

25 secs

8. Mat A says:

2 minutes, facepalmed

9. Jonathan Flowers says:

A few seconds. Knowing you it wasn’t going to involve calculating catenaries!

10. Bala Narayanaswamy says:

30 seconds. And BTW, it’s probably also worth mentioning as a part of the question that the curve is a Catenary and not a Parabola ;-). B. ‘Nary’ Narayanaswamy / New Delhi / India

1. what curve 😉

11. drgeraint says:

12. Kate says:

30 seconds…

13. Jenny says:

30 seconds

14. Five minutes pondering and scribbling with puzzled expression, then two seconds going “d’oh!” and feeling embarrassed about the five minutes.

15. Tom Ruffles says:

Nice to see you still using feet rather than these new-fangled metre units.

1. deepfield says:

Surprised to see that you are still using feet instead of the more logical and sensible metric units.

16. Wait…
So I spend about five minutes looking at quadratic equations, getting nowhere, then look at the comments for a hint and see that people are getting it very quickly.
Then I read the question again.
That’s… rather silly, actually. One day I’ll learn that the more mathematical the Friday puzzle looks, the less it probably is.

17. AMWhy says:

Time: however long it took to read the question twice

1. Me too!!!!
and then I recall that we rarely need math skill to solve the question so I read it again.

18. Anonymous says:

19. M says:

Tricky…

I was thinking about my maths:
– The shape of a hanging rope is that of a hyperbolic cosine.
– Then derive the length out of it.

Aaarghh…

Then read again and thought… Right. Not that hard.

20. benjaminsa says:

Solved it, 5 min 10s. 5 minutes doing calculus, 10 seconds going, oh wait… and 10 minutes hitting my head against the desk.

21. Paul H* LEEDS says:

Correction . * ABOVE * the ground ,
or how far from the TOP ?

22. got it in a minute. You just need what you learn in first grade’s math class to solve it.

don’t just stare at the graph, read the question

23. Anonymous says:

A few seconds because I did take a look at the picture only afterwards.

24. Edgar says:

No algebra nor geometry at all. It took me five minutes to realise I didn’t need it 😀

25. Goliath says:

Read the puzzle. Got scared away by all the math to do. Came back 20mins later. Some scribbling. Still afraid, turn away. Third attempt, 10 seconds.

26. hankwinner says:

The schematic threw me at first. I sort of got it straight away but it took me a minute or so before I realised I had the actual answer and that the puzzle wasn’t misstated.

27. Mike Hitchcock says:

Doh!

28. CH says:

Hmmm, normally get these quite quickly. I do have a solution involving ratios, however I’m not sure that’s actually correct, especially as people are saying it requires no arithmetic. Hmmm.

1. Anonymous says:

It does require arithmetic – only not much.

29. Took me about a minute of puzzling through things, and then AHA!

30. 2 seconds, then a minute of thinking “No, that’s a bit too devious even for you.”

31. Berhard says:

nearly instantaneously after i read the question for the second time…

32. Gareth Sefton says:

*facepalm*

5 minutes head scratching, and an instant of realisation.

33. The light dawned after the third reading. About 2 min total, I guess.

35. Paul says:

Even though there is a logical answer not requiring advanced maths, wouldn’t this answer assume that the fundamental laws of physics are violated?

36. Nath says:

I just don’t get it. How can it be so simple people are doing it in 5 seconds flat?

37. Nath says:

Ok i just got it.*shoots oneself in the head*

38. Dave says:

I lol’d after thinking the same as everyone else, but read through the comments going “I don’t get it… I don’t get it… I don’t – Oh.”

39. Read the question, went “Catenaries.. I know this”. Did the math, got the answer…

Embarrassed now, after 10 minutes.

40. Anonymous says:

41. Bleedin ell! About 2 hours of head scratching then the penny finally dropped

42. Timdifano says:

Had to start writing down the problem…then after a couple of minutes all became clear!

43. Anonymous says:

The poles are 80 ft apart

1. M says:

What is the size of your shoes?

2. Mort Canard says:

Has very very small feet but shoe size is still larger than I.Q.!

44. Got it. Sometimes you can’t see the monkeys for the dots. But then the dots become monkeys, and it all seems quite obvious.

45. I heard the puzzle on Car Talk, an American radio program, several months ago. It took me a couple of minutes while I was listening to the rest of the show and walking to work to get it.

So… either a couple of minutes while busy with other things, negative several months, or as long as it took me to read enough of the question to verify I’d heard it before and all the particulars matched, say 10 seconds. Take your pick.

46. Anonymous says:

You know what they say. Big shoes ……. Big feet

47. Ed says:

How Thick is the Rope?

48. Anonymous says:

About 5 minutes, and then I cursed myself after an AHA moment!

49. Dennis says:

About 5 minutes, and then and AHA moment.

50. Dennis says:

Hey, stop copying me!

51. Gerard Engelage says:

20 seconds. I don’t know if it would have been easier in metric units 😉

1. Berhard says:

no, it wouldn’t … definititvely…

52. Instantly. Then I had to go back to make sure I didn’t miss something. Nope.

53. Camel says:

15 seconds thinking about maths, quadratic equations etc, then another 1 second to realise I did not need any of them.

54. photon says:

The light came on after about 20 seconds

55. Phil says:

56. Bork Bork Bork says:

Sneaky sneaky! Took me a minute of conceptual figuring on how to set up the math problem. Then facepalmed as I actually looked at the numbers.

57. Lore says:

oh! how could yo be so fast!
my “facepalm” came a bit later!
Congratulations to you all!

58. Bill Hammer Cassell says:

Maybe about thirty seconds. Overthought it, thought about it again, and then went, “Oh. Heh heh.”

59. UXO says:

Not long – after the first reading, thought to myself “Hell, it’s been 20 years since I did a catenary curve problem”. Read the problem again and had the inevitable facepalm.

Wonder how long it’ll be NOW before I even think the word “catenary”?

60. Brian Engler says:

61. Joe says:

It took 10 minutes which was a bit too long. Duh!

62. Sparky says:

Car Talk (a US public radio show with a weekly puzzler) posed exactly the same question earlier this year.

63. Gus Snarp says:

This was pretty easy, and lack of mathematical skill was actually a benefit to me, I think. Instead of starting with some preconceived notion of how to solve the geometry, I just started assembling everything I knew and could easily discern from the information available, and the answer immediately became obvious. Then I spent some time trying to figure out how I must be wrong.

64. Michael Sternberg says:

About a minute. Didn’t get very far with sideways thinking 😉

65. SeleneBlueSky7 says:

I can’t form the equations to calculate it but my reason tells me an enstimation of an answer.

66. SeleneBlueSky7 says:

I can’t see how this could be accurately answered without mathematics.

1. Ben Hardwidge says:

You do kind of have to use mathematics, it’s just that the mathematics involved are facepalmingly simple once you realise!

67. Nathaniel says:

About 30 seconds, but admittedly most of that was trying to remember the word “catenary”.

(BTW, the curve of a suspended rope is a catenary, not a parabola.)

69. Anonymous says:

about 4 shameful minutes… of looking for algebra and calculus formulas for parabolas and catenaries, and then: D’oh!

70. Jimbo says:

It took me all day. Damn you, Wiseman!

71. No idea. Oh. Hang on. Got it. (He says, but not confidently).

72. I started out thinking about how to do an integral in calculus and then started thinking about how I could use a Newtonian approximation before thinking about how I could just do a linear approximation and then the solution became very very obvious. I might have overthought this one.

1. Michael Sternberg says:

Actually, the solution to the generalized problem (where the shortcut is no longer applicable) can be written in algebraically closed form (no approximation or iteration required).

73. SteveG says:

About 10 seconds to figure out no complicated Maths required, by thinking about the bounding cases.

75. NoAstronomer says:

Like several others I started out trying to work out how to work out the solution. At first I tried using an approximation to get some rough numbers. That’s when I realized my rough answer was the real answer. Total time about 5 minutes.

76. Jarl says:

Read the question, thought about picking up my old textbook in maths, thought about how hard (impossible?) this would be for people who have not done maths at this level. Read the question again trying to find an answer wich did not require any skills at math. Got it in about 15s total, while beeing slightly intoxicated. (guess I fail at life for spending 5 min writing a message online on a friday night while intoxicated)

77. Michael Sternberg says:

About a minute, but I too had to double back after the first routine approach, as so many else here.

Alas, same story as last week, sigh. The intent is clear, but there’s trouble if you take the words and the picture at face value. I’m on the fence if this makes the puzzle more engaging or tedious.

78. Jan says:

done in 1 minute

79. Tony says:

Yes got it. It’s a bit like a triangle with sides of x, y, x+y

80. Mort Canard says:

Another facepalm here!

Ten minutes of digging out catenary equations.

45 seconds of approximating trigonometric distance from top of pole to bottom of curve.

A couple of seconds of revelation! :~O

81. Jerry says:

Got this one immediately.

82. Steve says:

It took me about a minute, and I felt like a complete idiot when I worked it out.

83. guest says:

to get it exactly right I might need to know the diameter of the rope

1. UXO says:

If you’re getting that picky, you’d also need to know its bending modulus, the temperature, and the radii of the poles!

There’s a “right” (as in, “close enough”, conceptual), SIMPLE answer that doesn’t require that level of sophistication.

2. Michael Sternberg says:

… You’d also need to know the precise manner of fastening employed at the top of the pole, not to mention the radius of the planet you’re on.

3. Kelly says:

Well, at least Richard simplified it enough not to need earth’s gravitational constant or rotational speed.

4. Michael Sternberg says:

Not really, you know: while in a homogenous gravitational field all catenaries are similar to each other, I doubt that’s still the case in a radial field. Further, we need to know the azimuth of the line connecting the flagpoles, their latitudes, and the day length, all to account for the influence of centripetal forces. Shame that this of course makes symmetry fly out the window (in a strictly Newtonian manner I’m sure).

5. UXO says:

Well, you raise an interesting point: things are already complicated enough when considering just the Newtonian effects, but what about relativistic ones? We’d need to know not only the precise planetary gravity gradient, but also the distance from the primary and its characteristics, as well as those of any other large masses in the system. Then there are quantum effects, and don’t forget Brownian motion of the atmosphere!

Well, folks, that does it: the problem’s CLEARLY unsolvable.

🙂

84. Carl says:

Like most people, a minute or so of scribbling diagrams and numbers on paper, then… “Oh!”

85. Noel says:

This is the first puzzle I think I got. My daughter and I made a scale model. The answer comes very quickly that way.

86. Dave says:

I would say the diagram is a master stroke of misdirection

87. Tom (iow) says:

The word schematic is significant.

88. Pieter Delacourt says:

I got the answer after 1/8640 days, but needed an extra 1/17280 days for converting “feet” into metric units.
Best regards from the continent.

89. Lazy T says:

I started reading the comments before I thought clearly about the problem, and was happy to read that I didn’t need to use catenaries coz I’ve never heard of them.
I think I’ve got the answer without facepalming too, though I did a bit of face slapping before I got it.

90. David D says:

I’m sorry but these ‘puzzles’ are getting quite pathetic.

91. Anonymous says:

Thought it would involve parabola! Just did some maths, then thought parabola could also be ellipse. Solved for 5-10 minutes. Read comments, people reported 15 seconds solved! I admit to doing something complex. And there was the answer – 5 seconds.

92. Tom says:

TIMMY! That took me 5 minutes before the facepalm. I feel slow

93. MarkW says:

Very simple in the end. 90 secondss once I drew it.
Misdirection everywhere.

94. pata says:

secia ste sprosti

95. Danil says:

As math major, first idea was a catenary, which describes this situation exactly…

Second thought, lol, I’m embarrassed.

96. Dustin says:

Thanks for the puzzle Danil. I also agree that as a math student, every neuron was firing off, making the problem more difficult and quickly I wanted to dismiss degenerate answers.

97. sultan alshehri says:

Zero, or no distance