Problem of the week ski trip

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THORTHELABRADOR

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The recent snows have inspired the Math Club to plan a special trip to a ski resort. The Math club advisor and her friend have agreed to visit 5 local ski areas before making their selection for the club's visit. They plan to ski at each resort. The would like to maximize the time they have to spend on the slopes by minimizing the amount of time they spend driving.
As a club member,you are to help them plan their route. Assuming the roads are more or less equivalent, in which order should they visit the ski resorts? What is the total distance of their trip? Remember they must start and end their trip at school.
Use the following chart to help you determine the shortest trip that will start and end at school,and will include a visit to all 5 ski resorts. All distances are given in miles

.................school..........Big Boulder..........Camelback.........Fernwood.......JackFrost........Shawnee..........
school..........0....................16....................17.................23...............19.................18...............
Big Boulder.....16...................0.....................9..................30...............3..................24...............
Camelback......17...................9.....................0.................26...............11..................20..............
Fernwood.......23..................30....................26.................0.................34.................6...............
JackFrost.......19..................3.....................11.................34...............0....................26............
Shawnee........18.................24.....................20.................6................26....................0............

I hope the chart formats ok...last night it got all messed up. I think we should start at the farthest ski area and work our way back to the school....to maximize the ski time...and hopefully NOT sit in Pocono traffic with all the tourists from NY and NJ. So I think we should go to JackFrost 34 miles from school then to Big Boulder that is 4 miles from Jack frost...etc and end up at the school...I didn't add it all up yet I am going to do that in the morning -time for bed.
 


Did that sly guy give you a list of variables that you are supposed to take into consideration ?

How do you plan to quantify the change in driving time as a result of tour buses in some vicinity ?

I'm not familiar with the area, but this exercise begs use of the famous formula: distance can be represented as a product (average rate multiplied by elapsed time).

The instructions tell us to make an assumption: "the roads are more or less equivalent".

I'm thinking that you should ignore reality, and model this exercise as something like a scattering of six dots, each connected to the other with line segments (the lengths of which are in the chart).

These types of networks are studied a lot, and the dots are called "nodes".

If I were to begin refreshing my memory on these types of word problems, I would google keywords and quoted phrase:

nodes shortest route "word problem"

I mean, that's a start.

I'm thinking that it has something to do with summing up the elements in permutations.

In fact, perhaps you could convert the chart to the equivalent times, using some arbitrary average speed for the rate.

Then you might be summing a list of times, for each potential route (permutation).

Just some thoughts.

 
I at first thought I had to know the hours of operation to know that I was getting max ski time, but then figured even if I spent a nanosecoond at each resort that could really be my maximum time...so I dismissed that idea.
I then thought I would need to know if he wanted us to take actual road routes since these are all local resport and calculate the time say if route 123 has a speed limit of 35 mph until I hit the hightway and can go 65mph...but then again I figured I was reading too much into this problem. So I went back to what exactly was stated in the problem and what were actually questions written in the problem and am going with that.
I talked with my mom and she think he has too much time on his hands LOL....I told her he has 5 kids under age 7,teached, coaches track and cross country and runs himself.... where he comes up with these puzzles is beyond me...haha but they are fun.
 
This is a Traveling Salesman Problem. You can google this and find plenty.

These are best done with software.

I just tried a small LINGO program for this, but as soon as I introduced a 6 by 6 matrix, the wimpy demo version would not handle it.

I shortened it to the school and 4 resorts and it worked(5 by 5 matrix). Go figure. Sorry, all I have is the free demo version of LINGO and it

does not do that much. I got rid of Shawnee and it gave me a minimum distance of 69 miles to hit all resorts and come back to the school.
 
Not sure if I calculated right but I got a total of 89 miles. I left school and went to the farthest ski area and then the next farthest working my way back to school.
I think I liked his 6 letter word puzzle better lol
 
One algorithm could be to choose the "nearest" destination. Then

school ... 16...BB ....3 ....JF.....11....CB......20 .... S.... 3....FW ....23.....school = 76 miles

There are only 5! ways to arrange 6 things in a ring.

Denis could make an "exhaustive" program to find the most efficient way.
 
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