String Theory

Does cat's cradle put you to sleep? Here are some livelier, more sophisticated ways to pull a few strings

Knot a problem

A. | |

B. | |

C. |

**The building shown above has four floors and six elevators, but only one of the elevators reaches every floor. The yellow path shows one way to make a big loop over three floors by a combination of walking and using the elevators. The path is essentially the unknotted loop (A) shown at right, which is nothing more than a twisted circle.**

**1. [Medium]**Make a path that describes an overhand knotted loop (B) as shown at right. Within each floor your path must not cross or intersect itself, and you cannot use the same elevator more than once. What is the fewest number of elevator rides required? (One elevator ride may span several floors.) Hint: Model the path using a string or an extension cord.

**2. [Hard]**Can you make a path that describes the figure eight knotted loop (C) above? Hint: Use the same elevator rides as in the answer to problem 1.

**3. [Very hard]**Now make the overhand knotted loop (A) again, but this time avoid moving from floor to floor as much as possible. Can you make the figure without traveling more than a total distance of six stories in the elevators?

Hanging Tough

Suppose you hang a picture from two strong nails as shown at right. If you remove one of the nails, the picture will tilt.

**1. [Easy]**Can you wrap the wire around the nails in such a way that removing one of the two nails causes the picture to fall but removing the other nail leaves the picture hanging, although a little rakishly?

**2. [Medium]**Can you wrap the wire around the nails so that removing either of the two nails causes the picture to fall? Hint: The wire crosses itself three times, and the solution is symmetrical.

**3. [Very hard]**Can you wrap the wire around three nails so that removing any one nail causes the picture to fall? Hint: You'll need a loop of wire.

**4. [Medium]**Now try wrapping the wire around four nails so that removing one of two specific pairs of nails would make the picture fall. The picture should remain hanging if any one nail is removed. Can you wrap the wire around four nails so that removing any one nail would have no effect, but removing any one of four specific pairs of nails would cause the picture to fall? Hint: Start with the solution to problem 2.

Hold That Line

**1. [Medium]**Four people each hold two consecutive sections of a circle of string to make an octagon parallel to the ground as shown at right. The darkened lines show the length of string between each person's hands. Two people opposite each other raise their lengths of string while the other pair lowers theirs. Now everyone walks toward the center to meet his partner's hands, keeping the string taut. What three-dimensional figure is formed?

**2. [Medium]**Now six people hold the string to make a 12-sided polygon, a dodecagon, as shown at right. Every other person raises his length of string while their neighbors lower theirs. Everyone walks forward toward the center until all hands touch in pairs, keeping the string taut all the while. What three-dimensional figure is formed?

**3. [Hard]**Can you find a symmetrical way for four people to form the 12 edges of a cube with a large loop of string? Some edges will be doubled, with two pieces of string running the length of the edge.

**4. [Tricky]**Suppose three people hold a string to make the octahedron shown at right. How can a fourth person grab the string so that the octahedron becomes a cube? The original three people must continue to hold their sections, but they may move around.

**Solution**

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