If you were Wile E. Coyote, you could use the ladder and rope to build a crossbow and launch yourself across. Extra points for hitting the far side inches below the edge of the ditch and allowing Roadrunner to run across on your back.
So far I like ntucker’s idea the best. It requires the least amount of rule breaking. I don’t really like the idea of giving mass to the rope because once you do that you might as well fill up the ditch with rope as stated before.
My assumptions have been that you cannot alter what is presented:
- the ladder is 20 meters long, and made out of a material that is unbreakable.
- the area of the planet where the ditch is cannot be altered or penetrated in any way.
- The rope is simply a tool. To give it mass drastically changes the scope of the problem.
This pretty much reduces the problem to only physics and geometry.
I got down like this:
Tie end of rope to last rung on ladder. Lower the ladder into ditch. Walk around the earth with rope and repel down into the trench, the other end of the rope is braced not by friction but by the ladder.
Now that I am in the ditch I walk all the way back around to where the ladder is. Now I am stuck.
Why is “giving the rope mass” cheating? Rope has mass. It’s got friction, and a mile of rope laid out on the ground would probably actually make a decent anchor if it’s extended out in the opposite direction you’re pulling.
I consider this an order of magnitude more practical than filling a large ditch up with rope of an unspecified thickness, because it’s still doable if you specified the rope to be finite but very long, such as 5 miles.
(and LowerLip, I’m a computer programmer)
Okay, here’s my attempt
The rules that I’m following are that you can’t fill the ditch, you can’t embed anything into the side of the ditch, the rope and planet both have friction, that you can throw a small amount of rope about 6 meters, and that you have a lot of time available.
1: Walk around the planet and build the loop at the top of the ditch on your side. (A)
2: Lower a rope attached to (A) so you can get to the bottom of the ditch.
3: Hi, Opal
4: At the bottom of the other side of the ditch, build another loop around the world. (B) (It’s a good thing we have unlimited rope!)
5: build a loop of rope that will be long enough to go around the top of the other side, with a rope hanging off that can be climbed. (D)
6: Run a rope from (A) to (B) and stretch it taut. ©
7: Place the ladder at the midpoint of ©, perpendicular to © and secure with ropes to (A) and (B). The end of the ladder will then be ~8 meters from the top of the other side.
8: Using your phenomenal sense of balance (and maybe some tethers to keep you from falling to your death), stand on the very top of the ladder and throw a small part of loop (D) over the top of the other side. Done carefully enough, It will hopefully stay there due to friction.
9: Keep repeating steps 6-8 all around the planet until (D) is firmly around the planet.
10: Tighten the loop (D) and climb up the rope you left attached.
Excellent idea (steps 1-7). However, I don’t see how you’re going to tighten loop D once it’s all the way up on the ledge. But how about if you combine our ideas: build your taut rope going from point A to point B, then place the bottom of the ladder up high enough on the taut rope (yet still perpendicular) so that the top of the ladder is above the point halfway across, then swing from the end of it to the other side. One of the main drawbacks of my initial proposal is that it requires you to swing all the way up to horizontal, and even then you’ll necessarily be some distance away from the far edge. If you can get the point you’re swinging from above the midpoint and more than halfway across, you’re much better off. Here’s another picture. Add some guy wires going from the top and bottom of the ladder to points on ropes A and B (but off to the side), and the whole thing is pretty stable.
Note that this configuration has the problem of the ladder being in the way of getting a full swing going, but that problem can easily be remedied by leaning the ladder to one side or the other. With the guy wires, this should be achievable without sacrificing stability.
I think you’re right.
In order to avoid swinging into the ladder, you could set up another set of ropes from A to B that ran right beside the ladder and get a good running start on your swing. That should give you all you need to get across.
You don’t need to swing from a standstill. You could stand on the South ledge holding the rope and just swing Tarzan-style across. Assuming ideal conditions, you would reach up just as high as you started. This can be used in either ntucker’s first idea, or the modified one. You wouldn’t need to swing through the rest of the construction, either. Just stand off to the side a little. The rope would have to be slightly longer, but the same idea applies.
Here’s one that works using a frictionless surface (well, you have to be able to get around somehow, and you need the ladder to have friction, so let’s say no friction in rope/planet, excluding the ditch. It does assume elasticity in the rope, however.
1.Make two anchor loops (all around the planet), with one (anchor #1) slightly longer than the other.
2.Wrap the rope into anchor #2 on the other side of the planet, and tie to one end of the ladder.
3.Position the ladder perpendicular to the lip of the ditch, with about a foot overhanging, and hook anchor #1 over that end.
The whole setup should be : the overhanging part of the ladder is pulled down by the long loop, with the other end secured by the rope to anchor #2.
4.Get on the ladder.
5.Cut/untie the rope to anchor#2
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Go flying over the ditch as the overhanging end of the ladder pulls you down.
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Land on the other side.
I have two possibilities which might work if you are allowed to anchors.
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Assuming that the ladder is unbreakable but flexible I’m imagining a situation where you could use the rope to “bow” the ladder. Anchored properly (using the rope to lash it to a large rock or set of trees, etc.) you can create a large slingshot and/or catapult. You may not live after you’ve been launched but you could probably make it to the other side provided you weren’t too heavy and were able to generate enough force.
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Anchor a series of ropes (say 6 or so) at the top ledge. Lower yourself to the bottom of the trench along with the ladder. Place the ladder against the opposite side of the ledge and anchor the other end of the ropes as high up on the opposite trench wall as possible. Stand on the ropes and haul up the ladder. Position ladder on the ropes you have now anchored and climb up the ladder.
Grim
That does require the tension in the rope to be something like 60 times your weight, plus 30 times the weight of the ladder. If it’s a strong rope, I guess it could handle this, but when you’re setting it up, you have to pull on the rope with a force equal to this tension. Better do some warm-up exercises first.
I have discovered a truly remarkable proof which this margin is too small to contain . . . 
Here’ my guess:
I would reach the bottom using **Shade’s[\b] method, but would tie several ropes to his loop-around-the-planet to use later.
Then I would use the ladder to tie a rope around the other (destination) side of the planet 20+ m above the ditch floor. Tie the ropes from **Shade’s[\b] loop-around-the-planet to this new loop-around-the-planet. Use these ropes to balance the ladder 20+m above the ditch floor, leaning against the destination side of the ditch, and climb up the rest of the way.
I can’t provide a picture. Sorry. 
Nah, that won’t work. Sorrier!!
The thing I don’t like about my first method is that because of the angle of the ladder, the top end of it doesn’t quite reach to the center of the ditch. If it falls short of the center by X meters, then even if you swing all the way to horizontal on the other side, you’re still going to be 2X meters away from the edge. That could be quite a stretch.
Just in case, I suggest all solutions dangle a piece of rope into the ditch from the starting side, just in case you miss (and survive the 40m fall) and need to get up and try again. 
Using the catapult idea, you could construct Brave Sir Robin’s tightrope from steps 1-6, but leave it a little slack. Then stand the ladder up against the destination wall so that when it falls over, it will hit the slack line, pulling it taut. Then climb 3/4 of the way up the slack line, hold on, and pull on the piece of rope you cleverly tied to the top of the ladder. The ladder will fall over, hit the part of the rope below you, pulling it taut, and catapulting you to safety. Or directly into the face of the wall you want to climb. Or maybe nowhere if the ladder isn’t very heavy. See this picture for a rendition of this ridiculous plan.
At any rate, 40m is a long way to be thrown, so I don’t think catapults are very practical.
Is anybody else tempted to suggest more outlandish ideas just to see ntucker draw pictures of them? Those are great!
Remember that you yourself are about 2 meters (give or take) I think that’s enough to make up for the missing X due to the imperfectly stright ladder.
assumption 1. rope anchors allowed
assumption 2. normal v shaped ditch
solution:
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attach ropes to top and bottom rungs. The top rope should have a loop tied into it about 10 meters from the rung. a third rope should be attached to the bottom rung and threaded through the loop and back down to the top rung temporarily secured there with lots of slack.
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lower ladder till top is even with the surface of the planet and the ladder resting on the slope.
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while securing the bottom rung with the length of rope from the top of the slope, swing the top of the ladder over so that it fall/wedges onto the opposite slope somewhere near half way down. control the swing with the second rope and secure it taut at the top of the initial slope once the “top” rung is in place on the opposite side
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having secured the ropes with anchors at the top,climb down the rope resting on the near slope till you reach the ladder and walk across the ladder to the opposite slope.
5… secure the ladder at the opposite slope at the bottom rung with a short piece of rope and an anchor.
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Walk back to initial slope (this could have been done earlier) and cut the rope from the top of the initial slope.
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attach rope at this end of ladder at this point and bring the rope back to the opposite side.
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now that you are positioned at the secured top rung on the opposite bank, pull the third rope through the loop in the second rope mentioned in step 1, to begin rotating the ladder up from the initial slope. The loop had been position earlier on the still secure and taut second rope mentioned in step 1.
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When the third rope pull is no longer viable, pull on the more recently attached rope from the previously bottom rung which is now the top rung. you could also pull/push on the ladder sideways to comeplete the full rotation.
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the ladder should fall against the opposite side, secured at the bottom and within reach of the top so all you have to do is scramble up.
The slope is 44.72m. If you secured one end of the ladder halfway up, and then hoisted it up from there, the top would be 2.36m (7’9") from the edge of the ditch. That’s out of reach for a lot of people.
The highest up the Northern wall you could go is 25m, which you could do by anchoring the Southern end 15m from the bottom. Then the ladder would reach all the way to the North edge after you hoisted it up.
I’m not sure I trust rope anchors like you used in step 5, and I’m not sure the ditch is supposed to be triangular. But I think it’s a good idea.
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if the ditch was triangular, wouldn’t the angle have been specified?
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if you can anchor ropes in the ground, couldn’t you just walk up the wall by anchoring ropes, pulling yourself up, and anchoring another rope as high as you can reach?
How about using the ladder in a diving board type setup?
Wrap the rope aropund the planet a few feet from the ditch with one end of the ladder tucked under it and the other end extending out over the ditch. If the rope is tight and springy and you have good balance you can bounce on the tip of the ladder untill you have enough energy built up then lean foreward a bit into the last jump and boing! (followed by much cursing and a splat but it worked)
Surely the ditch has a rectangular cross-section, but even if it were triangular no angle would need to be specified as we know the base and height are both 40m (I’m assuming it’s an isoceles triangle you have in mind). Anyway, to my mind if this is anything other than rectangular then that should have been specified.
I’m not sure that we can anchor ropes at any points on the far side of the ditch, not even at point “B” at the foot of the far side – running a taut line between A and B, what stops B moving towards the near side?