Measuring 45 Minutes Exactly with 2 Ropes

[||spoon||]

?? 45 minutes exactly ??

hey guys,

Will try explain this as best I can, sorry if it is not clear.

Your aim Is to measure 45 minutes exactly.

You have two ropes, which burn for 1 hour exactly each, however the rate at which they burn is not constant so you can't simply measure 3/4 of the length and burn to that point.

Obviously you can light the ropes to start them burning.

Good luck!

-Spoon

P.s u don't have a separate clock, watch or other way of measuring time!

 

[Jimmy Snyder]

This puzzle has been posted on the brain teaser forum before.

Spoiler

Light both ends of one rope and one end of the other. When the rope with two ends lit burns up, 1/2 hour has passed and there is 1/2 hour left on the other rope. At that moment light the other end of that rope and it will be consumed in 1/4 hour for a total elapsed time of 3/4 hour or 45 minutes.

 

[redargon]

Or...

Spoiler

Fold one rope in half. Light both ends of the normal length of rope and when it is finished burning 1/2 an hour is passed. Then immediately light the two ends (well, three ends )of the folded rope, when it is finished burning 1/4 of an hour has passed. In total, 3/4 of an hour have passed.

 

[Jimmy Snyder]

redargon, suppose the folded rope burns from one end to the middle in 59.99 minutes and from the middle to the other end in .01 minute.

 

[davee123]

Or...

Spoiler

Fold one rope in half. Light both ends of the normal length of rope and when it is finished burning 1/2 an hour is passed. Then immediately light the two ends (well, three ends )of the folded rope, when it is finished burning 1/4 of an hour has passed. In total, 3/4 of an hour have passed.

Nope-- not quite:

"the rate at which they burn is not constant so you can't simply measure 3/4 of the length and burn to that point"

So one half of the folded-in-half rope might be done after 1 minute, and the other half might be done after 29 minutes. There's no guarantee that half the rope will burn in half the time.

DaveE

 

[redargon]

Nope-- not quite:

"the rate at which they burn is not constant so you can't simply measure 3/4 of the length and burn to that point"

So one half of the folded-in-half rope might be done after 1 minute, and the other half might be done after 29 minutes. There's no guarantee that half the rope will burn in half the time.

DaveE

um, yes it will. It doesn't matter at the speed at which each half burns, but rather the total time it takes for the entire rope to be burned. If one half is done after 29 minutes and the other after 1 minute, that's still half an hour in total.

 

[redargon]

redargon, suppose the folded rope burns from one end to the middle in 59.99 minutes and from the middle to the other end in .01 minute.

I see what you're saying, that makes sense. I thought there might be a second solution, but it appears the first solution is the only one.

Sorry Davee, I think you were on to the same flaw, but didn't explain it as well as jimmysnyder.

 

[neoteny]

Apologies for reviving an old thread, but...

With such a strange rope, there's no guarantee that when lit simultaneously at both ends it will burn out in 30 minutes.

 

[Rogerio]

Apologies for reviving an old thread, but...

With such a strange rope, there's no guarantee that when lit simultaneously at both ends it will burn out in 30 minutes.

After we lit both ends A & B, they will burn up to the point P, somewhere between A and B.
Of course the both parts AP and BP take the same time to burn out.
And, of course, AP + PB = AB would take 1 hour to burn out.
So, each part has to take exactly 30 minutes to burn out.

 

[neoteny]

After we lit both ends A & B, they will burn up to the point P, somewhere between A and B.
Of course the both parts AP and BP take the same time to burn out.
And, of course, AP + PB = AB would take 1 hour to burn out.
So, each part has to take exactly 30 minutes to burn out.

That argument works only if we know that PB = BP. But with such a strange rope maybe this symmetry doesn't hold?

 

[BobG]

That argument works only if we know that PB = BP. But with such a strange rope maybe this symmetry doesn't hold?

No symmetry is required for PB to equal BP. It's similar to this scenario:

I left on my journey from Allaboleil to Bentajomine at 6:00 AM and it was such a difficult and tiring journey that I forgot to check the time when I finally arrived. The next day, I left Bentajomine to return to Allaboleil at 6:00 AM and I was so happy to arrive back home that I forgot to check the time. I don't know how long either leg of the journey took, but I'm sure that somewhere on my return trip that I was in the exact same spot at the exact same time as I was the day before on the first leg of my journey.

This true even if the first day's travel took 18 hours and the second day's travel took 5 minutes or vice versa. There doesn't need to be any symmetry.

 

[neoteny]

In your scenario it's easy to show that the "exact same spot" (let's call it Puddletown) exists, but it doesn't follow that the time taken to travel from Puddletown to Bentajomine on day 1 is equal to the time taken to travel from Bentajomine to Puddletown on day 2. Those times need not be the same. Similarly the time the rope takes to burn from P to B need not be the same as the time it takes to burn from B to P.

(I'm assuming that when you say the two scenarios are similar you mean that A is equivalent to Allaboleil, B is equivalent to Bentajomine, and P is equivalent to the "exact same spot".)

 

[BobG]

In your scenario it's easy to show that the "exact same spot" (let's call it Puddletown) exists, but it doesn't follow that the time taken to travel from Puddletown to Bentajomine on day 1 is equal to the time taken to travel from Bentajomine to Puddletown on day 2. Those times need not be the same. Similarly the time the rope takes to burn from P to B need not be the same as the time it takes to burn from B to P.

(I'm assuming that when you say the two scenarios are similar you mean that A is equivalent to Allaboleil, B is equivalent to Bentajomine, and P is equivalent to the "exact same spot".)

I suddenly understand what you're talking about. Actually, you didn't even misword it - I just missed your point.

The solution is correct only if the rope is commutative. While it can burn at different speeds at different portions of the rope, it has to burn the same speed in both directions for the solution to be correct. However, if there's something about the rope that causes different parts of it to burn at different speeds, then there's no guarantee that the something different could cause the rope to burn at a different speed if it burned in the opposite direction.

For example, forest fires usually move faster uphill than they do downhill. Or perhaps the speed of burning is because of the direction of the threads in the rope. Perhaps frayed ends tend to lie in one direction or another and the frayed ends catch a spreading flame faster than the smooth part of the threads. And so on.