Yesterday I saw a column in the middle of town.  It was exactly 200 feet high and 16 feet 8 inches in circumference.  Someone had wrapped a spiral garland around the column exactly five times.  What was the length of the garland?

As ever, please do NOT post your answers, but do say if you think you have solved it 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).

118 comments

  1. Solved. 10 seconds debating what complex math to use to solve it, 2 seconds of inspiration and then 15 seconds to do a quick calculation for the answer!

  2. Solved it, about 20 seconds worth of thinking why anyone would work in Imperial, followed by some abuse hurled at you for that, and then a quick diagram drawn on paper, and a calc.
    Total of about a minute and a half.

    1. Feet and inches are surprisingly common here in England. One wonders why the unit ‘miles’ remains popular in many parts of the world (they, strangely enough, break down into these smaller units).

      There’s no accounting for parochialism.

  3. Not 10, not 20, but 30 seconds, thinking and execution. I now know why I attended high school. For this very day.

  4. Do we assume that the garland runs from tip to tail and is evenly spread (like the garlands in the picture) rather that tightly coiled and all bunched up (like an unstretched spring)?

  5. Pleasingly elegant/straightforward solution once I’d finished being distracted by pi. Took about a minute including making clumsy iPhone calculator errors. Thanks!

  6. Just some basic math and a little abstract thinking, you don’t even need to convert units, just use the numbers arbitrarily and add the units at the end

    1. That’s what I was expecting, but than I realized, I can’t just use the numbers, first I must know how many inches there are in one foot 😉

  7. Quickly saw a method. Made the calculation (needed a calculator). Got an answer which suggested that there was an easier method. About a minute.

  8. I got it. Did not take too long. I am not used to imperial, just like joe.

    Nice inputvalues. It gives an answer that looks like the inputvalues themselves.

    1. Haha – I didn’t actually twig about that relationship between the original numbers and the answer until I read your post. You are right.

      Logically, I would say that wouldn’t be the case, but the maths gives that answer, so I will be intrigued to see what the ‘correct’ solution is…

  9. Well 15 secs – and I will not calculate the actual length – I just can’t work with imperial units without calculator 😉

  10. The columns that are shown each has two garlands each spiralling around the column two and a half times.

    Also, being slightly pedantic, an ionic column should correctly have a height of about 9 times the diameter of the base. The shaft would taper slightly, affecting the result.

    1. Greek columns also often bulge slightly, which owing to an optical illusion creates the appearance of it being straight-sided, whereas a straight-sided one would tend to appear pinched.

      The fact that this particular square root will come out as an integer is a lot less well known than the canonical famous one. I wonder how many people instantly spotted it, as opposed to going through with the calculation.

    2. If the buldge at the bottom was symetric with the tapering at the top, then the answer would be the same.

    1. it’s because: height = (n^2-1)/2 circumferences,
      where n = number of complete turns in the helix.
      😉

    2. Well spotted martjin. Took me a couple of minutes, including a data entry error.

  11. I worked out how to do it in less than half a minute. Actually doing the arithmetic took another minute.

    If it had been a 60m high column and 5m in circumference, I could have done it a lot quicker, and without reaching for pen or calculator!

  12. 30 seconds to visualize it it, and 30 seconds with a calculator.

    Interesting answer, and I’m wondering if there’s a quicker, simpler way to do it.

  13. A couple of seconds to figure out how to do it and a couple of minutes wrestling with imperial units. Nice puzzle.

  14. Got it.

    Hint: a cylinder has zero Gaussian curvature.

    Has not calculated the answer though as it is against my religion to use imperial units.

    1. Now that you mention Gauss, another hint: Instead of maths, go to the toilet. Unroll the toilet paper. Take the cylinder and scissors…

    2. Not sure what you all are talking about; all I used was Pythagorean Theorem. I got what I think is a very reasonable answer.

  15. takes within a minute if you know the formula (because of all the multiplication by 12), more if you need to derive the formula

  16. Yep, think I’ve got it. Seems too easy in some ways, but questions are easy when you know the answer, aren’t they?

  17. Goodness! This is an old one; at least 100 years old. Use of imperial units probably means it’s partly been brought up to date, so don’t complain.

  18. Hmm, I got a very ugly number (imperial or not), so based on the comments I think my answer must be wrong….

  19. About a minute to work it out and type it in. Rolled off the cuff, to mix metaphors.

    I don’t agree that the units make this particularly harder.

    Surprising solution!

  20. I did this the hard (Pythagorean) way with pen and paper in about five minutes, only to see a blindingly obvious answer that I’m still not sure if I believe…!

    1. I too solved it with help from the greek mathematician, but I can’t figure out why the answer is what it is.

  21. Aha! I’ve figured it out, and so I have another column which will give you a similarly nice answer:

    Column height = 200 metres
    Column circumference = 50 metres
    Spiral goes around the column 3 times.

    How long is the garland?

    1. Nice Point…

      Or it is just a spiral gland, horizontally spriralling around the foot of the column, which in turn simplifies the calculation (when assuming a very small diameter of the gland ;-)… It is not said in the question that the gland is wrapped as shown in the picture….. from top to bottom..

  22. Or, try:
    column height of 200 metres
    Column circumference of 5 metres
    Spiral goes around the column 9 times.
    How long is the garland?

  23. I enjoyed that! Ten seconds of horror at such a nasty problem, one flash of insight, about thirty seconds of mental arithmetic. Yeah, I’m old and British enough to do feet and inches in my head.

  24. I instantaneously knew how to solve it, so i lost interest to acutally calculate and cope with the imperial system… Then i couldn’t avoid to ask myself which diameter the gland has, and if the space it is located in is acutally euclidic..

  25. I got a straight forward answer and a too complicated one. One is slightly more than a furlong the other is a good few chains and poles and perches more. I think both maybe wrong.

  26. Spotted the method immediately. Took a couple of minutes to do the maths, mostly because of the imperial units.

  27. As soon as it got mathematical my brain dribbled out of my ears and ran and hid in the corner. Counting thousands of pounds at my work and balancing spreadsheets/ledgers/stocktaking etc. and I’m fine – show me a mathematical puzzle, hell no.

  28. I think I have it. I remember a way of calculating the length of a helix using parametric equations, but somehow I doubt that’s the answer Richard is looking for. I found a much simpler and more elegant solution using some basic geometry.

  29. Ah. Spent a long while trying to calculate it in feet and dealing with annoying rounding errors. Try inches, its much easier!

  30. I had no clue how to do this, read some of the responses and came up with my own (simple) solution. Have no idea if it’s correct but it IS a nice round number in inches!

    If that’s not the solution then I’m ‘screwed’.

  31. I don’t think it’s got anything to do with maths. Maybe I’m just in denial as I hate numbers. Think it’s more to do with what is actually being done.

  32. I got the answer almost immediately. There seems to be some unnecessary information in this one. I will have to wait until Monday to see if I’m right.

  33. Hey mittfh, you’re not supposed to give the solution in your comment. Yes, we’re all very impressed that you figured out something requiring middle school geometry, bravo.

    1. I don’t think I gave away any more information than many other commentators, who’ve mentioned converting units, a certain Greek chap, and having spent ages on geometrical solutions before noticing an alternative. Besides which, I was slightly vague and deliberately omitted several intermediate steps.

      I have no idea whether my calculations are correct, and of course they assume the garland has a negligible diameter.

    2. I agree there are a couple of other comments that need moderating, but yours was the worst offender. But yes, Ivan and Michael need their wrists slapped too.

    1. The Friday problems are usually ambiguous. Usually you have to say “given that this is supposed to be a problem of interest that I can solve fairly easily, what is the best resolution to the ambiguity”. Sometimes the whole point is the ambiguity and we can have fun arguing about it. Without it, there’d be much less of of interest.

  34. I wouldn’t even TRY to figure this one out, as it would be an impossibility for my mathematically retarded brain.

    However, I did notice that there are 2 garlands on the picture snaking up from the base in the columns in the picture …

  35. Real garlands, of course, stretch.

    But we’re in the land of weightless rods and frictionless pulleys here, I can see.

  36. Insanely easy 😦

    Figured out how to do it by the time I got to the end of the puzzle. All that remained were the calculations.

  37. I’m worried that my answer is wrong now, took 10 seconds for a method, slightly longer to do the maths and 5 minutes wondering at the result because it looks neat…

  38. Oh, frak it, this bugs me anyway. Been watching some things but still thinking why I can’t wrap my mind around calculations that involve three-dimensional things. Obviously multiplying involved. But exactly how? Maybe Pi needed too?

    1. Yes, all the guys they say that they solved it in less than 30 seconds did forgotten to use the PI !……….

  39. Richard.

    Why couldn’t you use something as small as a toilet roll. So the dumb ones like me can wrap a piece of string around it.

    I’m having trouble finding a 200 foot pole

  40. Easy to solve, i think….
    Have to checkout whether the picture is showing the column exactly or its just for distraction.

  41. And it’s to be done from top to bottom not wrapped round the middle giving a shorter length? Kinda want to give up.

  42. The penny finally dropped this morning, a full 36 hours after I had read the question, and about 35 hours 45 minutes after I had stopped thinking about it. The brain is a weird thing!

  43. 12 inches in a foot, right? And I assume the garland is going from the top to the bottom (if it’s just coiled around at the bottom, it would be trivial)

    If so, slightly over 1 minute for me.

  44. Only took 2-3 minutes. Was most unsure that I was getting the bloody imperial units right though. Had to keep using Google Calculator to check the feet/inches conversion.

  45. 5 min reeling from reading the problem the first time

    10 min working against my dad to get the answer first

    5 min convincing him I was right

  46. I’m good at math, but not that good. I told my computer how to do the problem, and had it come up with the answer!

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