# A rope-cutting problem

Here's another oldie from a book, slightly paraphrased to make it more quantitative.

You're in a room with a ceiling exactly 100 feet high. Two thin but sturdy ropes are hanging from fixed hooks at the ceiling and just touch the ground, close enough that you can grab both of them at the same time comfortably. You can climb these ropes, but there is nothing else to climb on. Equipped with only a knife, how much of the rope can you cut down and still end up on the ground unharmed if you can only fall 20 feet without injuring yourself?

• I suppose 120 feet is too obvious. Commented May 16, 2014 at 3:07
• You can get way more than 120 feet.
– user88
Commented May 16, 2014 at 3:08
• I'm assuming you can tie ropes together as much as you want without them losing length or sturdiness? Commented May 16, 2014 at 3:40
• @IQAndreas Yes.
– user88
Commented May 16, 2014 at 13:27
• Should the question either indicate 1) how far apart the hooks are, or 2) that you are able to move them? As written, I imagine hooks as being stationary, so it seems important to know how far apart they are. Commented May 16, 2014 at 14:33

## 5 Answers

All 200 feet, as long as they're no more than 100 feet apart.

Take one end of one rope. Climb up the other. Tighten the first rope as much as possible, cut (or unhook) the one you just climbed up (keep ahold of it) and swing on the intact rope. Climb to the top of that one. Put the cut rope through the hook. Tie yourself a harness of some sort on one end of it, keep hold of the other, and cut/unhook the rope originally there (again keeping hold of it). Lower yourself down until you are near the end of the rope you are attached to. Tie the spare rope to the end of it, and keep lowering yourself. Once within your safe fall zone, undo the harness and let go that side of the rope, and once you're down pull it through the hook.

As it has been clarified that "the hooks are close enough that you can comfortably reach both of them," the description can be somewhat simplified. Climb up rope A, unhook B, tie it into a harness and re-hook it (such that you can lower yourself), unhook A and continue as above (lower halfway, tie A to the end of B, lower the rest of the way).

You can cut 199 feet of the rope. Climb to the top, cut one rope, and cut all but 1 foot off the other. You don't need to fall... If you have 1 or 2 feet at the top, you can make a loop to hold on to.

If you want to get to the ground, you can cut 200 feet. Cut one rope and feed it through the hook, and tie one end around your waist while holding onto the other end. Cut the second rope, and tie it to the first. Then, holding one end, lower yourself down by letting it out.

• I can tie my waist and before lowering myself, cut the remaining 1 feet of the rope too! got all of 200 feet, not 199! Commented May 16, 2014 at 3:17

So far as people have bested the solution I could come up with, I suppose I'll post the one that was in the book as well (again paraphrased to account for some changes I made):

Suppose the ropes are called rope A and rope B. Climb up rope A and cut about 99 feet off of rope B, leaving just enough to tie a loop with the remaining top portion. Tie this loop. Slide back down rope A, and tie one end of the cut portion of rope B to the bottom of this rope. Now, taking the other end, climb back up to the top, and feed that end through the loop until it's close to touching the ground. Then, swing over to the double rope you've just made, and slide rope A off of its hook, letting that end fall to the ground as well. Climb down this double rope which is hanging about 6 inches off the ground, and then pull on one of the ends to get it all down, which is about 199 feet.

Of course, if you assume that you can feed the rope through the hook like a pulley, as other solutions have done, it also becomes possible to get all 200 feet, by eliminating the "forming a loop" step:

Climb up rope A and take rope B off of its hook. Slide down and tie rope B to rope A. Taking the other end, climb back up the the hook and slide the end in your hand through the hook until it touches the ground. Then, swing over to the double rope, and slide rope A off its hook, letting that end fall to the ground as well. Climb down your double rope, and pull on one of the ends to slide the other end off the hook, getting all 200 feet of the rope down.

I've since found the old puzzle book that this puzzle came from (Giant Book of Mensa Mind Challenges, 1998 ed., section 1, puzzle 196). The solution in there is presented as follows:

First, tie the lower ends. Then climb up the first rope and cut the second one, close to the ceiling, leaving an end long enough to form a loop. (You can instead cut it off entirely if you slide the rope through the hook.) Hanging from the loop, cut the first rope at the hook. Be careful not to drop it. Then slide the rope through the loop until it's even. Climb down the double rope, and once on the floor, pull one end to get it all.

This solution involves grabbing hold of the loop or hook and hanging from it while working with the other rope, which seems a bit dangerous. I prefer my strategy (which is the same except doesn't involve hanging 99.5 feet off the ground) which involves one more up/down trip but never puts you in a situation in which you might conceivably end up stuck hanging near the ceiling if you're not careful.

• I think your second solution is more practically plausible way to do it (to get 200 feet), compared to other answers, which require you to tie a rope while you are still clinging to a rope. I mean, how can you "create a harness" with one hand? Commented Sep 4, 2014 at 7:28
• Some people can do crazy things with one hand. Like solve a Rubik's Cube. Or solve a 4x4x4 Rubik's Cube.
– user88
Commented Sep 4, 2014 at 16:04
• @justhalf I can easily tie a bowline one-handed, so it is very reasonable to be able to do so. And it's not like I spent a lot of time figuring out how to do so either. Commented Sep 30, 2014 at 22:16
• Haha, ok, but for me it seems very difficult to hold on a rope in one hand and your legs (not to say it's 200 feet high), and knot a tie using the other hand (and later use it to support your own weight!). Seems like a good stunt. Joe's second suggestion is much much easier, in my opinion. Commented Sep 30, 2014 at 22:43
• I would imagine that you could hook your legs to trap the rope between them and then just hook your arms around the rope leaving both hands free while your legs support most of your weight and your balance is kept by the rope in the crook of your elbow or some such. Shouldn't be that hard if you can climb up the rope already. Commented Oct 2, 2014 at 14:33

An approach which would be much more dangerous, but would be usable even if there were only one rope would to climb each rope (or the rope) and slice partway through it while leaving it strong enough to hold about twice one's weight, and then slide very carefully down it rope until twenty feet from the bottom. Lift up on the rope from underneath until one has reached a spot about ten feet from the end. Grasp that spot firmly and let go of the other place one is holding. Depending upon the elasticity of the rope it may be necessary to adjust one's choice of locations, but it should be possible to momentarily tug on the rope which exceeds one's own weight by a factor of at least four, thus snapping the rope above.

You can cut 170 and up feet.

• Cut the 2 ropes in half length-wise (they will still be 200 feet long.)
• Then cut the rope A or B.
• Afterwards go 20 feet up on the remaining rope.
• Cut as high as you can reach with your knife. This will vary depending on arm height.
• You fall 20 feet (even though you cut more rope, that is above you.) but am unharmed.

You can also cut as much rope as you want if you have protection where you fall.

• How do you grab the rope to cute it lengthwise?
– user88
Commented Jun 6, 2016 at 20:31
• @Joe Z. Take it off the hook, cut it lengthwise, then put it back on. Commented Jun 6, 2016 at 23:55
• @Joe Z. Also you said cute not cut. Commented Jun 6, 2016 at 23:56