Archive for September, 2017

Hiking Routes, Square Roots, and Trail Ratings

Not all those who wander are lost.

You may have read that line in Gandalf’s letter to Frodo Baggins in The Fellowship of the Ring, but nowadays you’re more likely to see it on the t-shirts and bumper stickers of hikers.

Hiking is a popular sport, and the 47 million Americans who reported that they’ve taken a hike in the past 12 months (Statista) had a lot of different trails to choose from: American Trails maintains a database of over 1,100 trails, and Backpacker‘s list of America’s Best Long Trails offers an impressive 39,000 combined miles. Plus, there are thousands of miles of trail not on either of those lists. With so many options, how are you supposed to choose?

Tour Du Mont Blanc

A wealth of information is provided for most hiking trails. But while some information — like distance and elevation gain — is absolute, other information leaves room for interpretation. What does it mean when the Craggy Pinnacle Trail just outside Asheville, NC, is described as a “moderate” hike? The Explore Asheville website says,

Moderate hikes could range anywhere from a few to ten miles with an elevation gain up to 2,000 feet.

By those standards, a three-mile hike with a 10% grade would be considered moderate. No, thank you.

Unfortunately, there is no standardized system for determining trail difficulty. Most of the time, the trail rating is a nebulous qualitative combination based on an examination of the terrain, trail conditions, length, elevation gain, and the rater’s disposition.

But I tip my hat to the good folks in Shenandoah National Park who have attempted to quantify this process. Their solution? The simple formula

r = \sqrt{2gd}

where g is the elevation gain (in feet) and d is the distance (in miles). The value of r then corresponds to a trail rating from the following table:

Numerical Rating Level of Difficulty Estimated Average Pace (miles per hour)
< 50 Easiest 1.5
50-100 Moderate 1.4
100-150 Moderately Strenuous 1.3
150-200 Strenuous 1.2
> 200 Very Strenuous 1.2

Elevation gain is defined as the cumulative elevation gain over the entire hike. So if the hike climbs 300 feet over the first mile, then descends 500 feet over the next 2 miles, then goes back up 200 feet to return to the start, the elevation gain is reported as 300 + 200 = 500 feet.

Old Rag is one of the most popular hikes in northern Virginia. Known for the half-mile rock scramble near the top, this trail boasts an impressive 2,415 feet of elevation gain over 9.1 miles. Applying the formula,

r = \sqrt{2 \cdot 2415 \cdot 9.1} \approx 209.6

which means Old Rag’s level of difficulty would be “very strenuous.”

This formula could lead to several activities for a middle or high school classroom:

  • Draw an elevation map depicting a trail on which any type of hike (from easiest to very strenuous) would be possible, depending on how far a person hiked.
  • With distance on the horizontal axis and elevation gain on the vertical axis, create a graph that shows the functions for easiest to very strenuous hikes. (See Figure 1.)
  • If you were on a trail with an average elevation gain of 300 feet per mile, how long would you have to hike for it to be considered a moderately strenuous hike?
  • If one 5-mile hike is rated “easiest” and another 5-mile hike is rated “strenuous,” what’s the minimum possible difference in elevation gains for the two trails?

Students could also do a comparison between this trail rating formula and the geometric mean, if you wanted to go really crazy.

Feel free to drop some of your ideas into the comments.

Figure 1. The graphs for various difficulty ratings, using the Shenandoah trail rating formula.

Just as every good hike comes to an end, so must this blog post. But not before we laugh a little.

As it turns out, there’s a math joke about hiking…

An actuary has been walking for several hours when the trail ends at the edge of a river. Having no idea how to cross, she sees another hiker on the opposite bank, and she yells, “Hey, how do I get to the other side?”

The man across the river — a math professor — looks upstream, then downstream, then thinks a bit and finally says, “But you are on the other side!”

It’s a math joke about hiking as much as any joke about any topic is a math joke, if you insert the correct professions.

There’s a great non-math joke about hiking, too…

A fish is hiking through a reservoir when he walks into a wall. “Dam!” he says.

And there is a very mathematical list about hiking, which might be considered a joke if so many of the observations weren’t true…

Eight Mathematical Lessons from the Trail

  1. A pebble in a hiking boot will migrate to the point of maximum irritation.
  2. The distance to the trailhead where you parked remains constant as twilight approaches.
  3. The sun sets at two-and-a-half times its normal rate when you’ re trying to reach the trailhead before dark.
  4. The mosquito population at any given location is inversely proportional to the effectiveness of your repellent.
  5. Waterproof rainwear isn’t. But, it is 100% effective at containing sweat.
  6. The width of backpack straps decreases with the distance hiked. To compensate, the weight of the backpack increases.
  7. The ambient temperature increases proportionally to the amount of extra clothing in your backpack.
  8. The weight in a backpack can never remain uniformly distributed.

Go take a hike!

September 23, 2017 at 5:01 pm Leave a comment

The Remote Associates (RAT) and Close Associates (CAT) Tests

The Remote Associates Test (RAT) is a test used to determine a person’s creative potential. When given a collection of three seemingly unrelated words, subjects are asked to supply a fourth word that is somehow related to each of the three stimulus words. For example, if you were given the words cottage, swiss, and cake, you’d answer cheese, because when combined with the three stimuli, you get cottage cheese, swiss cheese, and cheesecake.

To try your hand at one of these tests, you can head to, or you can just keep on reading.

The verdict is out as to whether a high score on the RAT actually means you’re more creative. But what’s not in doubt is how much I love to solve, and to create, these items. It’s a good game for long car rides, and my wife, sons, and I can amuse ourselves for hours by creating and sharing them with one another.

For your enjoyment, I present the following mathematical RAT test: The fourth word related to the three stimulus words in each set is a common math term. Enjoy!

  1. attack / acute / fish
  2. inner / around / full
  3. phone / one / mixed
  4. powers / rotational / point
  5. up / on / hot
  6. tipping / blank / selling
  7. even / duck / couple
  8. black / Bermuda / love
  9. inkhorn / limits / short
  10. sub / hour / tolerance
  11. world / television / infinite
  12. ball / camp / data
  13. Dracula / head / sheep
  14. field / dead / stage
  15. town / off / meal
  16. air / hydro / geometry
  17. common / X / fudge
  18. key / bodily / dis
  19. disaster / 51 / grey
  20. ball / S / hairpin

The Close Associates Test (CAT) is a similar, yet completely fictitious, test that I just made up. Each item on a CAT test contains three words which do not appear to be unrelated in the least; in fact, they are so closely related that finding the fourth word they have in common is somewhat trivial. The following mathematical CAT test will not measure your creativity, though it might reasonably determine the depth of your mathematical vocabulary. Good luck!

  1. acute / obtuse / right
  2. scalene / equilateral / isosceles
  3. sine / logistic / regression
  4. real / irrational / whole
  5. proper / improper / reduced
  6. convergent / infinite / divergent
  7. in / circum / ortho
  8. Pythagorean / De Moivre’s / Ramsey’s
  9. exponential / differential / Diophantine
  10. convex / concave / regular
  11. golden / common / test
  12. square / cube / rational

RAT Answers:

  1. angle
  2. circle
  3. number
  4. axis
  5. line
  6. point
  7. odd
  8. triangle
  9. term
  10. zero
  11. series
  12. base
  13. count
  14. center
  15. square
  16. plane
  17. factor
  18. function
  19. area
  20. curve

CAT Answers:

  1. angle
  2. triangle
  3. curve
  4. number
  5. fraction
  6. series
  7. center
  8. theorem
  9. equation
  10. polygon
  11. ratio
  12. root (or number)

Feel free to submit more triples for the RAT or CAT test in the comments.

September 6, 2017 at 7:24 am Leave a comment

About MJ4MF

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

MJ4MF (offline version)

Math Jokes 4 Mathy Folks is available from Amazon, Borders, Barnes & Noble, NCTM, Robert D. Reed Publishers, and other purveyors of exceptional literature.

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September 2017

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