## 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?

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.

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!

## 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 remote-associates-test.com, 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

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

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

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

## My Insecurity Over Security Codes

Every time I attempt to access one of my company’s applications via our single sign-on (SSO) system, I’m required to request a validation code that is then sent to my smartphone, and then I enter that code on the login page.

It’s a minor nuisance that drives me insane.

The purpose of the codes are to provide an additional level of security, but given how un-random the codes seem to be, it doesn’t feel very secure to me. This screenshot shows some of the codes that I’ve received recently:

Here’s what I’ve observed:

• Every security code contains 6 digits.
• The first 3 digits in the code form either an arithmetic or geometric sequence, or the first 3 digits contain a repeated digit.
• Similarly, the last 3 digits in the code form either an arithmetic or geometric sequence, or the last 3 digits contain a repeated digit.

As an example, one of the codes in the screenshot above is 421774. The first 3 digits form the (descending) geometric sequence 4, 2, 1, and the digit 7 appears twice in the second half of the code.

I believe the reason for these patterns is to make the codes more memorable to those of us who have to transcribe them from our phones to our laptops.

This got me thinking. The likelihood of someone correctly guessing a six-digit code is 1 in 1,000,000. But what is the likelihood that someone could correctly guess a six-digit code if it adheres to the rules above?

If you’d like to answer this question on your own, stop reading here. To put some space between you and my solution, here’s a security-related joke:

“I don’t understand how someone stole my identity,” Lily said. “My PIN is so secure!”

“The year of Knut Långe’s death,” Lily replied.

“Who is Knut Långe?”

“A King of Sweden who usurped the throne from Erik Eriksson.”

“And what year did he die?”

“1234.”

(Incidentally, Data Genetics reviewed 3.4 million stolen website passwords, and they found that 1234 was the most popular four-digit code. The researchers claimed that they could use this information to make predictions about ATM PINs, too, but I don’t think so. All this shows is that 1234 is the most commonly stolen password, and therefore this inference suffers from survivorship bias. Without having data on all the codes that were not stolen, it’s impossible to make a reasonable claim. But, I digress.)

To determine the number of validation codes that adhere to the patterns I observed, I started by counting the number of arithmetic sequences. With only 3 digits, there are 20 possible sequences:

• 012
• 024
• 036
• 048
• 123
• 135
• 147
• 159
• 234
• 246
• 258
• 345
• 357
• 369
• 456
• 468
• 567
• 579
• 678
• 789

But each of those could also appear in reverse (210, 975, etc.), giving a total of 40.

There are far fewer geometric sequences; in fact, only 3 of them:

• 124
• 139
• 248

And again, each of those could appear in reverse, giving a total of 6.

Finally, there are 10 × 9 × 8 = 720 three-digit numbers with no repeated digits, which means there are 1,000 ‑ 720 = 280 numbers with a repeated digit. (Here, “number” refers to any string of 3 digits, including those that start with a 0, like 007 or 092.)

Consequently, there are 40 + 6 + 280 = 326 possible combinations for the first 3 digits and also 326 combinations for the last 3 digits, which gives a total of 326 × 326 = 106,276 possible validation codes.

That means that it would be about 10× more likely for a phisher to correctly guess a validation code that follows these rules than to guess a completely random six-digit code. But said another way, the odds are still significantly against a phisher who’s trying to steal my code. And quite frankly, if someone wants to exert that kind of effort to pirate my access to Microsoft Word online, well, I say, go for it.

## 8-15-17

Today is a glorious day!

The date is 8/15/17, which is mathematically significant because those three numbers represent a Pythagorean triple:

$8^2 + 15^2 = 17^2$

But August 15 has also been historically important:

But as of today, August 15 has one more reason to brag: It’s the official publication date of a bestseller-to-be…

Like its predecessor, this second volume of math humor contains over 400 jokes. Faithful readers of this blog may have seen a few of them before, but most are new. And if you own a copy of the original Math Jokes 4 Mathy Folks, well, fear not — you won’t see any repeats.

What kind of amazing material will you find on the pages of More Jokes 4 Mathy Folks? There are jokes about school…

An excited son says, “I got 100% in math class today!”

“That’s great!” his mom replies. “On what?”

The son says, “50% on my homework, and 50% on my quiz!”

There are jokes about mathematical professions…

An actuary, an underwriter, and an insurance salesperson are riding in a car. The salesperson has his foot on the gas, the underwriter has her foot on the brake, and the actuary is looking out the back window telling them where to go.

There are Tom Swifties…

“13/6 is a fraction,” said Tom improperly.

And, of course, there are pure math jokes to amuse your inner geek…

You know you’re a mathematician if you’ve ever wondered how Euler pronounced Euclid.

Hungry for more? Sorry, you’ll have to buy a copy to sate that craving.

To purchase a copy for yourself or for the math geeks in your life, visit Amazon, where MoreJ4MF is already getting rave reviews:

For quantity discounts, visit Robert D. Reed Publishers.

## Mo’ Math Limericks

I’ve posted limericks to this blog before. Quite a few, in fact.

But a friend recently sent me The Mathematical Magpie, a collection of math essays, stories and poems assembled by Clifton Fadiman and published by Simon and Schuster in 1962. Coincidentally, one section of the book is titled Comic Sections, the name of a mathematical joke book written by Des MacHale in 1993. (I contacted Professor MacHale several years ago, and he suggested that we swap books. Best. Trade. Ever.) Des MacHale is Emeritus Professor at the University of Cork, a mere 102 km from Limerick, Ireland… which brings us full circle to today’s topic.

The Mathematical Magpie contains quite a few limericks, one of which you have likely heard before:

There was a young lady named Bright,
Who traveled much faster than light.
She started one day
In the relative way,
And returned on the previous night.

Despite a variety of other claims, that limerick was written by Professor A. H. Reginald Buller, F.R.S., a biologist who received £2 when the poem was published in Punch, and he “was more excited at the check than he was later when his book on fungi was published.”

You may not, however, be familiar with Professor Buller’s follow-up limerick about Miss Bright:

To her friends said the Bright one in chatter,
“I have learned something new about matter:
As my speed was so great
Much increased was my weight,
Yet I failed to become any fatter!”

Here are a few other limericks that appear in The Mathematical Magpie:

There was an old man who said, “Do
Tell me how I’m to add two and two?
I’m not very sure
That it doesn’t make four —
But I fear that is almost too few.
Anon.

The topologist’s mind came unguided
When his theories, some colleagues derided.
Out of Möbius strips
Paper dolls he now snips,
Non-Euclidean, closed, and one-sided.
Hilbert Schenck, Jr.

A mathematician named Ray
Says extraction of cubes is child’s play.
You don’t need equations
Or long calculations
Just hot water to run on the tray.
L. A. Graham

Flappity, floppity, flip!
The mouse on the Möbius strip.
The strip revolved,
The mouse dissolved
In a chronodimensional skip.
Frederick Winsor

And though it’s not a limerick, this one is just too good not to include for your enjoyment:

A diller, a dollar,
A witless trig scholar
On a ladder against a wall.
If length over height
Gives an angle too slight,
The cosecant may prove his downfall.
L. A. Graham

Finally, I leave you with a MJ4MF original:

With my head in an oven
And my feet on some ice,
I’d say that, on average,
I feel rather nice!

Got any math poems or limericks you’d like to share? We’d love to hear them!

## Just Sayin’

Heidi Lang is one of the amazing teachers at Thomas Jefferson Elementary School. When she’s not challenging my sons with interesting puzzles and problems, she’s entertaining them with jokes that make them think. On her classroom door is a sign titled Just Sayin’, under which hangs a variety of puns. Here’s one of them:

Last night, I was wondering why I couldn’t see the sun. Then it dawned on me.

That reminds me of one of my favorite jokes:

I wondered why the baseball kept getting larger. Then it hit me.

Occasionally, one of her puns has a mathematical twist:

Did you know they won’t be making yardsticks any longer?

And this is one of her mathematical puns, though I’ve modified it a bit:

When he picked up a 20‑pound rock and threw it 5,280 feet, well, that was a real milestone.

I so enjoy reading Ms. Lang’s Just Sayin’ puns that I decided to create some of my own. I suspect I’ll be able to hear you groan…

• He put 3 feet of bouillon in the stockyard.
• When the NFL coach went to the bank, he got his quarterback.
• She put 16 ounces of poodle in the dog pound.
• The accountant thought the pennies were guilty. But how many mills are innocent?
• His wife felt bad when she hit him in the ass with 2⅓ gallons of water, so she gave him a peck on the cheek.
• Does she know that there are 12 eggs in a carton? Sadly, she dozen.
• When his daughter missed the first 1/180 of the circle, he gave her the third degree.
• She caught a fish that weighed 4 ounces and measured 475 nm on the visible spectrum. It was a blue gill.
• When Rod goes to the lake, he uses a stick that is 16.5 feet long. He calls it his fishing rod.
• What is a New York minute times a New York minute? Times Square.
• I wanted to dance after drinking 31 gallons of Budweiser, so I asked the band to play the beer barrel polka.
• The algebra teacher was surprised by the mass when she tried to weigh the ball: b ounces.

And because this post would feel incomplete without it, here’s probably the most famous joke of this ilk:

• In London, a pound of hamburger weighs about a pound.

## MORE Jokes 4 Mathy Folks

I know, I know.

You remember the day that you bought Math Jokes 4 Mathy Folks. You headed directly home from the bookstore and read it cover to cover. Then, once the tears of laughter had dried, you read it again. And sure, you were a little concerned that if you read it a third time, well, you might be accused of neglecting your family. But social reputation be damned… you’re a mathy folk, and neglecting people is what we do. So you returned to the first page and gave it one more go.

That day was several years ago.

Today, MJ4MF occupies a position of honor on your bathroom shelf, and while conducting your business you occasionally open to a random page, hoping to rediscover an old chestnut. But alas, you’ve read it so many times, you have every joke memorized, and the cover is falling off.

So, now what?

Well, don’t worry. You’ve waited patiently, and your patience is about to be rewarded. Announcing the release of the second volume in the MJ4MF franchise…

Head over to Amazon to order a copy today! Officially, it isn’t available until August 15, 2017 (bonus points if you know why that date was selected as the publication date), but you can get it now, and you’ll have plenty of time to memorize the jokes before the first day of school.

(And while you’re there, you should probably buy a replacement copy of Math Jokes 4 Mathy Folks, too. Get a new one with its cover intact. You don’t want to look like someone who doesn’t take care of your books, do you? Of course not. And besides, purchasing another copy for you will boost the sales ranking for me. Win-win.)

So, what will you find in this new collection? Over 400 jokes, from every branch of mathematics.

 Pentagon Hexagon Oregon

An excited son says, “I got 100% in math class today!”

“That’s great!” his mom replies. “On what?”

The son says, “50% on my homework, and 50% on my quiz!”

What is PA + PN + LA + LN?

A (P + L)(A + N) that’s been FOILed.

Heck, there are even jokes about other counting systems…

What happened in the binary race?

Zero won.

And what won’t you find in this new collection? You won’t find a single one of the 400+ jokes that were in the original Math Jokes 4 Mathy Folks. That’s right, this collection is 100% entirely new!

Don’t delay! Be the coolest kid on your block by ordering a copy of MORE Jokes 4 Mathy Folks today!

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.