Indiana Jones and the Golden Rope by Pradeep Mutalik
Indiana Jones is hot on the trail of the legendary gold of the Aztincas. With a comfortable head start on his evil pursuers, he stands by the side of a tropical mountainside deep in the jungles of the Andes, at the threshold of the fabled rock-cut mountain cavern of King Auzoctemuma. He enters, and is dazzled by the sight that greets him. The cavern is huge, and in its ceiling are embedded two large metal rings about a foot in radius and a foot apart, from each of which hangs a thick rope made of an alloy that is almost pure gold! Indy’s pulse quickens as he realizes that he has reached the end of his quest.
Each of the two magnificent ropes, which are of identical width and length, reach all the way from the ceiling ring to the floor, which is a distance of 108 feet (108, as is well known, is a mystical number in many cultures). The ropes are so thick that Indy can just about get his fingers around them and yet they are flexible enough to be tied into a tight knot that will not unravel. The whole ring and rope system is strong enough to support the weight of dozens of men.
Indy estimates that he can scale either rope all the way to the top in about 20 minutes, and then slide down in almost no time. (He has half an hour to leave before his evil pursuers arrive). He has with him a diamond-edged knife from an earlier adventure, so he knows he can straddle the metal rings and cut either rope. Jumping from that height would of course be fatal, but Indy is confident he can safely leap on to some loose earth when his feet are 10 feet off the floor. As we all know, Indiana Jones is a master of the knee-bend-and-roll technique of jumping. (Don’t try this at home!). Indy’s outstretched hand reaches 8 feet high when he stands upright on the cave floor.
So here are your challenges. Try either one:
1. What is the greatest length of golden rope that Indy can make off with? Describe exactly what he would need to do to achieve this.
2. When Indy starts climbing the ropes, he finds it hard work as they are slick. He can ascend 20 feet in a minute, but then he has to rest for a minute, during which he slides down 10 feet. In how much time does he reach the top (assume he starts climbing with his hands at a height of 8 feet)?
And for you math-heads here are some more challenging questions.
In actual fact, Indy is human and does get tired a little. So only his first 1-minute ascent is exactly 20 feet. He recovers during the rest period, but not one hundred percent. His second 1-minute ascent is a fixed percentage of the first one, and the next is the same fraction of the previous one, and so on. He always slides down 10 feet during the 1-minute rests. In addition to this, he can draw on his adrenaline reserves and make one “heroic spurt” where he can climb twice as much in a particular minute as he would have normally, but from that point on he also needs to rest for 1.5 times as long every time, and therefore slides down a proportionally larger distance.
What is the minimum “recovery percentage” that he needs to make the climb in half an hour? If you forget about time, what is the minimum “recovery percentage” to make the full climb at all? How long does it take then?
And while Indiana Jones gathers his large length of golden rope, you word-lovers are going to make a long rope or chain of words too. Consider the following sentence:
As you can see, at least two letters at the start of every word are the same as the ending letters of the previous word. Can you make such word-chain sentences that make sense? You get more credit if your sentence is topical.
The Indiana Jones and the Golden Rope puzzle was unusual in that the main puzzle did not involve mathematics or abstract logic, but as befits an adventurer like Indy, practical strategy. In this puzzle, Indiana Jones finds two 108-foot-long golden ropes hanging down from two 1-foot radius metal rings a foot apart, embedded in the ceiling of the rock-cut cavern of King Auzoctemuma in the Andes. Your task was to determine how much rope he could make away with using his diamond-edged knife to cut it off, assuming he has time to climb the rope to the top once (which he estimates he can do in 20 minutes). You can read about the detailed conditions here.
Versions of this puzzle can be found in books and on the Internet, and the “official” answer is based on realizing that there is a set of manipulations that allow Indy to make away with the entire length of both ropes. Briefly, Indy ties the ropes at the bottom, climbs up to the top, cuts off one rope at its ring, pulls the cut end through its ring till the knot between the two ropes is at the top, cuts off the second rope at its ring, grabs the combined rope with both hands, one on either side of the ring, shimmies down the rope with his weight equally distributed and then pulls the combined rope through the ring to the floor. This solution was found by many posters. The very first poster, Moses, even described the kinds of knots Indy should tie and precisely estimated the time each step would take.
This may well be the official solution in all those books and sites, but TierneyLab readers are smarter than that. Several readers including Len, a science teacher, pointed out that 108 feet of thick golden rope would weigh more than a ton, and castigated me for perpetuating bad science for the sake of “good” math. Well, mea culpa. Although there was some wiggle room in the wording — I had said that “it was a thick rope made of an alloy that was almost pure gold,” — I had not thought about these details, which I should have, because it was supposed to be a practical problem, not just one of abstract logic.
So let me make amends. Let’s look at some of the physical parameters of this problem.
The solution just described could only have been possible if the rope weighed something like 50 to 100 pounds rather than a ton, in which case it would be impossible to pull it up through its ring. Reader Jesse calculated that a 50 lb rope, made of pure gold, would only be two millimeters thick. Amazingly, a pure gold rope this thick would support about 200 pounds before it started yielding (the yield strength of hard gold is 200 megapascals, which converts to about 30,000 pounds-force per square inch). To be as thick as described, the rope would have had to be made of some other conventional material with this much gold fiber weaved in. Such a rope would certainly support a dozen men. But how much would it be worth? At current prices (about $950 an ounce), the 100 pounds of gold in the two ropes would be worth about $1.5 million. Of course, as an ancient artifact prized for the gold-fiber rope braiding technique of the Aztincas, it would have been worth far more, and we can assume that the famed archaeologist Dr. Indiana Jones only wanted to preserve it from vandals in order to later restore the cave hall of the most famous king of the Aztincas.
All this can be imagined post-hoc, but what if the ropes really had a significant proportion of gold, as the original puzzle plainly implied? As several readers calculated, a rope of pure gold, with no volume lost due to weaving, that was as thick and long as was described, would weigh about 2,500 lbs. A couple of readers suggested that Indy just cut off as much of the ropes as he could carry and then hightail it out of there. But that’s for wimps, right? Indy needs to do better, so let us imagine that the ropes actually had a significant proportion of gold and weighed 600-900 pounds or more, (making the gold alone worth closer to $15-20 million). Could Indy still get all the rope? Reader Hugh Crawford had the following ingenious suggestion:
He’d cut off about six feet of rope from one of the ropes, and tie the two ropes together seven feet above the ground. Then he would climb to the top of the other rope carrying the six foot length. At the top he would run the six foot length through the two rings and tie it to the two ropes just below the rings. Then Indy would cut the two ropes from the rings and descend to the ground. There he would cut the shorter rope as high as he could reach above the knot, and pull the other rope end which would pull the rope through the two rings to the ground.
So, by carefully splicing his 6-ft piece to the two heavy ropes using his knife, Indy could conceivably get the ropes to the ground and emerge unscathed, if he took care to get out of the way of a ton of falling gold. Hugh’s solution also explains another puzzling feature of the puzzle: why did Indy take so long to climb a hundred feet? Top rope climbers can do this in a couple of minutes. Well, if he had a 50-75 lb loop of gold rope tied around his waist, it would take longer wouldn’t it?
There were a couple of other questions in the original puzzle:
If Indy ascended 20 feet in a minute, but then he had to rest for a minute, during which he slid down 10 feet, how long would he take to reach the top?
It’s easy to see here that in 2 minutes Indy goes up 10 feet. It’s tempting to reason that therefore it would take 20 minutes to climb 100 feet. If you did that, you would have banged your head on the ceiling far more often than is good for you! The trick here is to see that it will take 16 minutes to reach 80 feet. When you ascend 20 feet in the next minute, you reach the top, and the climb ends. So the answer is 17 minutes.
The second question sketched a more realistic scenario:
Suppose Indy tired a little and climbed a fixed fraction of his previous effort from one 1-minute ascent to the next. In addition, he could use one “adrenaline burst” during which he could climb twice as much as he did normally, but then would be drained, and would slide down 1.5 times as much during his rests. You had to determine what his minimum “recovery percentage” needed to be to climb up in half an hour, and to make the full climb at all.
The answers were 95.985% to climb up in 29 minutes, and 95.965% to climb the rope at all, in which case it would take 31 minutes, as determined by several readers including Tom E., Joshua, rusty and Sailesh whose excellent account you can check for details. The answer requires repeatedly multiplying the recovery fraction by itself for every one-minute ascent (effectively raising the recovery fraction to a power equal to the number of ascents), and using the adrenaline-fueled effort only at the end, which is the only time it is useful. Note how a difference in recovery percentage in the second decimal place results in a difference of 2 whole minutes! Even more surprising, if the recovery percentage is less than about 96% percent, Indy cannot reach the top at all! The reason is that when you multiply the recovery fraction (about 0.96) by itself many times (17 in this case) the product falls to less than one half. So Indy cannot even climb 10 feet (half of 20) in his 17th one-minute ascent, and from then on slips down more than he can climb up. This might be termed the Fractional Law of Diminishing Returns: when you take a proper fraction, no matter how close to one, and multiply it by itself many times, it falls down to zero pretty rapidly. This is the reason why, in repetitive mechanical processes, things run down without external replenishment, no matter how close to 100% the energy recovery from one cycle to the next is. Thus, a ball that bounces to 96% of its release height, will bounce to less than 50% on the 17th bounce. After a hundred bounces, the height will only be about 1.5% of the original. When returns are diminishing, no matter how by how little, they fall down to near zero pretty darn quickly.
Where did the name of the king Auzoctemuma come from? Readers Nathan, Tom E. and LW correctly pointed out that the name of the king comprised “Au,” the chemical symbol for gold, and an anagram, or quasi-Spoonerism of “Moctezuma” the name of an Aztec emperor (better known to us as Montezuma).
“Wait a minute,” you say, “how can Indy haul away close to two thousand pounds of golden rope?” Ah, here the original problem said nothing, so we are free to use our imaginations. This is Hollywood after all! Why, he could have a gasoline-powered heavy-duty winch lowered to the site by a cargo helicopter, which was arranged for him by a general in the local armed forces whose life he had saved earlier in the movie. The winch would gather up the rope, be hoisted up, and finally, Indy himself would perform another rope climb, this time into the helicopter. Indiana’s pursuers would, of course, just arrive on the scene, and he would dodge their bullets, and unerringly take out their leader while precariously clinging to the wildly swinging cable dangling from the helicopter.
Of course, if Indy could have arranged for this elaborate escape, why didn’t he have the foresight to equip himself with a 108 ft rope to begin with? (If he didn’t have one he could have perhaps tied six of his spare bullwhips together.) This would have rendered our problem irrelevant. Well, why didn’t he? Don’t ask! After all, as I said, this is Hollywood!
So we’ve got the aerial stunts, the adventure, the escape shoot-out, a significant part of the plot and the score, and even the obligatory logical snafus. We’re all set to script the next Indiana Jones blockbuster! All we have to do is get George Lucas and Steven Spielberg to read this column. George, Steve… are you listening?
Last edited by Rocket Surgeon : 08-08-2009 at 01:14 PM.
Reason: This has to be the largest Spoiler ever!