[Guest]

Easy

Two cords

You are given two cords that both burn exactly one hour, not necessarily with constant speed. How should you light the cords in order to determine a time interval of exactly 15 minutes

Medium

One cord

You are given one cord that burns exactly one hour, not necessarily with constant speed. How should you light the cord in order to determine a time interval of 15 minutes? (Hint: solve the corresponding riddle from part "Easy" first.)

I think I figure out the easy one :-) , but I can not find any method to do medium one?

Could anyone help me ?

[Guest]

Light the first cord from both sides and 2nd from only one side.The first one will get burnt in half an hour due to doubled average rate of combustion (which will work even if the speed of combustion is not constant). So we'll know when half anhour lapses. After the first cord gets burnt, extiguish the second cord. Now the time taken by the remaining second cord to burn if lighted from both sides would be 15 mins. I'm still trying the medium one.

[Guest]

Burn one cord from both ends and another from one end. First cord completes burning in 30 mins. Now light the free end of the second cord. It takes 15 minutes to completely burn off.

cut the cord in half along its axis. they will take 30 minutes each. If you burn any cord from both ends, you'll get 15 minutes.

I haven't got it yet. Challenge Accepted.

[Guest]

The total for all this will be 15 minutes.

Light the cord from both ends and also from the middle. This effectively turns the cord into 2 cords each burning both ends. Whenever one of the cords is finished, then light the remaining cord in the middle, thus turning it into 2 cords burning from both ends. Repeat. When all is burned, that will have been 15 minutes. This works because even though the cord may not burn at the same rate, the average will still be the one hour divided by the number of burning places, so all that is required is to make sure there are always 4 places burning. I checked a few scenarios as well and it works for the ones I tested.

Extra credit, do you think it matters if it is lit in the middle or if it can be lit at any point between the burning ends? I technically already answered that question.

Edited May 13, 2011 by Nana7

[Guest]

I would disagree with nana7 because i

n the case she has mentioned say for example 1 half takes 10 mins to burn while the other half takes 50 mins to burn. If the cord is burnt from between the net time required would be 25 mins as one half would get burn in 5 mins while the other half in 25 mins.. I had already thought of that!!

[Guest]

I would disagree with nana7 because i

n the case she has mentioned say for example 1 half takes 10 mins to burn while the other half takes 50 mins to burn. If the cord is burnt from between the net time required would be 25 mins as one half would get burn in 5 mins while the other half in 25 mins.. I had already thought of that!!

Taking the case where 1/2 would normally burn in 10 minutes and the other in 50. Burning each half from both ends would reduce that to 5 minutes for one and 25 for the other as you say. However, there is a further step. Once the first end has finished burning, then the other half needs to be lit from its center (or any point between its ends). It has already burned for 5 minutes so it has 20 left to go, which is reduced to 10 minutes for each of its halves if they burn at the same rate, which yields 15 minutes overall time for all cord pieces to burn. If they do not burn at the same rate, say one has finished after only 5 minutes, which means the other half had 20-5 or 15 minutes and now has 10 minutes remaining. Lighting it in its center would burn those two halves in 5 minutes if they burn at the same rate, for a total time from each step of 5+5+5=15 minutes. If they do not burn at the same rate, light the remaining piece in its middle. Any scenario is going to result in 15 minutes overall because the original cord needed 1 hour to burn from 1 end but it is constantly being burned from 4 ends, burning it 4 times as fast for 15 minutes.

[Guest]

Understood!!!

[Guest]

Ignite both ends one one cord and one end of the other. Even with a variable burning rate the cord lit at both ends will burn through in 30 minutes. At that point, ignite the other cord's unlit end. As it had already been burning for 30 minutes, it will take 15 minutes to burn from both ends through the remaining cord.

With only the one cord I do not believe one can present a valid solution.

Splitting the cord may create two cords, but doing so may create two cords of different variable burning rates that may not even add up to 30 minutes. Changing the thickness of the cord in such a manner may change the length of time the cord burns through in toto. Thus splitting the cord is not a solution.

Igniting the cord at both ends and in the middle is also not a solution as one can not know where the 'middle' of the cord is in relation to the burning rate. If one lit the cord in its center, it might take only 10 minutes for one section to burn through with the remaining section taking 20 minutes, for example. And with no timer but the burning cord, one can not even ascertain that.

[Guest]

I'm still trying to figure this one out, I like some of the answers but I would have to say they are all incorrect ... only because the problem states "exactly 15 minutes

". There would be a few seconds lost on all the solutions ... if it can be "about 15 minutes" then some of the solutions are dead on!!

[Guest]

I do not think there is a solution that does not involve altering the cord and assuming that its properties remain intact.

There is no timing mechanism for you to "know" it has been 15 mins. With the two chords problem, you had to use the "other" chord as a method to "time" your chord. No matter how many times you light the chord you can never know how long it has been or how much time is remaining to be burnt. We can say for certain instances, that we can light here or there and get 15mins, but in reality we never truly know how long it has been without an external timing source. edited: (besides 30 mins)

Edited May 13, 2011 by cogitodp

[Guest]

Wow, my first post and I have realized I'm wrong. I have an algebraic way of proving that this is possible.

let x = the quicker or shorter amount of burn time

then (30-2x)/2 would be the amount of time it takes the other "half" of the chord after you light it in the middle

so,

x + (30-2x)/2 = 15

x + 15-x = 15

x = x

that will teach me to post prematurely.

[Guest]

I think Nana7 got it!

It may take more then three steps or as little as one. The key is to keep the single cord burning from four end, so when one segment goes out start the remaing somewhere in the middle.

Glamar

[Guest]

Thanks for Nana7's great ideas.

I looks every agreed his solution.

Me too

Thanks again.

[Guest]

Taking the case where 1/2 would normally burn in 10 minutes and the other in 50. Burning each half from both ends would reduce that to 5 minutes for one and 25 for the other as you say. However, there is a further step. Once the first end has finished burning, then the other half needs to be lit from its center (or any point between its ends). It has already burned for 5 minutes so it has 20 left to go, which is reduced to 10 minutes for each of its halves if they burn at the same rate, which yields 15 minutes overall time for all cord pieces to burn. If they do not burn at the same rate, say one has finished after only 5 minutes, which means the other half had 20-5 or 15 minutes and now has 10 minutes remaining. Lighting it in its center would burn those two halves in 5 minutes if they burn at the same rate, for a total time from each step of 5+5+5=15 minutes. If they do not burn at the same rate, light the remaining piece in its middle. Any scenario is going to result in 15 minutes overall because the original cord needed 1 hour to burn from 1 end but it is constantly being burned from 4 ends, burning it 4 times as fast for 15 minutes.

Nana7's solution is almost correct, but not quite...

In your example, you kept demonstrating where the halves burned at the same rate. I shall try to demonstrate the method requires one more step. Let us say that 1/2 of the cord would take 8 minutes and the other 52 minutes. Igniting the cord in the middle would halve each segment's burning time to 4 minutes and 26 minutes. When the 4 minute segment finished burning, the other segment would have 26 - 4 = 22 minutes remaining. Lighting its center would give two new segments with a total burn time of 22/2 = 11 minutes. Yet, in your example, each segment burns concurrently respective to each other. In order that 11 more minutes pass (as 4+11=15, which is the sought goal), both ends would have to burn consecutively. Therefore, immediately upon igniting the middle of the the remaining cord for the second time, both ends of one of the segments would need be immediately extinguished and then relit when the other segment finished burning.

Edited May 13, 2011 by Dej Mar

[Guest]

I think when you say they must be consecutive burns, you are mixing steps 1 and 2. Step 1 takes 4 minutes and step 2 takes 11 minutes (assuming the second two halves burn at the same rate). Those 2 steps do need to be consecutive. But the burns in step 2, in which the 22 minute cord is lit in the middle, must have its two new halves burning at the same time, and will add the needed 11 minutes either by the end of step 2 if they burned at the same rate or by the end of a future step if they did not burn at the same rate.

If in step 2 you light the middle of the 22 minute cord and then extinguish the burns on one of the new halves, you have no way of knowing when 11 minutes have passed, no way of knowing if the half you let burn did so in less than 11 minutes or if it took longer like even 20 minutes to burn. By allowing both halves to burn concurrently, you are guaranteed that either 11 minutes exactly have passed if both are done at the same time, or that less than 11 minutes have passed if only one is finished since one must be less than/equal to 11 and the other greater than or equal to 11.

Also, if in step 2 you extinguished one cord, then you no longer have 4 simultaneous burns taking place, which is required for this method to work since all we are interested in is speeding the burn up from 60 minutes to 15 minutes and burning 4 ends does that. It does not really even matter how or when you choose to light what, just as long as there are 4 spots burning at all times.

Nana7's solution is almost correct, but not quite...

Edited May 13, 2011 by Nana7

[Guest]

I think when you say they must be consecutive burns, you are mixing steps 1 and 2. Step 1 takes 4 minutes and step 2 takes 11 minutes (assuming the second two halves burn at the same rate). Those 2 steps do need to be consecutive. But the burns in step 2, in which the 22 minute cord is lit in the middle, must have its two new halves burning at the same time, and will add the needed 11 minutes either by the end of step 2 if they burned at the same rate or by the end of a future step if they did not burn at the same rate.

If in step 2 you light the middle of the 22 minute cord and then extenguish the burns on one of the new halves, you have no way of knowing when 11 minutes have passed, no way of knowing if the half you let burn did so in less than 11 minutes or if it took longer like even 20 minutes to burn. By allowing both halves to burn concurrently, you are guaranteed that either 11 minutes exactly have passed if both are done at the same time, or that less than 11 minutes have passed if only one is finished since one must be less than/equal to 11 and the other greater than or equal to 11.

Also, if in step 2 you extenguished one cord, then you no longer have 4 simultaneous burns taking place, which is required for this method to work since all we are interested in is speeding the burn up from 60 minutes to 15 minutes and burning 4 ends does that. It does not really even matter how or when you choose to light what, just as long as there are 4 spots burning at all times.

I realized that after I had posted. So I am back to that there is no solution. The method you posted has a similar problem. You need the time as if both burn consecutively, but to get that time both need to burn concurrently. As you can not have it both ways...no solution.

[Guest]

You do not need to know any times for any individual steps. We are simply assuming times for what if scenarios to verify it works by seeing if it works or fails if the times worked out a certain way. Not sure what you mean by not having it both ways. The burns within one step are concurrent within that step (but NOT concurrent with other steps), and are consecutive with other steps, I do not see what the problem is.

I realized that after I had posted. So I am back to that there is no solution. The method you posted has a similar problem. You need the time as if both burn consecutively, but to get that time both need to burn concurrently. As you can not have it both ways...no solution.

[Guest]

You do not need to know any times for any individual steps. We are simply assuming times for what if scenarios to verify it works by seeing if it works or fails if the times worked out a certain way. Not sure what you mean by not having it both ways. The burns within one step are concurrent within that step (but NOT concurrent with other steps), and are consecutive with other steps, I do not see what the problem is.

My apologies. Your method does indeed work. It was an error in my math.

[Guest]

I haven't read any solution, the mine for de medium is:

I divide de cord in two parts. Burn in the middle both and they are burning a four times normal speed. When one of them reaches the final at any side (given their irregular burning) I burn the opposite for maintaining the four times speed. Then they last 15 minutes.

[Auramyna]

All you have to do is make sure that 4 places on the cord are alight at a time. So if you light at each end and once in the middle and if one half burns out first, you then light in the middle of the other half. This way, it is burning at 4x the speed hence (60 mins/4 =)15 mins.