On Friday I posted this puzzle….

You have two ropes and some matches. Each rope, if lit at its end, will burn for 60 minutes. But the rate of burning is not regular, so cutting a rope in half doesn’t result in a burn time of 30 minutes.

How can you use the ropes to time exactly 45 minutes?

If you have not tried to solve it, have a go now.  For everyone else the answer is after the break.

Light both ends of the first rope, and one end of the second rope, at the same time.  After 30 minutes, the two burning ends of the first rope will meet, and the second rope will have 30 minutes  left to burn. At that point, light the remaining end of the second rope. 15 minutes later the two burning ends of the second rope will meet, and you will have timed 45 minutes.

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

1. Helena Handcart says:

That is fiendishly clever. I am fiendishly stupid. Great puzzle this week. Thank you your Profness.

1. I totally agree with Helena. The simplicity is awesome.

2. Alternatively, use the ropes to lassoo (spelling?) someone who has a watch….

3. Gareth says:

Good puzzle!

Now use the two ropes to time exactly 48 minutes… 😉

4. Or alternatively you could set fire to one rope and keep an eye on your watch…

5. Barry Goddard says:

I am glad that Richard’s answer vindicates mine. I received some grief from armchair solvers because I had dared to go beyond assumptions and find an inventive answer. Now I can be happy.

1. Steve Jones says:

Well, to continue the grief, maybe you would put up your solution here and explain how in any way it works and how Richard’s answer vindicates it. As it is, I can only conclude you’re either a fantasist, or a wind-up merchant.

nb. I first heard this problem on a trekking holiday in Ethiopia, on the way down from the country’s highest peak, Ras Dashen. The solution hasn’t changed.

2. The Lord Manley Fan Club says:

Steve,
You should appreciate that Barry Goddard is a world-class wind up merchant (even if he confuses the words “vindicates” and “contradicts”).
You should therefore treat him with the respect and deference that is is due.
TLMFC

3. Steve Jones says:

I’ll satisfy myself that Barry G as the Salvador Dali of the logical puzzle world, and just lives in a world of alternative perceptions of reality. I’m clearly in a generous mood for a Monday.

4. Anonymous says:

ATTENTION ctj
On Friday you posted as follows –
“there are at least two legitimate solutions to the puzzle that give exactly 45 minutes, but Barry Goddard’s “Y” is not one of them.”
Any chance of telling us what the two are?

5. Steve Jones says:

I don’t know what cjt had in mind for his second solution, but there is a way of timing 15 minutes with just one rope, albeit based on the implausible idea you can cut a piece of rope in two and light the cut ends instantaneously.This is how it works.

1) light one rope at both ends
2) wait until the rope burns out (30 minute elapsed)
3) instantly cut the remaining rope into two (doesn’t matter were) and light the ends.
4) wait until one rope burns out. If both do this simultaneously, 45 minutes will have elapsed and you are finished, otherwise go back to 3

You’ll see that for this to work, the second rope has to burn through in 15 minutes, or 4 times the rate of a single burn. As, at all times, the second rope has 4 flame fronts burning, then the burn rate has to be 4 times faster. That’s assuming, of course, that you can cut rope and set it burning instantaneously…

6. Barry Goddard says:

I am used to my fluidly intuitive logic being alien and sometimes frightening to others. It comes from my deep appreciation of the logic and truth and beauty inherent in the pursuit of astrological truth. It can be hard to follow by others. I will bear that in mind if I chance to comment on Richard’s blog in future.

Thank you all for discussing, It is truely by community communication that we all progress towards the truth and light.

7. Anonymous says:

Barry
In the same way that you seem to confuse the words “vindicate” and “contradict” you now seem to be confusing the words “frightening” and “tedious”.
I would recommend purchasing a good dictionary

8. jh says:

Everybody makes mistakes sometimes. It is wise to admit it when one is wrong.

9. Anonymous says:

Barry, Steve,
In connection with your (Barry’s) website, I can think of nothing more apposite to say than to quote from those noted philosophers Beavis and Butthead – “Beware the Klingons near Uranus”

6. Khoái nhất cái cảnh người yêu ôm nhau rồi đi ăn tối hjhj mong mọi người đều có ny nhé http://wapnam.com hân hạnh được post bài viết này

7. I tried to burn 45.76271186440678 copies of your 59 seconds book one after the other, but that took me all weekend! );(

8. Eddie says:

Very subtle puzzle – all about the time not the distances along the rope!

9. Convinced myself that the burning rope thing was a red herring. Bah.

10. Mervulon says:

The solution is, of course, entirely predicated on the premise, which is 100% hypothetical and realistically impossible to ascertain beforehand.

1. Steve Jones says:

Agreed, but then many of these sorts of puzzles have implausible physical properties and ridiculous scenarios. Things like rivers that flow at exactly the same rate and/or are dead straight of the same width, infinitely thin ropes, groups of replies where, magically, you know precisely one is a lie and much else.

You just have to let that stuff wash over you. They are logic problems. You just take the premise and try and produce solutions which fit the premise and introduce as few other assumptions as possible. Sometimes there are sensible alternatives to Richard’s solutions, and occasionally there are better ones. However, in this case I’ve never come across an alternative which doesn’t introduce unwarranted assumptions.

nb. there are other sorts of problems which do introduce such naturalistic phenomena (like knowing that tree trunks grow outward, and not up (save at the shoots), but this really isn’t of that sort.

11. Lazy T says:

Not such a useful method of timing things in public places tho’, it would breach the smoking ban.

12. So hear is the issue I have always had with this solution:
It is stated that the rope burns irregularly, so doesn’t that mean that by lighting both ends of one rope you do not insure that each side will burn for 30 minutes? what if the first half burns faster than the second half or any other variation in between. Granted I am no great logician so there may be something I have missed.

1. RD says:

Quite right. The second rope has been partially burned when you begin using it to measure time elapsed … so, based on the premise in the question, RW has not provided a valid answer.

R

2. RD says:

Stupid response from me …please be kind …

3. dave says:

If they burn at different speeds then they do not meet in the middle but do meet after 30 minutes.

13. Cheryl says:

I thought you could light both ends and the middle of the first rope, and also light one end of the second rope at the same time.
When the last piece of the first rope was gone, 15 minutes would have elapsed, and there would be 45 minutes left on the second rope.
One end of the first rope might burn out first, due to the uneven distribution, but I believe that if you wait for the whole thing to burn out, that will be 15 minutes. Can someone check me on that?

14. Cheryl says:

Ah, never mind. I guess that doesn’t work after all. For example, imagine a rope where the first half burns in 58 minutes and the second half burns in two minutes; if you light it in three places, the whole thing will burn out in 29 minutes, not 15.
My solution would work with a rope of even consistency, but not the cheap crud that we’re burning in this example. Oops.

Great puzzle, one of my favourites so far.

16. ctj says:

on friday, i thought i came up with a different solution, but it turned out to be the same. so i was only able to come up with one.

a fuse like the rope in this puzzle can exist in the real world. say you are using a fuse to light a rocket in poor weather. the fuse needs to be long enough that you can light it from a safe distance, but you don’t want the fuse to burn too slowly, in case wind or rain extinguishes it. but the fuse can’t burn too quickly at the other end, to ensure enough contact with the rocket motor for it to fire.

so you have a fuse that starts out fast, but ends slow. you also need to know how long the fuse burns altogether, so you aren’t wasting your time if the rocket turns out to be a dud.

17. Alex says:

How does someone prove that rope lighted in both ends would burned in half time quicklier than rope lighted from one end?

1. It’s fairly easy to deduce. Firstly we know the burn rate is unaffected by direction as the problem doesn’t specify a given end. It should then be obvious that if we light both ends at the same time, it will take 30 minutes for the flame fronts to meet. If this isn’t obvious, do this little thought experiment. Imagine you had a timer and you started one end of the rope burning. After 30 minutes, you stopped the rope burning (you can do this, it’s a thought experiment). Now consider the remaining rope. There must still be 30 minutes burn time left in it mustn’t there? Also, as the time taken will be the same from whichever end you light it, then do your thought experiment again and light both ends of the rope. The two flame fronts will meet at 30 minutes.

Now there are plenty of real world reasons that this might not be exactly true. With two ends burning, as the two flame fronts came closer, they might affect each others rate of burn. Also, rope will not light instantly, so there is variability there. Indeed, how could anybody possibly know how long it would take for a rope to burn if the rate is irregular in the first place? However, this is not a problem in the real world of burning ropes. It’s a logic problem disguised as a physical one. You simply go with what we’ve been told, and make the fewest assumptions and assume all the apparently physical actions, like lighting ropes, happen instantly.

18. “Firstly we know the burn rate is unaffected by direction as the problem doesn’t specify a given end.”

Could you explain the above sentence further? I’m guessing you mean we should assume that the ropes will take 60 minutes to burn no matter which end we ignite. I agree that’s a resonable assumption, but even given that assumption I believe there’s a flaw in your proof.

1. Steve Jones says:

I did indeed mean that. However, where is the flaw in my proof? I’ve never heard of anybody disagreeing with this particular aspect.

2. The flaw is that, even given that assumption, it’s possible for *sections* of a rope to burn faster in one direction than the other, and that can break the solution.

Imagine a rope AB, with a slight asymmetry of construction such that it takes 29 minutes to burn from A to B, but 31 minutes from B to A. Take a similar rope A’B’ and join B to B’. Now we have a rope which takes an hour to burn when lit from either end, yet when lit from both ends will burn out in 29 minutes.

Is it reasonable to posit a rope that burns faster in one direction than the other? I think it is — after all, this is a logic puzzle about a rope with an irregular burn rate. In my experience, most people do not find it reasonable.

3. Steve Jones says:

This is where you bring in Occam’s razor. The assumption that the rate of burn at any one point is the same in each direction is a great deal simpler than the one where the burn rate varies with direction at any point, yet in such a way that the total burn time is 60 minutes in each direction.

However, the killer point is that with “two different rates in different directions at any one point” assumption, the problem is insoluble.

There are any number of reasons why the solution won’t work in the real world, but to repeat, it’s a logical problem dressed up as a physical one, and you go with the simplest set of assumptions which fit the premise. If you are after absolute proof, then the problem would have had to have been specified in far more detail than it was.

4. The OP, Alex, might be disappointed that the proof he requested requires an extra assumption about the burn rate, or a meta-assumption that the problem is soluble. It’s a logic problem, agreed, and as such it needs only a modest addition, such as that the burn rate at any point on the rope depends only on its thickness at that point, for a rigorous proof to be possible. Occam’s razor cuts both ways, if you’ll forgive the pun. Who’s to say whether placing an additional constrainst on the burn rate is “simpler”?

19. Alex. says:

I’m angry I made a post 2 days ago. It is missing. Is this site censored?