I have had my mind wrapped around this problem for about two weeks. I have decided to give up because I believe the solution is “cheap”. I would like, if possible, to hear a true solution. Here is the problem:
You’re on a perfectly spherical planet except for a large ditch that runs around the equator. The ditch is 40m deep and 40m wide. You are on one side of the ditch with an unlimited supply of rope and a 20m ladder, and you need to get to the other side. How do you do it?*
Cheap solutions I have found include:
Throw the unlimited rope into the ditch until the rope level rises to the top and walk across.
Break the ladder in half vertically and turn it into a 40 meter pole with little steps (securing it together somehow with the rope).
I have a perfectly acceptable way to get into the ditch using the ladder and the rope without jumping or falling but I can’t get out now. Help!!!
Okay, getting down is relatively easy: you lay the ladder flat on the ground, with only one centimeter of it sticking over the lip of the ditch. Tie the rope to that last cm, and gently lower yourself from the rope. The weight of the rest of the ladder will form a cantilever (I think that’s the right word.)
Once you’re at the bottom of the ditch, walk out into the middle of it and pull the rope. Catch the ladder.
Stand in the middle of the ditch. Start swinging the ladder around in a horizontal circle. (Clockwise, let’s say.) Use the height of your shoulders – say 1.5 m – as the height of this circle. Swing the ladder faster and faster. The far end will just miss grazing the ditch’s north and south walls.
Faster and faster! Build up some real speed! You’ve been eating your Wheaties! Now, when the ladder is pointing due east, push down on it, hard enough so that it will just miss grazing the floor of the ditch when it is pointing due north. Now pull up on it as it spins, so that, when it is pointing due south, it points at a greater than 45 degree angle. Let go of it! It will sail up over the lip of the ditch and land up on the southern hemisphere. Reverse the first step to climb out. (You never untied the rope, right?)
Oh, you say you came from the south in the first place? Um…
Nice idea, but I think that swinging a 20 meter ladder around wouldn’t be easy for even the tallest of peoples.
I’d say screw the rope. Break out the rungs of the ladder, chip off the ends to make them sharp, and jab them halfway into the wall at about 3 foot intervals, pulling out the lower ones as you climb. Use the rope as a harness (waist to highest rung) if this climb will be difficult for you.
First, break the ladder in half as you described before. Now tie the rope to the end of one half of the ladder.
Take this half (the one with the rope) and drive it into the ground as deeply as you can. Take the rope that is tied to this ladder and start twisting it. The rope will start to compact as you twist it. It will wind and coil and curl. Aren’t you glad you hammered one end into the ground? Use the other half of the ladder for leverage if you need to. When you’re satisfied with your winding, grab onto the rope and let it unwind. The rotational motion of it springing back will be enough to hurl you across the ditch.
I can think of a less practical but more amusing way of getting down: Walk all the way round the edge of the ditch laying rope. Make a loop and thread the rope through it, making a noose, hold the end an walk round the planet kicking the noose into the ditch. Pull the noose tight (pull 80 pi m of rope). Walk round the pole, throw the rope over and climb down it, it being anchored to the noose round the ditch at the other side. (Or maybe tie another length of rope to the noose a few miles along and hang onto both, so the ropes can’t slip off the pole).
But I can’t think of a fun way of getting up the other side.
OK, after mulling it over over dinner, here’s a non-facetious answer. Actually, a hint, 'cos I’m too lazy to do the calculations, which are left as an exercise to the reader.
Tie four ropes to the top of the ladder. anchor two at the edge of the ditch, and one away from it. tighten all three so your ladder is standing upright say a meter away from the edge. Tarzan-swing back and forth on the fourth, getting more and more speed each time. When you have enough energy built up in your swing, slide down the rope as you clear the lip of the ditch. Slide down the rope to about 45 meters from the top of the ladder, and when you clear the other lip, you should be just about stopped so you can let go of the rope and pop up onto the other side.
As I said, I can’t be bothered to do the rotational-kinetic energy calculations, but there’s an approach…
No problem. Grab a rung and shove it down until the ladder sticks. Jump up and down on the lowest step. When the lowest rung is in threat of hitting the ground, break it off and repeat with a new one until the ladder’s pretty deep. Or pile that unlimited amount of rope on it until the weight forces the ladder down. Of course, I’m assuming neither that the ladder nor the planet is made of solid adamantium.
Or - take the rope and make a gigantic “X” pattern. Maybe forty miles across. When the Zetans come to yell at you for copyright infringement, whack 'em with the ladder, steal their ship, and fly to the other side of the ditch.
Somehow I imagine digging in the ground isn’t allowed, otherwise the “dig a ramp” solution would be the obvious one.
Here’s a crazy idea. Tie a loop in the end of the rope. Now tie the point 20m from the end of the rope to the top rung of the ladder. Holding on to the other end of the rope, set the bottom of the ladder against a point a few feet down on the wall on your side of the ditch. Lower the top of the ladder towards the center of the ditch, holding onto the “unlimited” end of the rope, until the ladder is almost horizontal. This will provide a cantilever that (almost) reaches the center of the ditch and has a 20m rope with a loop hanging from it. Now take the “unlimited” end of the rope and walk a few miles, letting it out as you go, but keeping the tension on it. After a while, you’ll have enough rope out so the friction will hold the ladder up on its own. Keep walking for a while (it needs to hold you up too, plus some extra force for you swinging on it).
Now go back and climb out to the end of the ladder, shimmy down to the end of the rope, and start swinging. Eventually you will swing up close to the far edge, and if you’re really lucky and/or an action adventure hero, you’ll be able to leap onto the other side of the ditch.
Piece of cake.
Note that this method assumes the ground and the inside edge of the ditch are not frictionless surfaces.
And since a picture is worth a thousand words, and my explanation probably sucks: here’s a picture
One big gotcha with this method is that the ladder is precariously held there, and while you’re swinging toward the far side, you’ll be pulling on the end of the ladder, possibly causing the amount of force exerted on the wall by the foot of the ladder to decrease enough that the foot of the ladder loses its traction, making the whole thing fall down. However, if the ladder itself weighs enough (and a 20m ladder is substantial), it might have enough downward force to keep the foot anchored solidly.
Another gotcha is that at this scale, wind resistance might slow you down and keep you from being able to swing high enough to reach the other side.
Oooh, I really like the cut of your gib, ntucker. I’ve been trying to come up with a solution like yours. Two things that might be small improvements:
You can use a second length of rope, tied to the bottom rung of the ladder, anchored in the same way, to make the ladder somewhat less precariously perched. If you’re not allowed to cut the rope, you can do effectively the same thing with just the one length, but it’s more difficult to describe.
If anchor by friction doesn’t work, there’s another way to anchor the rope. Walk around the planet at the equator, leaving rope in the ditch as you go along. When you get back to where you started, tie it off, making a loop that encircles the equator. Make sure the loop is too small to leave the ditch. Then tie a piece of rope to some point on the loop, and walk due south carrying this second rope with you, until you get to the south pole. Keep walking, and when you get back to the ditch on the other side, you’re carrying a well-anchored rope.
Yeah, shade mentioned the “loop of rope around the world” method. I was trying to come up with a way that didn’t involve walking around the whole world.
Anchoring the foot end of the ladder in my scheme is a good suggestion.
Yes, but combining our methods seems most satisfactory to me. I feel the planet has to be there for something and your solution is perfect for getting across the ditch.
Aha! If the planet is frictionless use my method to anchor the rope. If not, use yours. THey complement each other.
