Posers and Puzzles
11 Feb 07
12 Feb 07
Originally posted by PBE6An interesting (if rather complex) proof. But does it tell me how to find a valid starting point given a distribution?
Here's my stab:
Let the fuel distribution around the track be g(t), and the rate of fuel consumption be R. The mileage the car gets out of the amount of fuel picked up between any two points (n-1) and (n) is:
int(g(t)/R)dt from (t(n-1)..t(n))
And the distance actually traveled is:
delta L = t(n)-t(n-1)
Since the car must get more mileage out of ...[text shortened]... must always be able to make the trip given the judicious selection of a starting location.
12 Feb 07
Originally posted by PBE6Impressive for sure. Are you a physicist by any chance?
Here's my stab:
Let the fuel distribution around the track be g(t), and the rate of fuel consumption be R. The mileage the car gets out of the amount of fuel picked up between any two points (n-1) and (n) is:
int(g(t)/R)dt from (t(n-1)..t(n))
And the distance actually traveled is:
delta L = t(n)-t(n-1)
Since the car must get more mileage out of ...[text shortened]... must always be able to make the trip given the judicious selection of a starting location.
12 Feb 07
2 edits
Originally posted by XanthosNZI think that's a nice way of saying "longwinded", hehe 😵. I prefer short and simple solutions too, but I couldn't think of one. 😕
An interesting (if rather complex) proof. But does it tell me how to find a valid starting point given a distribution?
All my solution really tells you is that it's always possible to travel the circuit, but I think it suggests a method to check for a starting point. If the fuel distribution is discrete, check each set of micro-trips starting from any point with fuel (but preferably from the beginning of the largest block of fuel - not a guarantee, but a greater likelihood of success). If the distribution is continuous, draw a line across the graph of g(t) at g(t)=R, and start checking from each lump that peaks above the line (again starting with the largest one); the car won't be able to go far if g(t) < R, and this effectively turns a continuous set of starting points into a discrete set for easier checking.
What's the "short and sweet" solution for the original problem and this follow-up question?
12 Feb 07
Originally posted by PBE6Here's the method for finding a starting point (with either discrete fuel points or not):
I think that's a nice way of saying "longwinded", hehe 😵. I prefer short and simple solutions too, but I couldn't think of one. 😕
All my solution really tells you is that it's always possible to travel the circuit, but I think it suggests a method to check for a starting point. If the fuel distribution is discrete, check each set of micro-trips starting ...[text shortened]... 's the "short and sweet" solution for the original problem and this follow-up question?
1. Pick any point on the track and imagine starting from there.
2. Graph the amount of fuel that you will have throughout your circuit. These values could be negative.
3. If the graph doesn't go negative at any point you have found a valid starting point. If it does then the absolute minimum of the graph is a valid starting point (this is also true in the first case but obvious).
This method relies on a proof that a circuit is always possible but is a very elegant way of finding valid points once that is done (as you did earlier).
12 Feb 07
Originally posted by XanthosNZDo you mean maximum rather than minimum?
Here's the method for finding a starting point (with either discrete fuel points or not):
1. Pick any point on the track and imagine starting from there.
2. Graph the amount of fuel that you will have throughout your circuit. These values could be negative.
3. If the graph doesn't go negative at any point you have found a valid starting point. If it d ...[text shortened]... ible but is a very elegant way of finding valid points once that is done (as you did earlier).
12 Feb 07
Originally posted by XanthosNZI can see that maximum doesn't work, but I couldn't see how minimum would work either.
No I don't. Think about it.
Maybe the point immediately after the absolute minimum.
Even then, consider the situation where all the fuel is at a single point.
I pick a starting point exactly halfway round the track, and then I have 2 minimums of -5 L at opposite sides of the track, but only 1 of them is a valid starting point
12 Feb 07
Originally posted by aging blitzerNo you don't. Here's what the graph would look like in the case you mentioned (with track length 1000m for argument's sake):
I can see that maximum doesn't work, but I couldn't see how minimum would work either.
Maybe the point immediately after the absolute minimum.
Even then, consider the situation where all the fuel is at a single point.
I pick a starting point exactly halfway round the track, and then I have 2 minimums of -5 L at opposite sides of the track, but only 1 of them is a valid starting point
http://img360.imageshack.us/img360/3319/graph3ua6.gif
We have one minimum at -5L which would indeed be a valid starting position (the only valid starting position).
Note: Perhaps it's best to make a comment that I am ignoring the usual mathematical crap that goes along with discontinuities.
12 Feb 07
Originally posted by PBE6Your intervals, n,n+1, n+2, etc. I assume these are all equal intervals.
Nope, just a guy with a lot of time on his hands on a Sunday night. I am an engineer by training, though.
Would it change the results if they were not equal? I was alluding to that in my admittedly rambling post, like maybe a fibonacci series
or some power function with limits suitable for this problem, any differance?
12 Feb 07
Originally posted by sonhouseThe distribution of fuel is g(t) according to PBE's notation. That means it could be uniform or non-uniform. n and n-1 are the only two points used.
Your intervals, n,n+1, n+2, etc. I assume these are all equal intervals.
Would it change the results if they were not equal? I was alluding to that in my admittedly rambling post, like maybe a fibonacci series
or some power function with limits suitable for this problem, any differance?
Seriously, try to comprehend before you post.
12 Feb 07
Originally posted by XanthosNZAs it turned out, I actually asked that question because I did not know the answer so you have a problem asking questions? Bit harsh there Xanth.
The distribution of fuel is g(t) according to PBE's notation. That means it could be uniform or non-uniform. n and n-1 are the only two points used.
Seriously, try to comprehend before you post.