Homework help!!

[2011]

1. You have been given an infinite number of ropes. All ropes are of varying length. If you burn any single rope such that it is lit from one end (and allowed to go on till it reaches the other end), the total time the rope takes to get burnt is one hour. Some parts burn faster than the others. Now a person comes along and claims that he can use this ‘burning ropes ’ method to exactly measure time of one hour and 15 mins (75mins). He doesn’t use any other method, source or tool (besides burning and observation) to measure time. How would you do it if put in the same situation? [Hint: You can burn ropes simultaneously] <br style="color: rgb(40, 40, 40); font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 22px; "><br style="color: rgb(40, 40, 40); font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 22px; ">2. You are given a stone, weighing 40 kg. Your job is to break it in four different parts such that they can then be used in a balance to accurately weigh all objects in the range from 1-40kg. What would be the individual weight of the four parts? (Assume that all objects to be measured have integer weights, i.e. no decimal numbers)<br style="color: rgb(40, 40, 40); font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 22px; "><br style="color: rgb(40, 40, 40); font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 22px; "><br style="color: rgb(40, 40, 40); font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 22px; ">3. Using this image: http://i.imgur.com/3PPsW.png<br style="color: rgb(40, 40, 40); font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 22px; ">Design a track through the lands so that you cross each of the seven connections labeled, only once. All connections have to be completely traversed exactly once. The track must be uni-directional, i.e. you cannot double-back and move the way you came. (This one seems impossible)

Help on any questions would be appreciated

[DrRocket]

1. You have been given an infinite number of ropes. All ropes are of varying length. If you burn any single rope such that it is lit from one end (and allowed to go on till it reaches the other end), the total time the rope takes to get burnt is one hour. Some parts burn faster than the others. Now a person comes along and claims that he can use this 'burning ropes ' method to exactly measure time of one hour and 15 mins (75mins). He doesn't use any other method, source or tool (besides burning and observation) to measure time. How would you do it if put in the same situation? [Hint: You can burn ropes simultaneously] <br style="color: rgb(40, 40, 40); font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 22px; "><br style="color: rgb(40, 40, 40); font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 22px; ">2. You are given a stone, weighing 40 kg. Your job is to break it in four different parts such that they can then be used in a balance to accurately weigh all objects in the range from 1-40kg. What would be the individual weight of the four parts? (Assume that all objects to be measured have integer weights, i.e. no decimal numbers)<br style="color: rgb(40, 40, 40); font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 22px; "><br style="color: rgb(40, 40, 40); font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 22px; "><br style="color: rgb(40, 40, 40); font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 22px; ">3. Using this image: http://i.imgur.com/3PPsW.png<br style="color: rgb(40, 40, 40); font-family: helvetica, arial, sans-serif; font-size: 14px; line-height: 22px; ">Design a track through the lands so that you cross each of the seven connections labeled, only once. All connections have to be completely traversed exactly once. The track must be uni-directional, i.e. you cannot double-back and move the way you came. (This one seems impossible)

Help on any questions would be appreciated

 

If this is homework, you need to show some thought or preliminary work on the problems. If not this is not the right forum. If this is homework it would be interesting to know the nature of the class. The problems do not seem to have much in common.

 

That said, in #1 I see nothing obviating the possibility that all ropes smolder on the lit end for a period arbitrarily close to one hour then burn rapidly to meet the 1 hour period for complete burning -- in essence a simple time delay signal of one hour with nothing happening qalong the length of the rope during that period. In that case no matter how you configure the ropes, from any single ignition you get only signals spaced at one hour intervals. Lighting multiple ropes would only introduce sets of signals with one hour intervals interspersed depending on the unknown ignition spacing. In this case I don't see how the specified task can be completed. You might as well have bells that ring at one hour intervals.

 

If you could use the known interval to calibrate a linear device then you could do something. But no other tools are allowed.

 

#2 is possible with a judicious choice of division of the stone plus the knowledge that only integer weights between 1 and 40 kg are possible. A little thought and some trial and error will get you to a solution.

 

#3 is, as you guessed, impossible. Google "Euler circuit" -- this is a well known class of problems in graph theory. The most well known similar problem is the "Konigsberg bridge problem."