Challenge Problems

For my every class I try to prepare a challenge problem to stretch the minds of my students. Here is a problem I took from Adam A. Castello’s website:

There is a ceiling a hundred feet above you that extends for- ever, and hanging from it side-by-side are two golden ropes, each a hundred feet long. You have a knife, and would like to steal as much of the golden ropes as you can. You are able to climb ropes, but not survive falls. How much golden rope can you get away with, and how? Assume you have as many hands as you like.

The next problem I heard from my son Sergei:

You are sitting at the equator and you have three planes. You would like to fly around the equator. Each plane is full of gas and each has enough gas to take you half way around. Planes can transfer gas between themselves mid-air. You have friends, so that you can fly more than one plane at once. How do you fly around the equator?



  1. Alexander:

    Second problem:
    Fly planes 1 and 2 1/4 of the way. Refill plane 1 and ditch plane 2.
    Fly plane 1 1/2 of the way. We are currently at 3/4 of the way around.
    Fly plane 3 1/4 of the way BACK. Refill plane 1 and complete the trip.

  2. Alexander:

    First problem:

    Climb to the top. Cut both ropes, leaving a foot or so of one of them hanging. Tie the hanging part in a noose knot. Tie the two cut parts to each other at one of the ends. Thread the tied rope through the noose. Climb down holding both hanging ends of your rope. When on the ground, pull on one end of your rope until you pull it all the way through the noose. 199 or so feet of gold-rope.

  3. Alexander:

    Just read Costello’s solution. I also though to leave parts from both ropes and tie them together, but figured a noose may be more efficient.

  4. Gregory Marton:

    So when you “cut both ropes”, I suspect you fall. A noose is no good. The trick here is that you need a knot that makes a sturdy loop without you needing access to the ends of the rope. Such a knot is the climber’s friend, the butterfly.

  5. Alexander:

    I don’t know much about knots. Why is a noose no good?

  6. Asher:

    Given an infinite number of hands, I would cut off a very large quantity of hands and climb the resulting pyramid to the top, where I would steal the entirety of the rope.

  7. Martin:

    Second problem:
    Two friends fly two planes for a distance of 1/4 and stop; one flies clockwise the other one flies counter-clockwise.
    Then you fly clockwise around the earth;
    at 1/4 you stop at the other plane and transfer all the gas to your plane;
    at 3/4 you stop at the other plane and transfer all the gas to your plane.

  8. AKC:

    Alexander, a noose knot will tighten and clinch whatever is within the loop restricting the movement of the rope you are climbing down with.

  9. Max:

    “Assume you have as many hands as you like” Cut off as many hands as necessary to reach 100 feet, attach the knife to the top hand with a bit of rope, and cut both from the top. You will receive 200 feet of rope. (Attach the hands to each other with ligaments, conveniently found within the hands.)

  10. Max:

    @Asher A genius plan, I had not seen that answer before I wrote my own.

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