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A rectangular strip of 1 mm thick paper has length = 500 m and width = 1 m. It is lying on a horizontal floor and you are allowed to rotate and fold this inelastic paper as often as possible.

What is the maximum height it can reach above the floor, while still in contact with it?

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  • $\begingroup$ Is paper durability and weight considered in this situation? $\endgroup$ – Joe-You-Know Jun 27 '18 at 16:33
  • $\begingroup$ I think durability and weight are not factors, though if concerned may consider light and durable. $\endgroup$ – Tom Jun 27 '18 at 16:36
  • $\begingroup$ ...so it's a piece of wood, then? $\endgroup$ – Chowzen Jun 28 '18 at 1:42
  • $\begingroup$ Inelastic in a flexible way i.e. unstretchable is all that's needed. $\endgroup$ – Tom Jun 28 '18 at 2:00
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Answer:

500.001 m

Explanation:

Stand the paper up on one corner so that the line between the corner touching the floor and the opposite corner at the other end is perpendicular to the floor.

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  • $\begingroup$ Of course, the real question when doing this is how would you keep it stable. $\endgroup$ – Joe-You-Know Jun 27 '18 at 16:48
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    $\begingroup$ You might need to brace it, but it certainly shouldn't bend (in fact you couldn't fold it if you tried) because it is 'inelastic'. $\endgroup$ – Penguino Jun 27 '18 at 21:52
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Since we're allowed to fold this paper as often as possible, folding it repeatedly makes the thickness exponentially grow. So folding it 20 times gives us just over a kilometer (.001m x 220 = 1048.576m) and it just keeps doubling. Good luck actually doing it though.

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  • $\begingroup$ After 10 folds, it's half a meter wide, and a meter thick. How are you imagining that you're going to fold it again? $\endgroup$ – jasonharper Jun 28 '18 at 0:54
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    $\begingroup$ @jasonharper Carefully $\endgroup$ – Veskah Jun 28 '18 at 1:16
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    $\begingroup$ For each fold increasing amounts of the paper are needed for the edges of the folds (which produce the vertical rise). As the paper can't be stretched, no two points in it can be further separated than the distance given in osdavison's answer. Nice answer on MSE. $\endgroup$ – Tom Jun 28 '18 at 2:09
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If we are only using the paper and no other objects, you could.

Fold the 500 m side so you have a have a 90 degree and a 499.5(or longer) m side and a 0.5 m(or shorter, what ever the paper can stand with) base, so it would appear as a triangle without a hypotenuse.
enter image description here

But,

The edges do not have to be 0.5m or 499.5 m, they could be as long or as short as the paper could stand alone with. They could be 0.1 m and 499.9m.

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The answer,

500 m

Explanation:

If we use a cable or elevator to lift ourselves up, we can effectively rotate it so that the length becomes the height. After that, we just build a structure around this slim piece of paper to hold it in place. Since only the 1 mm thickness by 1 meter wide is touching the floor, it'll be hard to maintain, but is possible. Unless you rip the paper, which I don't believe we can do, this is as tall as it can get.

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