QuoteOriginally posted by: bhutesBut then, the analogy maynot be that strong ... rope-world is not really angle-world.If quatro friend found a solution (even though, hard) ... the analogy will be disproved. (because if 1 hour can be trisected .. then given any-minutes ropes .. it could be trisected)agreed. I'm looking forward to the solution too.

QuoteOriginally posted by: vplanasAttach a weight in one of the ropes so that you have a pendulum (with unknown characteristic frequency). Put it to oscillate. Burn the second rope and count the number os oscillations in that hour.Now you have a calibrated clock to measure any time.(of course, assuming no friccion problems).I love this answer. I think it's the leader so far (completely meets the spec of the problem, and it doesn't require 'infinite cutting'), and it gets extra points for lateral thinking.Here are a few, progressively more ludicrous (not making fun of vplanas's excellent idea, just joining in the fun)Point one end of the rope at the sun, and start burning the other rope.When the other rope is done burning, point the other end at the sun.Trisect the angle.When the sun has crossed your new measure, 20 minutes has elapsed. Use one rope to demonstrate the magical 'burning in one hour, without uniformity' ability to people.Use that demonstration to sell the other rope to someone from the amazed crowd.Use the funds to buy a watch.Tie a rope around your chest and the other end around something secure, try to keep this rope out of sight.Tie a noose around your neck and tie it up, with enough slack that if you fall, the chest rope will catch you.Stand up and yell crazy stuff about killing yourself until the police come.Make a demand that they wait exactly 20 minutes. You might also get to demand a pizza or something.-t.

There are clouds, there are no people and the nearest town is at 300km.Seriously, i haven't found a solution with finite cuts. Maybe it doesn't exist!

QuoteOriginally posted by: quantorThere are clouds, there are no people and the nearest town is at 300km.Seriously, i haven't found a solution with finite cuts. Maybe it doesn't exist!Vplanas's solution doesn't require the sun, people, or a nearby town.-t.

Make the two ropes overlap as below: <----------------- rope 1 --------------->1a ----------------1b -------------------1c |--------------------|----------------------|-----------------------| (approximate)--------------------2a -------------------2b --------------------2c---------------------<------------------- rope 2 ---------------->now light the rope simultaneously at 2a , 1c, and 2c.Does this make sense?(soory about the terrible diagram, the spacebar doesn't work)

QuoteOriginally posted by: tristanreidQuoteOriginally posted by: quantorThere are clouds, there are no people and the nearest town is at 300km.Seriously, i haven't found a solution with finite cuts. Maybe it doesn't exist!Vplanas's solution doesn't require the sun, people, or a nearby town.-t.But it does not work inside a spaceship travelling with unknown accelerated motion.Henny, I do not see how can you get there the solution. the ropes do not have any kind of uniformity so the middle of a ropa (point 1b) it's not neccesarely the middle of the burning time).

QuoteHenny, I do not see how can you get there the solution. the ropes do not have any kind of uniformity so the middle of a ropa (point 1b) it's not neccesarely the middle of the burning time).vplanas, you are right. wrong thinking. I am beginning to think this is impossible in the sense that you cannot exactly represent 1/3 in decimal.

Tristran and vplanas,I like the cutting solution and the pendulum solution a lot - nice...

Thank you, Etuka. I also like the tristan construction.I don't know if the analogy with the trisection of an angle can be applied here. actually, it would be more like the trisection of a segment in a metric space with unknown metric. vplanas

QuoteOriginally posted by: vplanasThank you, Etuka. I also like the tristan construction.I don't know if the analogy with the trisection of an angle can be applied here. actually, it would be more like the trisection of a segment in a metric space with unknown metric. vplanasI think that this analogy -"trisection of a segment in a metric space"- is farther off than the analogy with the "trisection of an angle"(Though, I am somewhat, but not fully, agreed to the angle analogy too) -- rope starts looking like an angle, if we start looking at the rope as an angle . In other words, if we restrict our ability to manipulate the rope in only 2 ways (1) use the rope in whole, (2) bisect the rope, then the rope will behave just as an angle (you can do both those things with the angle). In fact, rope is a restriced angle, because there is no pi equivalent (or we have to treat infinity as the pi in rope-land). But, yes, there is no solution to 20 minutes ... if the rope if bound by only two types of manipulations (as stated above).(As hongyi defined ropeland, it's defined by allowed manipulations on the rope .... not existence of instruments corresponding to a compass and a straight-edge) -- that is what I find very restrictive.We may not impose such additional hard constraints on a problem ... which then only lead to prove that the problem has no solution under "the self-imposed hard constraints"

Quote ... rope starts looking like an angle, if we start looking at the rope as an angle . Brilliant! You are probably right. The angle analogy is better than the segment analogy.I'm not a fanatic of analogies either. I don't see why tristan solution shouldn't be right. I think it's fine, unless you are not comfortable with an infinity steps process.Maybe one can prove that there's not a finite number of cuts to measure 20 minutes with the ropes, but i suppose that if you do need at least two ropes, there's a different solution than tristan's.

In defense of the rope-angle analogy, I think it is somewhat more than an analogy. To be more precise, think of it as a homomorphism from the group of rope manipulations to the group of actions on angles using a compass and straightedge. If the homomorphism is injective, then it follows that a twenty-minute burn would rigorously imply the existence of an angle trisection. As bhutes states in his last post in clearer fashion, the question is whether injectiveness holds.QuoteOriginally posted by: bhutes(Though, I am somewhat, but not fully, agreed to the angle analogy too) -- rope starts looking like an angle, if we start looking at the rope as an angle . In other words, if we restrict our ability to manipulate the rope in only 2 ways (1) use the rope in whole, (2) bisect the rope, then the rope will behave just as an angle (you can do both those things with the angle). In fact, rope is a restriced angle, because there is no pi equivalent (or we have to treat infinity as the pi in rope-land). But, yes, there is no solution to 20 minutes ... if the rope if bound by only two types of manipulations (as stated above).As I mentioned before, there is a third manipulation - (3) divide the rope into pieces (without being able to define the burn time of each piece). This corresponds to using the straightedge to divde an angle into subangles (without control of the exact values of the subangles).Quote(As hongyi defined ropeland, it's defined by allowed manipulations on the rope .... not existence of instruments corresponding to a compass and a straight-edge) -- that is what I find very restrictive.We may not impose such additional hard constraints on a problem ... which then only lead to prove that the problem has no solution under "the self-imposed hard constraints"I sympathise with this viewpoint. However, the whole point of the rope problem is that we are restricted to a specific set of actions - otherwise, we'd soon be allowing rulers, clocks, etc. And I believe a case can be made for the abovementioned three manipulations spanning all permissible actions. All actions correspond to igniting or extinguishing a burning rope (at one or both ends).We can start or stop burning a rope either i) when a specific event occurs or ii) at a timing which cannot be pinned down. The first case corresponds to subtracting one angle from another, since the only possible specific event would be a rope burning out. The second corresponds to subdivision of an angle into unknown subangles. (Cutting a rope arbitrarily also correspnds to the second case.)This may not allay bhutes concerns, but it's my point of view.

QuoteOriginally posted by: hongyiIn defense of the rope-angle analogy, I think it is somewhat more than an analogy. To be more precise, think of it as a homomorphism from the group of rope manipulations to the group of actions on angles using a compass and straightedge. If the homomorphism is injective, then it follows that a twenty-minute burn would rigorously imply the existence of an angle trisection. As bhutes states in his last post in clearer fashion, the question is whether injectiveness holds.I agree with this ... if injectiveness is proven to hold ... there is no solution to 20 minutes.The problem is: Angle geometry and the capabilities of compass / ruler are precisely defined (as axioms or otherwise). And these capabilities (specified in mathematical terms) go into the proof of angle-trisection impossibility.Rope geometry is either undefined .. or loosely defined. Again, if we start defining the "abilities" of Ropes in terms of what we know about angles .. you are guaranteed to have an injectiveness. (In that case, Rope will have a subset of all possible actions do-able by compass/rules on angles -- there is no pi).---------------------------------------------------------------------A solution to this teaser (if somebody did get it ... -- there are doubts about that already!! --), essentially lies in breaking the axioms of angle-geometry and enabling our ropes with manipulations, beyond what a compass or ruler could do.Maybe, that's where the tease is involved .... think of such a manipulation, which does the job .... and would be acceptable to most wilmotters, as a rightful capability of a rope. (eg. I was earlier not comfortable to "allow" the "ropeland-compass" to have infinite burnings capability ... others were fine, allowing this. Maybe, a majority would allow this capability !!!)But yeah, defining a rope in terms of an angle -- and it's not too difficult to do, given a striking similarity -- proves the trisection impossiblity for the ropes, unmistakably.So ....... if someone does put in a solution, it'll be interesting to see, what additional manipulation of rope was used

Bhutes, I guess we're on the same page, except that I'm rather more confident that we'll not be able to come up with any more manipulations than those already mentioned.An objection to the pendulum solution: In order to measure 20 minutes exactly using the pendulum method, the pendulum has to swing precisely a whole number of times (actually a multiple of 3) in 1 hour, which is improbable (in fact, a priori, the probability is zero).

QuoteOriginally posted by: hongyiBhutes, I guess we're on the same page, yes, we are.