# Putnam Exam Prep Questions

erraticimpulse
"You are locked in a 50x50x50-foot room which sits on 100-foot stilts. There is an open window at the corner of the room, near the floor, with a strong hook cemented into the floor by the window. So if you had a 100-foot rope, you could tie one end to the hook, and climb down the rope to freedom. (The stilts are not accessible from the window.) There are two 50-foot lengths of rope, each cemented into the ceiling, about 1 foot apart, near the center of the ceiling. You are a strong, agile rope climber, good at tying knots, and you have a sharp knife. You have no other tools (not even clothes). The rope is strong enough to hold your weight, but not if it is cut lengthwise. You can survive a fall of no more than 10 feet. How do you get out alive?"

Is it just me or does this question seem to have a simple answer? You just cut the two ropes from the ceiling, tie them together via knot tying skill, and then again onto the hook.

Homework Helper
I guess the problem is how to cut the ropes down from the ceiling without falling.

neurocomp2003
yeah...the porblem is how to cut the 2nd rope without killing yourself
some of the putnam problems are extremely easy(usually 2-3) so don'ot worry if you come across one...its to get people comfortable in the exam(well that's what i found -remember to use recursion and stats.).

the solution to the problem i think is to get to one of the stilts...though i can't see how the room is supported if the stilts don't cover all four quads.

erraticimpulse
heh yeah I missed that :tongue2: I only considered the height of the stilts and not the room.

Homework Helper
No, I have a solution that only uses ropes. It gives about a 2.5 ft fall from the ceiling and a 5 ft fall from the tower, if that helps.

Gold Member
Cut 50 feet of rope from the ceiling. Tie it to the hook. Climb down the rope. Swing to the stilts. Climb down.

Gold Member
How tall are you in the problem?

erraticimpulse
Hmm... I think I know the answer. Hey StatusX, without giving it away can you tell me if cutting some of the rope, cutting it lengthwise, and taking it up with you to do something is apart of the solution? Nothing says that the rope can't carry half your weight if you use both ropes.

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erraticimpulse
any more insights from anyone?

mruncleramos
I use my extraordinary power of teleportation to get out.

Homework Helper
No, you don't cut the rope lengthwise in my solution. I don't see how that could help anyway, since you would need at least two halves to support your weight at any given time, which is the same as one uncut rope. Here's another clue: there are two cuts and two knots in my solution, plus another knot tied to the hook. By the way, I just realized there's actually 2.5 extra feet of rope for the drop from the ceiling. And those numbers are all a little arbitrary (I picked them to be realistic to a person's size, but you could use the same basic idea but change the numbers a little and stay within the acceptable ranges).

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Staff Emeritus
Gold Member
It's often good to try and set bounds on what you must do, can do, and can afford to do.

An example of each:

(1) When you make your final cut, you must be hanging from at least 40 feet of rope.
(2) You can cut 10 feet off of one rope and jump back down without involving the other rope.
(3) You can afford to leave 10 feet of rope hanging from the ceiling.

All of those are almost obvious, right?

Then, once you've set down enough of these constraints, you try to piece together things you can do and afford to do until you get the solution, or you try to identify and correct assumption that's blocking you from seeing the answer.

Staff Emeritus
Gold Member
Next hint in white. (don't look)

Well, we've determined:

(1) You must have 40 feet of rope above you when you make your final cut and drop 10 feet.
(2) You must be able to retrieve all but 10 feet of the rope after doing so.

...

erraticimpulse
Firstly I really do appreciate the hints and advice. However, I don't see what your solution is. You do know that you can't just pull the ropes out of the ceiling right?

This solution stems from the idea that the rope cannot hold your body weight if it is cut lengthwise at any point. So the result will be the climber at the top holding onto both ropes with a portion of the ropes cut lengthwise and retied at the top. Here's my solution albeit complicated.:

1. Cut a foot or so from one of the ropes, cut it in half, and then cut both halves lengthwise.
2. Climb one of the ropes while carrying the spliced ropes, the knife, and the end of the rope you are climbing.
3. Tie the end of the rope a foot or so down from it's source in the ceiling thus creating a large loop.
4. Cut a little below the tie and then retie it using one of the spliced ropes.
5. Now cut the other rope and retie it in the same manner using the spliced rope.
6. At this point grab both of the ropes in your hand(s) below the points where the splices are tied on.
7. Now untie the one rope so it no longer has a loop in it.
8. At this point climb down making sure to be using both ropes at the same time.
9. When you reach the bottom climb each rope separately to use your body weight to snap it since the rope cannot handle your body weight where it has been cut lengthwise.

practically a hundred feet of rope to play with

Homework Helper
I think that would work. This was my idea:

1. Climb up to the top of one of the ropes, and cut the other one about 5 feet from the ceiling, making sure to hold on to the 45 foot segment.
2. Pull up the loose end of the rope you're hanging on and tie it to one end of the 45 foot segment.
3. Tie a loop at the end of the 5 foot segment hanging from the ceiling.
4. Now swing over to the 5 foot segment, and while hanging on, cut the end of the other rope as high up as you can (you now have a 95 foot length of rope), and put this end through the loop.
5. Pull it through until an equal amount is hanging out each end of the loop.
6. This gives about 47.5+5 feet from the ceiling, more than enough to climb down.
7. At the bottom, pull the rope down, tie it to the hook, and use it to climb down the tower.

Staff Emeritus