Challenge: Escape from the 3rd story.

[knot4u]

You must escape from a building by exiting a window that's three stories up.  You have in the room with you a knife, two bed sheets made of strong fabric, a pillow case made of strong fabric, and a post that may be used as an anchor. There is nothing else in the room.  You are naked.  The landing below is concrete.  There is nobody else within a 10 mile radius.  There is a store on the next block filled with things you want and need.  

In short, how do you use the bed sheets and pillow case to get you down?  Here's a solution I found that seems pretty good, but maybe there's a solution that's better or more detailed.

http://answers.yahoo.com/question/index?qid=20070810125205AAqZmRW

1. Forget the pillow case.
2. Tear each sheet long wise into 8 equal width strips. i.e tear in half, then tear each piece in half twice.
3. Knot 5 strips together using reef knots to make 3 long strips about 36 feet long. You will have one piece left over to anchor the rope at the top.
4. Twist each strip, then tie the three ends together.
5. Start braiding the twisted strips together (I hope you know how to do a braid).
6. Tie one end to a secure object.
7. Climb down.

[KnotMe]

The math was confusing me for a while until I reviewed the link where they had 2 sheets x 8 = 16 strips = (3x5) +1. But your problem says you only have one sheet, so maybe in your problem you need the pillowcase.  Too bad because I was going to suggest that you could use it for a breechclout and a bandeau if you needed one. 

To help others play the game, here's some measurements:

Flat Sheet Sizes:
    Twin--66 x 96 inches (or 167 x 243 cm)
    X-Long Twin--66 x 102 inches (or 167 x 259 cm)
    Full--81 x 96 inches (or 205 x 243 cm)
    Queen--90 x 102 inches (or 228 x 259 cm)
    King/California King--108 x 102 inches (or 274 x 259 cm)

Pillow Cases:
Standard Case: 21" x 32"
Queen: 21" x 36"
King Case: 21" x 42"

How tall is a story:
Although 10 feet (3m) is a common estimate for a story height, every building is unique and story heights can vary widely.

A story describes the distance between two levels of a structure and can vary depending on the building, it's function, etc. Most homes have story heights of 9-10 feet. Commercial buildings, such as office buildings or retail stores, can have story heights ranging from 10-15 feet or more. Big-box stores, such as Wal-mart or Home Depot, have story heights of 20-25 feet.

[knot4u]

Oops, I've changed it to two sheets.  I don't want that to be a limiting factor here.

[Rrok007]

Why forget the pilliowcase? You've got a knife. Cut holse and use it as makeshift coverings since you're nude.

[drnihili]

The braiding seems pointless.  Tear each sheet in half lengthwise, tie together to make one long sheet and secure to the anchor.  That should get you at least 20 feet below the window, so hanging from the end means you have less than a 5 foot drop to the ground.  If you're worried about that, tear the sheets in thirds.

[knot4u]

The braiding seems pointless.

The twisting and braiding improves strength, yes?  It's an effort to emulate a rope.

[KnotMe]

I checked up on the Mythbusters prison escape episode.  They used full sheets (but then they didn't have a limit on the number of sheets they were using) and just knotted them together.  They braided hanks of hair and then braided the braids.  They twisted up rope/cables of toilet paper.

window 3 stories up = 9m

sheet 243cm more or less standard length => 2m + tying length (for the sake of argument)
2 sheets whole = 4m
split both in half = 8m.  short a metre but if even a short adult human is 150cm and you consume 50cm for the knot to your anchor and you're holding onto the end of the sheets there still wouldn't be drop to the ground.

half a sheet should probably not need the reinforcement of braiding.

so drnihili is right and you can still use the pillowcase for clothes. 

[knot4u]

half a sheet should probably not need the reinforcement of braiding.

so drnihili is right and you can still use the pillowcase for clothes.  

Well, if you took out the "probably", then maybe I wouldn't braid.  Since my life is at stake and there's no particular rush, I'd go ahead and spend an hour or so on braiding.

I don't know much about rope making, but I'm learning.  Twisting and braiding increases strength, right?

[Wed]

Twisting and braiding increases strength, right?

Twisting preloads each fibre with a force. So it could be said that it actually decreases strength.

However, the twisting increases friction between fibres and keep them all together. This communicates the overall load to all the fibres in a cross section. That way, no parts of the rope is idle, while others are overloaded.

[DDK]

Twisting and braiding increases strength, right?

Twisting preloads each fibre with a force. So it could be said that it actually decreases strength.

However, the twisting increases friction between fibres and keep them all together. This communicates the overall load to all the fibres in a cross section. That way, no parts of the rope is idle, while others are overloaded.

I think the comment about no part being idle is key.  For example, doesn't the trick of ripping of a phone book in half require that most of the pages are "idle" and not supporting any load?  I think a decent braid would be an improvement as I wonder if the load might have more of a chance to be concentrated in one part of the strip in fewer wider strips and possibly start a tear.  And along those lines, does the lay of the weave of the sheets come into play on how one ought to cut the strips?

Some preloads are beneficial like in the preloading of concrete with compressive stress (using steel cables).  Concrete, which is very strong in compression, behaves poorly and easily cracks under tensile stress.  I would agree that the preloads (mainly due to the strand being non-straight?) on the individual braided fibers is likely not beneficial.

DDK

[Wed]

I think the comment about no part being idle is key.

It is. But it isn't enough. A successful rope need more key features. You need the fibres to squeeze each other in order to stay put. Take a wad of cotton for example. it's fibres are a few cm each. It will fall apart (left example) unless there is sufficient friction perpendicular to the string you form from it (Middle example).

Strongest possible rope would be absolute parallell fibres with equal load. This however is rather difficult to obtain. Modern materials allow for long enough strands. But equalling the load on all is impractical. It is sometimes seen on circus when a long haired lady twirls from the ceiling suspended by the scalp.

Each strand could be rather tough. But they still have a maximum before break. And if the break of one comes before the next can share the load, the overall strength is pitiful at best (rightmost example).