Minimum Number of Trips
Date: 07/01/2001 at 15:58:54 From: B. Hastings Subject: Minimum number of trips puzzle Hello, The problem is that of a fully laden lorry that can carry fuel and supplies for a trip into the desert of up to 400 miles. The base camp on the edge of the desert has an unlimited amount of fuel and supplies. By setting up strategic supply dumps along the proposed route, it is possible to travel more than 200 miles into the desert and return. How many trips would be needed to penetrate 600 miles and back? (Note: return required.) My son is in Year 8 in the UK, and is 13 years old. He used trial and error, and found a very acceptable answer. I enjoy puzzles, and felt sure there must be a formula, with the distance between supply dumps as a variable, to calculate the minimum number of journeys and/or the minimum distance travelled (i.e. fuel consumption). Starting at the most distant point, there must be a relation between the distance between successive supply dumps and the amount of fuel to be dumped there using a geometric series. When I had solved the basic puzzle, I intended to extend the solution to a variable lorry capacity and a variable end point, but I couldn't even solve the initial problem. I look forward your reply with anticipation. Barbara Hastings.
Date: 07/02/2001 at 16:33:55 From: Doctor Jaffee Subject: Re: Minimum number of trips puzzle Hi Barbara, Let's assume that 1 pound of fuel and supplies is required to travel 1 mile. I found a solution that required a total of 11,800 pounds of fuel and supplies. I am curious how my solution compares with your son's. My solution was based on the premise that the lorry driver needs to deposit 800 pounds of supplies and fuel at the 400-mile marker. From there the driver could travel 200 miles to the edge of the desert, then back 200 miles where there would stil be 400 pounds of fuel and supplies waiting. This would be sufficient to get him home. So, if he started at the base camp with 11,800 pounds, he could load the lorry, drive 100 miles, deposit 200 pounds of fuel and supplies, then drive back to base camp. He would then repeat this process until 5900 pounds of fuel and supplies were deposited, at which time it would become his new base. Then, starting from his new base he would load the lorry, drive 100 miles deposit 200 pounds of fuel and supplies and return to base. He would repeat this process until a total of 3000 pounds of fuel and supplies were deposited and his new base camp would now be established at the 200-mile marker. He would then follow the same procedure and deposit 1500 pounds of fuel and supplies at the 300-mile marker and finally deposit 800 pounds of fuel and supplies at the 400-mile marker. Now, establishing a formula is challenging. If the capacity of the lorry is x pounds of fuel and supplies, then you need to deposit x pounds at a distance of x miles from the end of the trip. However, in this case twice as much was required because of the looping nature of the trip; that is, the lorry went from the 400 miles marker to the end, then looped back to the 400-mile marker in one trip. So, however many pounds of fuel and supplies are required at that last marker, approximately twice as much is required at the previous stop. That's where the complication is. In the problem your son solved, 400 pounds of fuel and supplies is the capacity of the lorry. At alternate stops 100 pounds less than twice the amount required at the next stop must be accumulated. If the capacity of the lorry were 500 pounds, it would work very differently. There may be a generalized formula that will solve all problems of this nature, but personally I think it is easier to work it out logically and systematically as I have done, and as I expect your son did. I've enjoyed working on this problem; I found it quite fascinating. I hope my response has been helpful, and please write back if your son's solution is significantly different from mine, or if you want to discuss it some more. - Doctor Jaffee, The Math Forum http://mathforum.org/dr.math/
Date: 07/22/2001 at 13:14:49 From: B. Hastings Subject: Re: Minimum number of trips puzzle Hi, Dr Jaffee, My son's solution was similar to yours. Assuming 1 unit of fuel and supplies per mile, he started from the idea that he needed 400 units at the 400-mile mark to reach the 600-mile point and return to the 400-mile supply dump. He then said that he needed to make 3 round trips from the 300-mile dump to the 400-mile dump, depositing 200 units per trip - since only making two trips would leave him stranded at the 400-mile point. To enable him to make three trips from 300 to 400, he said he needed to make seven trips from 200 to 300, fifteen trips from 100 to 200, and 32 trips from Base to 100. This would ensure enough extra units at each supply dump to get him home. His total consumption was therefore 12200 units. I thought this was a good solution, particularly for a 13-year-old. His teacher said the solution required only eleven trips, but he could not remember how she did it. I think I have found a way that only takes 6400 units and 30 trips. If the lorry makes 15 round trips from base to 100 miles depositing half its load each trip, there will be 3000 units at the 100-mile point. The lorry sets out for the 16th time from the base, and arrives at the 100-mile point 3/4 full. The total at the 100 m.pt. is now 3300. The lorry then makes 7 round trips to the 200 m.pt. using 1400 units, depositing 1400 units and having 500 left at the 100 m. pt. It fills up so there are now only 100 units left at the 100 m.pt., and sets out for the 8th time from the 100 m.pt.. At the 200 m.pt.,the lorry is 3/4 full, so the total at the 200 m.pt. is now 1700 units. The lorry makes 3 round trips to the 300 m.pt. using 600 units, depositing 600 units at the 300 m.pt. and leaving 500 at the 200 m.pt. The lorry fills up so there are now only 100 units left at the 200 m.pt. and sets out for the 4th time from the 200 m.pt. At the 300 m.pt. the lorry is 3/4 full, so the total at the 300 m.pt. is now 900 units. The lorry makes 1 round trip to the 400 m.pt. using 200 units, depositing 200 units at the 400 m.pt. and leaving 500 at the 300 m.pt. The lorry fills up so there are now only 100 units left at the 300 m.pt. The lorry sets out for the 2nd time to the 400 m.pt., and is still 3/4 full when it reaches it. It takes on 100 units, completes its trip to the 600 m.pt. and back, and is empty when it gets back to the 400 m.pt. It takes on 100 units and drives back to the 300 m.pt.. It takes on the 100 units left there, and drives to the 200 m.pt.. It takes on the 100 units left there, and drives back to the 100 m.pt., takes on 100 units and drives back to base. I tried to work with supply dumps at different distances, but couldn't find a better answer. Since my solution uses exactly half the total of yours, either I have made an elementary error or I have saved fuel by not transporting the fuel necessary for the journey home to the 400 mile point. All the supply dumps are empty at the end of my trip. I'm glad you found this problem as intriguing as I did. I have always enjoyed puzzles. I think that you are probably right in saying that trial and error is the best way to solve problems like this. I just wondered whether my son's teacher was introducing a new topic by letting the children tackle a puzzle, and then explaining how it should be approached, but apparently not. I hope your colleagues enjoyed this problem. Thank you for your help. Barbara Hastings.
Date: 07/26/2001 at 20:14:45 From: Doctor Jaffee Subject: Re: Minimum number of trips puzzle Hi Barbara, Bravo! Your solution was very efficient; I went through it step by step and it solved the problem and used just a little more than half of the resources required in my solution. Reading your solution gave me a fresh perspective on solving problems of this nature. I'm glad I was able to help you and I appreciate the new insights you have provided for me. - Doctor Jaffee, The Math Forum http://mathforum.org/dr.math/
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