http://www.maa.org/features/elvisdog.pdf
One for the math people. A bit of a long read but pretty interesting - we did a ‘lab’ on this in class today, playing with different speeds for the dog’s swim and run on the beach.
I don’t have a dog, and calculus was a very long time ago, but this reminds me of the “travelling salesman problem.” Basic idea is that given a list of cities to visit, the salesman wants to go to each of them with the minimum amount of driving. As I recall, humans are pretty good at approximating solutions to problems like this, despite the fact that working them out mathematically is ferociously complicated.
For real people and the number of cities that is practical for most trips, the problem scale is small enough that an exact solution may be doable with some patience. Much beyond that and the problem belongs to a class of problems called NP-Complete, for which the time required for an exact solution using the best known algorithms rises exponentially with the number of cities. That is, the number of cities is part of the exponent (rather than the base) in the time-to-solve equation. Lots of heuristics are known for various special cases but since in almost all cases the exact solution isn’t known, how good a solution you have is usually a bit hard to say.
Chris
I suppose there must be x! possible solutions, with x representing the number of cities, but the thing that seems to throw a monkey wrench into solving it is that the values for the distances change with every pair of cities.
In real life, I imagine that the arrangement of highways (no direct routes from some cities to others, and a tendency for highways to provide obvious routes between some clusters of cities) makes it easier to approximate solutions.
This comment in the article reminded me of my experience at IMFL 2005: “On the other hand, since he runs considerably faster than he swims, another option would be to minimize the swimming distance.” When you return to the water for your second loop in the swim, you have to decide how far to run along the beach before entering. For a relatively weak-swimmer/strong-runner like myself, the optimal solution involves a longer run than it would for the rest of you.
Maybe if I ever do the race again I should refresh my memories of calculus first. 
Maybe if I ever do the race again I should refresh my memories of calculus first.
I struggle just to figure out my pace when I’m in the middle of a race.
Do you think they’d let you bring along a waterproof scientific calculator, as long as you didn’t use it for propulsion?
LOL, I was thinking the same thing as I typed my post!
The “traveling salesman” problem, BTW, reminded me of a course I took in graph theory.
I struggle just to figure out my pace when I’m in the middle of a race.
Brush up on your mental math. I can recite derivatives of inverse trig functions at the end of a 20 mi run yeah I still have no idea how THAT came into my head Sunday.
Brush up on your mental math. I can recite derivatives of inverse trig functions at the end of a 20 mi run yeah I still have no idea how THAT came into my head Sunday.
And suddenly, the pocket-protector set, heartbroken by Danica McKellar’s recent nuptuals, has a new poster girl. 
awful hypoglycemic hallucinations, I tell you.
I’m merely glad that an Exercise Physio student is taking Calculus II and thus getting a real education.
Good on you Tiger…
I struggle just to figure out my pace when I’m in the middle of a race.
Brush up on your mental math. I can recite derivatives of inverse trig functions at the end of a 20 mi run yeah I still have no idea how THAT came into my head Sunday.
Jeesh, you’re sounding like my wife. She’s a math teacher, so she yells at me when she sees me reach for a calculator to figure out simple stuff.
As for what mental feats I can accomplish in the heat of mid-race hypoxia, I think that the best I can do is something like “That guy is waving a flag and pointing to the left. Maybe I should start running down that road instead of going straight.”
“I can recite derivatives of inverse trig functions at the end of a 20 mi run yeah I still have no idea how THAT came into my head Sunday.”
I’m too busy doing pace calculations and extrapolations (“how does this compare with BQ pace, if I adjust for disparity in distance, hills, terrain, etc.?”) to do that during a run. But one night, many years after I left school, I did have a dream in which I recalled Taylor’s Theorem. (It was in my head when I woke up.)