The Fencers
Three visitors to Puzzlaria are having beer in the Puzzlaria tavern and each is boasting about how fast and efficiently he can build a fence around a field. Zaux is interested in their anecdotes and challenges them to fence in the largest possible area using a unit length of fence. Visitor A makes a circle with his piece of fence and says that since a circle is the perfect shape, his design is the best. Visitor B fences in a square shaped area and immediately regrets it. Visitor C sets his up in a straight line and waffles on about infinitesimal calculus, trying to prove how it is an infinite fence enclosing an infinite area. (Don't ask me how; I am mathematically challenged!)
Cam happens to drop by and laughing at the three, beats them all with his own ingenious design. Zaux is only too happy to declare him the winner. Can you figure out what Cam did?
posted by Jin at
6:25 AM
20 Comments:
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my math may be wrong, but I keep coming up with a semi-circular enclosure with the diameter closing the semi-circle. It seems to provide significantly more area than a circular enclosure using the same amount of fencing.
Sorry, Zaux, your math is wrong.
Assume length of fence is p.
A circular fence of circumference p has radius (p/2*pi), and encloses area p^2/(4*pi). For a fence of length 100 ft, this encloses an area of 2500/pi = 795.77 sq feet.
A semicircular fence of fence length p ends up with a radius of p/(2*pi+2) and encloses an area of (p^2*pi)/(8*pi^2+16*pi+8). For a fence length 100 ft, this encloses an area of 228.941816 sq ft.
The ratio between these two areas is 4/pi+2/pi^2+2, or about 3.475 times larger for the circle.
My ToM answer to the what Cam did is that he made a circle from his fence, but reversed the outside and inside of the area enclosed. In other words, he declared that the fence was actually enclosing the entire surface area of the planet EXCEPT the circle that looked like it was inside.
I like that answer Ross, why not take it a step further, and make a semi-circle for less area being excepted ?
Oooh, good point. Unfortunately then, we can use even more pointy shapes to end up with even less area, down the the limit of an infinitely pointy star that encloses zero area ...
Or maybe just a straight line to achieve the same thing ? In which case visitor C has it already
I had a moment and re-checked ... Ross, you're absolutely right ... a semi-circle enclosure is not a good answer
If the height of the fence is irrelevant, then you could make several low height fences from the original. Making perhaps 5 very low fence pieces. Now bend that into a circle.
Or simply lay the fence down flat (eventually killing the grass).
Maybe Cam built his fence enclosing a nearby mountain, thus fencing a much greater area of land than the 2-dimensional area of the interior shape of the fence.
Without knowing the unit length, or if the height can be split to allow more, fencing in just the grass (laying the fence down) etc, he somehow made a Mobius strip fence ?
I would simply try to keep out of my enclosed area the smallest area possible, i.e. an infitesmal dot, with the fence provided. So I place first fence post and tightly as possible wrap(spiral) all the fence around the first fence post. The infinitesmally small dot is now outside of my enclosed area of the universe!
Cam
Looks like Ross and Cam think big by thinking small. I hope they've more or less got the official answer.
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Yup, Cam and Ross have the right idea. The published answer is that the winner wraps the fence tightly around himself and then declares himself to be on the outside. The solution given by Cam and Ross reduces the 'outside' space to just a dot !
Good job :)
I am way late for this. Reading the comments I was hoping for a practical answer.
True, a circle does maximize the area, but it does not maximize the practical aspects. A lot depends on the size, a lot depends on the building material. Straight building material would be the most practical, easier, and cost effective. But it's all depends on the size.
If you want to get real about it, then what about the legal aspects. I doubt that every "court" in the entire universe would uphold that Cam owned the entire universe, except for a very small part, just because he marked it out with some material and made the claim.
This post was good, and the answer was good, I shouldn't complain.
But I was a little disappointed at the lack of challenge.
I see no reason to be rude. It was a fair lateral thinking puzzle.
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