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Author: Subject: A puzzler: "Fun" with Averages
mayko
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[*] posted on 12-3-2016 at 16:34


Quote: Originally posted by aga  
Spending a few (tens of) minutes with a pen and paper, i get 90mph for 1 hour to work.

... provided that the route is not a straight line from A to B, but a circle where start and finish are both point A.


Ok, I might be losing my thouch. I implied but never explicitly defined a property called the topology of the space the question takes place in.

The intended topology was not necessarily a straight line segment, but something that is "line-sement like" (topologists would say that the space is homeomorphic to a line sement.) This would be a reasonable description of a single road, from point A to point B, on which Sue can drive back and forth, but cannot offroad. It might be a Euclidean, as-the-crow-flies straight line, or it could wiggle all over the place; all that's important for the problem is how much distance the road takes up.

There are actually lots of other topologies we could give him (for example, a grid-like network of roads, like in New York City). You give the example of a circular race track; a circle (or a thing homeomorphic to a circle) is a topologically different space than a line segment; for example, you can take a point out of a circle-like track while preserving a property called "path connectedness" (if we put up a roadblock on the circular track, Sue can still drive to any other point on it - not so with a line segment!). First, let's look at the most general description of the problem, then look at two alternate topologies.

For any particular part of the trip x, the speed $$S_x$$ is related to the distance traveled $$D_x$$ and the time taken $$T_x$$:
$$S_x = \frac{D_x}{T_x}$$

Thus, for the forward, return, and total trips, respectively:
$$S_f = \frac{D_f}{T_f}$$
$$S_r = \frac{D_r}{T_r}$$
$$S_t = \frac{D_t}{T_t}$$

Moreover, the total distance and time are the sums of the component distances and times:
$$D_t = D_f + D_r $$
$$T_t = T_f + T_r $$

Merging these identities and churning the algebra, I get the identity:

$$S_r = \frac{D_r}{\frac{D_r+D_f}{S_t}-\frac{D_f}{S_f}}$$

The intended interpretation fixes the forward and return distances as equal: $$D_r = D_f = 1mi$$ Combined with the constraint that $$S_t = 2S_f$$ the denominator implodes.





Let's suppose he has an ATV, though, and can offroad. This means the topology available is "flat-plane-like". In particular, there is no particular limit on how far back Sue can drive in order to complete the circuit. With $$D_r$$ no longer constrained, the above identity works out:
$$S_r = \frac{D_r S_t}{D_r-D_f}$$
Which has a meaningful solution so long as Sue takes a longer drive back than forward.




Now let's consider the closed-loop track. This constrains the return distance to some whole-number multiple of loops around the track : $$D_r=w D_f$$ (if he doesn't do this, he hasn't returned to the starting point, and hasn't "driven back" ).
Topologists often call w a winding number

Combing this constraint with the above identity, I get the relation:

$$S_r = \frac{w}{w-1}S_t$$

Note that the intended interpretation is more or less equivalent to w=1.







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mayko
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[*] posted on 12-3-2016 at 16:43


(PS - I'm sure you meant no ill will, but you should be aware that tr*nny is generally considered a pretty bad slur, and I would caution against using it casually, or at all.)

Often in economics, problems are by default written around women because it tends to shake the reader a bit; I can think of a couple where the riddle actually hinges upon unconcious sex prejudices. I sometimes use the boy named Sue for the same disorienting effect. I'm surprised IRC and I are the only Cash fans about- good to have something in common at last :P




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"Wubbalubba dub-dub!" - Rick Sanchez
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aga
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[*] posted on 13-3-2016 at 12:19


So the route is circular then ?

P.S. We all like Cash. Lots and lots of Cash every day is great.

Not so sure about the Johnny, whoever she is.
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[*] posted on 13-3-2016 at 13:54


Quote: Originally posted by aga  
Not so sure about the Johnny


https://www.youtube.com/watch?v=-OVHnWyDmdw

https://www.youtube.com/watch?v=ZE9KWU-ntBk

https://www.youtube.com/watch?v=KHF9itPLUo4

A few years ago I downloaded the movie "I walk the line" which was free but now all I see are scams and pay links. The movie would tell you about him and understanding it would give you an idea of this country as it was for generations. Alas, no longer.




"Science is the belief in the ignorance of the experts" Richard Feynman
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mayko
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[*] posted on 13-3-2016 at 15:47


Quote: Originally posted by aga  
So the route is circular then ?


Like I said, the structure of space was implied but not specified. Under the implied structure, there are no finite solutions. A circle-like course will work, though your proposed solution of "90mph for 1 hour" is off. (90 mph for 1 hr gives 90 mi, for a total distance of 91 miles and a total time of 62 minutes, ie a trip average of 87.6 mph ) However, 90 mph for 2 minutes (ie w=3) should do the trick.

Here's another example of a puzzle where (possibly unstated) assumptions about the structure of space can determine if an answer even exists:


Quote:

A city contains three utility plants: water, gas, electricity. For safety reasons, no utility lines can cross: no electric cables over water mains, etc. There are three factories which need all three utilities. Can you do this safely?


As it happens, on a sphere like planet Earth, this is not possible, for reasons related to the Four Color Map Theorem. However, it's totally feasible on Toroidia, the legendary doughnut shaped planet!




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aga
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[*] posted on 13-3-2016 at 23:48


Doh !

> drives for one mile at thirty miles per hour.

I wrote that down as 1 hour at 30mph.

There's a basic assumption gone wrong already : that i could read straight.
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