As is well known, Spider-Man's airborne flights through the city hang on a thread - well, a few threads. They keep him from falling down and help him steer. Spider-Man defies both physics and biology on many levels during his travels through the city, so instead let's imagine a man with a long whip. In honor of another series of motion pictures, we'll call him Indie.
In these problems, there is a series of obstacles and a goal. Indie stands on a board that resembles a skateboard but cannot tilt. Under the chassis are four metal wheels on universal swivels. He travels over a smooth frozen lake that has narrow wooden poles sticking up and out of the ice.
1. If he is going in a given direction and has not attached the whip to any pole (represented by dots in the figure), then he will keep going in a straight line. He cannot otherwise force the chassis to turn.
2. The whip is light so throwing it in any direction does not change his direction materially.
3. He can detach a whip from a pole at will.
4. He can pull on the whip or just let it rotate him, but he cannot push.
Consider this figure, an overhead view of Indie's situation in which the dot represents the only pole.
You want to go from the bottom to the top. You may start at the bottom at any angle. By using the whip, you need no more poles at all. The challenge is to get Indie from one end of a corridor that is partly blocked by two barriers. Assuming Indie can approach the corridor at any angle, how can he reach the other end without touching any barriers? Solution:
This figure shows a solution.
The whip connects briefly to one pole at a time.
He comes in at an angle and then throws the whip just at the proper moment to snag the pole.
Now here are two problems for you:
1. In this new corridor, does Indie need more than the two poles shown to go from the bottom to the top?
You want to go from the bottom to the top. You may start at the bottom at any angle. There are two poles. Do you need more poles?
2. In this problem, there are several L-shaped barriers (L-shaped) and several objects (positioned at the X's) that Indie wants to collect in a game called "Treasure Skater." What is the minimum number of poles he will need and where should he place them?