Same Distance Up the Hill


It does not matter what everyone has to say, the wife was right.

All about work, not friction and the first law of dining.


I think the woman is right, but there was a lack of communication/understanding. Let’s say the one thing she has convincingly determined is true: both routes are the same distance. Therefore, neither is really more circuitous than the other. The one she prefers simply has a more leisurely and, one must assume, steady grade. If the husband’s route is equally “circuitous,” being the same distance, then who knows what the grade is like on his route? It may be flat almost all the way, or even include some downward grade along the way, and have a very steep hill at the end, which is what the woman seems to be saying. Maybe she didn’t exactly say that, but I’m surprised the Knuckleheads didn’t see that was a possibility. At least they acknowledged that if what the woman said were possible, hers would be the better route. They were just in such a hurry to get the show over with, they didn’t stop to think it certainly was possible. Come on guys! Before you start calling somebody wacko you should make sure you’ve got your own ducks in a row!


that it’s hard to decide where to start.

She said that her husband said that the steep climb has a 60% gradient – that’s just about impossible. Engineering a public road to have a 60% gradient would be an act of colossial stupidity and very difficult.

Then she claims that the route that she likes is far less steep but the it’s the same length. That’s simply not possible.

Now let’s add the idea that the paths are 3.1 miles. That means that the 60% grade would create 1.8 miles of elevation change. I don’t recall where this caller lives, but how many short pieces of road in the lower 48 that have 1.8 miles of elevation change? I can’t think of any. The closest I can think of are some Jeep trails out of Bishop California that lead up into the White Mountains. You can get 5000+ vert on these ‘roads’.

Her description of the situation is so messed up that there is no way to guess what the actual facts are.

Add to that the stupid answer that Ray gave about it requiring the same amount of work to travel two paths of unequal length, and this has gotten far too silly.


Well rowed ducks do not guarantee non-wackoness.
Now, please go to my recent post ODD KEYLESS REMOTE PROBLEM and solve it. Surely this is not the unsolvable problem.


Shortest distance between two points is a straight line. So 3.1miles for both routes is BS ( assuming one way is straight up). The work is the same.


(I emailed an earlier version of this seperately, but now that I know there is a discussion group I am posting it.)

I don’t think she is as entirely whacko as claimed.

The apparent paradox in her story can be resolved looking at the trigonometry at small numbers of degrees and considering the inaccuracies of automobile odometers that is rounding to 1 decimal place.

There are four numbers underlying this problem.

  1. The entire trip measures 3.1 miles. So, the shortest measured drive is 3.050 miles and the longest is 3.149 miles.
  2. As you pointed out the actual grade of a hill that seems very steep is a very small number of degrees, let’s say 15 degrees.
  3. The woman said on her long route she did not perceive a climb. Let’s say thats 1 degree.
  4. The steeper trip is only at its steepest some percentage of the way.

The question is can we describe realistic scenarios where the perception would be there is no difference in distance, when in fact there is but it just is being poorly measured. With the figures this can happen easily.

Picture an overhead map of the two routes. The “as the crow flies” distance from the starting points to the top of the hill, as you might connect the dots between the two location on the map, is going to be a shorter distance than that experienced by one climbing a hill. Or put another way both of these people are traveling a greater distance than the “as the crow flies” distance. This distance determines the cases that must be considered.

Depending on other assumptions, from a real “as the crow flies” distance of anywhere from 3.0495 miles to 3.122 miles, the two different trip that she described could both happen and both show 3.1 miles on the odometer.

Assume the “as the crow flies” distance is 3.0495 miles and that the steep climb involves a 15 degree climb for 1/4 the trip and 0 for the rest and that the minimum climb trip is all at 1 degree the whole way. A little trigonometry show that the odometers on both cars could read 3.1 miles, by rounding. With the 1/4 assumption for the steep climb the cars would could both show a 3.1 miles reading anywhere from 3.0495 miles to 3.1224 miles of real “as the crow flies” distance.

Even assuming that the steep climb involves 15 degrees of climb for 3/4 of the trip and 0 for the rest and that the minimum climb trip is all at 1 degree the whole way. The odometers on both cars could read 3.1 miles, by rounding, if the “as the crow flies distance” is anywhere from 3.0495 miles to 3.0687 miles.

Now, although this I got these numbers with trigonometry, of course, the analysis is not about really about right triangles. If the situation had to fit into a right triangle, then the different angles would give different hieghts to the hill. But remember, the drivers are driving windy roads no a hypotenuse. The above combinations can work.

So, this might be a very special circumstance, but not an impossible one.

From a geometry teacher,

  • Bob Benjamin


OK be merciful. That was dumb. Saw the error of my ways.


  • Bob


I did some investigation of hills in the neighborhood of Poland NY via Google maps. Poland sits on a river bank; behind it is Buck Hill rising to over 1200’ in just under a mile via Buck Hill road. That’s a 13% grade on average.

Or you can follow a road that goes up the river, and turns to follow another river that comes down from Buck Hill. The slope would be much gentler.

The only plausible destination via either road within a reasonable distance of the cited 3.9 miles is the town of Russia. Here are the Google directions for both ways:

Start Poland, NY
End Russia, NY

Travel 3.7 mi ? about 8 mins

  1. Head east on N Main St/RT-28/RT-8 toward N Main St
    Continue to follow N Main St/RT-28: 0.7 mi 2 mins
  2. Head northeast on RT-8 E toward Sprague St: 0.5 mi 1 min
  3. Turn left at Buck Hill Rd: 1.0 mi 3 mins
  4. Turn left at Military Rd: 1.5 mi 3 mins

To: Russia, NY

Start Poland, NY
End Russia, NY

Travel 4.9 mi ? about 7 mins

  1. Head west on N Main St/RT-28/RT-8 toward Maple Leaf Ln
    Continue to follow RT-28/RT-8: 0.8 mi 1 min
  2. Turn right at RT-28: 1.8 mi 3 mins
  3. Turn right at Gravesville Rd: 0.6 mi 1 min
  4. Continue on Russia Rd: 1.7 mi 2 mins
  5. Slight right at Military Rd 318 ft

To: Russia, NY

Assuming Mara is no better at odometer readings than she is with grades, this seems to explain it. She really does go farther than her husband if they are starting from Poland. But according to Google, they both arrive at nearly the same time - perhaps Google makes some assumptions about speed when driving up a 13% slope her husband does not share?

But her odometer is really seriously out of whack if it’s under-reporting mileage by about 20% - I really can’t believe that she would fudge her reports to get one-up on her husband. Who ever heard of such a thing?

I concur, however, on gas mileage. Fighting gravity up a 15% grade is like lead-foot acceleration for its effects on fuel economy.


I think everyone is making this WAY to complicated. Just imagine a right triangle except the hypotenuse, zigzags.

Let?s assume she is correct about the distance and time of travel. I think we all agree that IF she were to travel in a straight line from bottom to top (on the hypotenuse of a triangle), the distance would be shorter then if she were to travel along the other two lines of the same triangle.

If you look at a right triangle who?s sides are 1? X 1? X 1 7/16?. The husband?s route is a total of 2?; while the wife?s route is only 1 7/16? unless her route zigzags enough to make it 2?. Think of it in 3D not 2D.

My conclusion is, he?s road is the more direct route (on a compass but not in elevation) while hers strays a bit on the compass but maintains a more consistent rate of ascent.

I am not much with words but my math (if I have expressed it correctly) is (I believe) irrefutable. Seeing as I am infallible, :slight_smile: I am sure that everyone here is now slapping their own heads asking why they didn?t think of something so perfect.

This still doesn?t address her question though. Which route uses less gas? I could, of course, give you all the correct answer but I would much rather watch everyone try and figure this out for them selves. How else will you learn if not by discussion?

Your thoughts?


Build it and they will come.