On Friday I set this puzzle…..

Imagine that you have a strip of fencing that is 1 meter high and 10 meters long.  It is totally flexible and so can be made into any shape, but you can’t cut it.  You are standing in the middle of a field.  What is the best way of arranging the fencing to ensure that it encloses the largest amount of land?

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

Easy.  Form the fencing into a circle, stand in the centre of the circle and declare that you are on the outside of the fence!  Did you come up with this solution?  Any others?

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79 comments

    1. Not so mad anyway. If you too the ‘real’ meaning of enclosed, then a circle would be the best way to get the maximum area with a given length of fencing.

    2. simontaylor, see my comment (#12 or so) about how there’s a better shape than a circle for the “real” meaning of enclosed. As long as you’re looking at more than just the “length” of fencing, but include the “height” too.

    3. Better to declare yourself to be Wonko the Sane, and the rest of the world mad, thus enclosing them in the assylum that exists outside of the fence.

    4. The answer doesn’t make sense on several levels. The first is that making a fence enclosure and then declaring yourself as being outside it when you are manifestly inside plays fast and loose with the definition of “enclosure”. Secondly, even if you were to allow that the enclosed element of the field is outside the closed path of the fence, then the best solution is not a circle. You would have to wrap the (completely flexible) fence round your feet to enclose the minimum area. Indeed if you can manage to stand on one leg, of tiptoes, so much the better. So the answer fails on its own terms.

      A more reasonable answer in keeping with any reasonable definition of “enclosed” is to fold the “totally flexible” fence along one of the diagonals (making a length, by Pythagoras’s theorem, of the square root of 101, or about 10.05 metres). Now form this into a circle with the folded diagonal along the ground (there is no minimum height requirement for the fence). That gives the largest enclosed area in the normal sense of the word.

    5. My solution actually encompassed more area.
      1: Form it into a circle
      2: Twist it into a Moebius Strip
      3: Declare all area to be inside the fence

    1. That was what I was thinking off…

      Either the answer is,
      – make a circle and that is the max of the surrounding area
      – OR declare the field outside the fence as the field surrounded by the fence, BUT then do NOT make it into a circle. Just make it flat…

  1. Did you prove that a circle is the shape with the smallest area that you can make with the fencing? Because I reckon that the circle will have area 7.96 m squared (3sf) whereas a 1×4 rectangle has perimeter 10 but area 4m squared. So you haven’t in fact enclosed the largest area.

    1. I was thinking of a rectangle with a 0.0000000000000000000001mm width (sent it to as arbitrarily small as your materials will allow) which will have an area approaching zero so we can do better.

      The circle is the best only if you are enclosing what’s on the inside, otherwise it’s the worst shape.

  2. Wouldn’t a less “area-efficient” shape create even more space on the outside? Assuming we are only allowed end to meet end (i.e. not allowed to wrap the fencing around and around to make a small circle) what is the least efficient shape?

    1. acually yes. a star shaped or (inner-) area- minimized wold apparently enlarge the encirleled “outer” area..
      That is without the inside out logig, Wiseman got the largest inner area, while,
      In case of the insild out logig, wiseman completely failed, as he “enclosed the smallest amount of land”….

      Tztztztzt… Nice try mr Wiseman, but finally, wrong.

    1. Or just put the fence into a shallow lake, or pond…. as you want to enclose the largest amount of “land”….

  3. Well, apart from this “Wonko the Sane” answer, I looked at the more traditional version of “enclosed”. Using the fencing on its long side, it gives you a circle of circumference 10m and height 1m, with an area of 7.96 square metres. I wondered how I could improve on that, and I did.

    Lay the fencing flat on the ground. It’s not enclosing anything at that point, but is resting just on top of it. So, now take every edge, and fold down at 90 degrees towards the ground a strip of 1cm. You now have a structure which is enclosed on all sides, has a height of 1cm, and a roof. It is 98cm wide and 9.98 metres long. It encloses an area of 0.98*9.98 = 9.7804 metres squared. And of course, if you reduce the lip you put on your fence down to an infinitesimal amount, you approach 10 metres squared for your answer.

    So, as long as you’re only wanting to enclose insects or bacteria on a perfectly flat surface, and you don’t mind the enclosure having a roof, then the answer is 10 metres squared, minus epsilon.

    Circles are for whimps.

    1. I like that answer. Mine was to fold the fence half diagonally to give it a length of sqrt(101) m, then place that in a circle enclosing just over 8 m2.

    2. no need to fold the edges. laying it flat upon the ground encloses it just as well, giving you the full 10 m^2.

  4. Is this like those “No Drinking Outside These Premises” signs? Meaning they don’t want you to drink anywhere else in the universe apart from in their shop.

  5. I went for tilting the fence down sideways to get some extra length out of it rather than totally dismantling it and putting it down one slat at a time – although that option could give you a pretty long fence.

  6. Typically (possibly deliberately) Richard messes up his clever answer by enclosing the maximum possible area inside the circle, thereby minimising the area outside. The correct answer would of course be to have a very thin area, just wide enough to stand in.

    1. @John loony

      You are correct! Wiseman got a little too cute in his answer here and didn’t think it through. The question states” What is the best way of arranging the fencing to ensure that it encloses the largest amount of land?” If you are going to declare that you are outside the enclosure then the SMALLER the area the better. So wrap the fence around you have it meet so the excess is flush against itself. If you are outside the area it encloses then the “amount of land ” it encloses is : Area of Earth- Area You’re standing in=Largest amount of “land”

    1. I meant you could bend the fence further, making it completely circular from sideways. than connect the two ends like another circle. This way only a small amount of land or air will be inside the fence. So you than own the whole universe except for the circular fence.
      My comment was only false on last sentence because actually youre already outside the fence that way.

    2. In my method, no air is inside the fence… my god what am I saying here? I sound like a right nutter.

  7. Make the fence into a long thin rectangle 4.9m*0.1m, with area 0.49m².

    Then ‘declare’ that 0.49m is a larger number than 100.

    1. Make the fence into a long thin rectangle 4.9m*0.1m, with area 0.49m².

      Then ‘declare’ that 0.49 is a larger number than 100.

  8. Do a little trouser cough and tell everybody that fences are, in fact, the only way to be happy. Crawl under your piece of fence and cry yourself to sleep. If anybody hears you do this, they will probably try and get you some help; you obviously have mental health problems.
    “I’m sorry mother, my pumpkin friends told me to do it!”

    1. Wrong, AI. Although Richard’s circle answer is clearly not optimal, his take on enclosing the outside is logically fine. Here’s why…

      Imagine a small fenced-off farm. The farmer keeps buying more surrounding land, and expanding the fence to enclose it. Eventually her farm grows to enclose an area of greater than 50% of the surface of the Earth. As the farm continues to expand, we approach the situation given by Richard’s answer.

      It’s an absurd scenario, but not illogical. The definition of an enclosure is not connected to the ratio between the enclosed and non-enclosed areas. For example, there are many zoos where the “enclosures” cover more area than the “non-enclosed” areas.

  9. Form a circle as previously described, but before closing it, give one fence post a half twist to making it a moibus strip. Then there is no need to declare yourself inside or outside; your are on both sides! (And yes, I did think of this Friday – and I also thought it unworthy.)

  10. Proclaiming that the exterior of the circle is the enclosed portion of land is irrational. Gib’s answer is better. Can a mobius strip be used to “enclose” a space?

  11. I considered something similar to Richard’s answer, but rejected it as cheating (preferring instead solutions that would enclose under 50% of the field’s area).

    The basic one is the inverse of Richard’s – form a circle; that will have a diameter of about 3.18m so enclosing an area of 7.96 sq m. However, considering that fields usually have hedges or fences on their perimeter, it’s also possible to take advantage of this. Head to a corner of the field, arrange the fencing to form a quadrant, and you’re now enclosing a much more respectable 31.83 sq m (from a radius of 3.367m). Keeping the fencing straight and cutting one corner of the field to form a triangle would result in an area of 25 sq m (the fencing being the hypotenuse, and the two sides of length sqrt 50). A half circle somewhere along one edge would have a radius of 4.183m and enclose an area of 15.915 sq m.

    Unless I’ve made a foul-up and need to revise my trigonometry and circle equations… 😀

    1. Why not read the question – I know that Richard has set a bad example by not answering the question properly – but I expected better of you mittfh

  12. Richard’s silly answer aside…

    I and a colleague were wondering whether the two ends of the fence could be laid on the ground, meeting at the former ‘top’ to make a V shape with 90-degree angles sticking out like two ends of a shirt collar. Since Richard specified that the fence was fully flexible, this might work, and would be a different shape from the circle, and would certainly have a greater perimeter.

    The question of whether this would have a greater AREA than a circle was one that I attacked with gusto, having a maths degree and being pretty confident I could do it. Unfortunately I failed. I think the Calculus Of Variations is required here, and the process is beyond me. If anyone can do it, please let me know!

  13. Wrap the fence around yourself?
    Nah. Roll it up, plant it. Instant space invader occupying earth.
    Where did that come from?
    i peeked, the 99%.

  14. I think that Richard should read the question, before he attempts to answer it.
    The one small mercy is that this question does not involve Richard’s “specialist subject” of probability.

  15. I figured that the circle was the best way to encompass the most area. I must admit the counting the inside as outside solution did occur to me as well.

    1. Don’t worry Neil
      Not counting the outside as the inside probably means that you are not an idiot.
      This not a problem.
      Stay calm.

    2. Oh dear.
      I have just realised that I misread what you wrote.
      Please ignore my earlier post above.
      Is there some old saw about keeping an open mind, but not letting your brain fall out….

  16. I think the lesson for us all here is that if you’re going to give a smart aleck answer, you’d jolly better well make sure your answer is correct, otherwise you look like a bit of a numpty.

  17. If you’ve ever taken topology or a class in General Relativity that references topology, this answer makes sense. Except for the circle part, a form fitting fence would be best, if you could enclose it over your head you could enclose the entire universe inside your fence.

  18. A circle of circumference 10 would have an area of approximately 7.9. A square with 2.5 m sides would have an area of 6.35. Wouldn’t a square leave “out” a smaller area, thus fencing “in” a larger portion of, you know, the universe?

  19. Gave this problem to Maths Club at school yesterday. They got the correct answer relatively quickly, given that they are yr 7 and 8. They realised that if you’re ‘enclosing’ the area outside the fence, then the area inside needs to be as tiny as you can make it.

  20. I agree with all who said or intimated mr. Wiseman was cheating. Before posting this I consulted with my husband ,the brilliant math man ,who makes his living with this talent & is triple 9 member who completely agreed it is -cleaning this up-hogwash and wrong. please forgive sloppy typing-only 1 good hand now.

  21. Something else to consider about Richard’s answer is that by declaring the outside to be enclosing land, well it’s not enclosing just land. it’s also enclosing all the oceans and lakes on the earth. So, really what you have is an enclosure of land where the borders are the coastline of whatever land mass you’re on, plus the fence.

    Hell, whatever animal you’re enclosing by a 1m fence probably couldn’t cross a river or ravine either, so your enclosure is really reduced even further.

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