## What’s in Your Pocket?

I recently received an email from adoring fan Alden Bradford:

Teacher: “Would you like a pocket calculator?”

Student: “No, thanks. I already know how many pockets I have.”

Thanks, Alden!

Of course, that reminded me of this gem from Spiked Math:

And one final pocket joke:

The department chair said to the math teachers, “I have good news, and I have bad news. The good news is, we have enough money for a new microwave in the staff lounge.” The teachers cheered! Then one of them asked, “What’s the bad news?” The chair said, “It’s still in your pockets.”

Ouch.

## No Bull — This is My New Favorite Fermi Question

It’s hard to say which emotion was strongest — awe, bewilderment, admiration, horror, fear — when I heard the following statistic:

McDonald’s sells 75 hamburgers every second.

But I’m a math guy, so there’s no doubt where my mind turned after that emotion passed:

How many cows is that?

Have at it, internet.

What do you get when you divide the circumference of a bovine by its diameter?

Cow pi.What is the favorite course at Bovine College?

Cowculus.A mathematician counted 196 cows in the field. But when he rounded them up, he got 200.

## Math Words for National Dictionary Day

Want to start today the right way? Say, “Good morning!” to Alexa today, and she’ll respond:

Good morning! It’s National Dictionary Day. Ever wonder what the shortest word is? Technically, it’s a toss-up between the single letter words

andI, but sinceais always capitalized, I’d sayIis just a little shorter.a

Is there anything more powerful than a language arts joke to get the day off to a good start?

I have no words to describe today. I do, however, have a ton of obscene gestures.

So, what’s the shortest **math** word? Technically, *e* and *i*, but if you don’t like constants, then you’ll have to settle for the three-letter words *set* and *box*.

And what’s the longest math word — at least based on the list at Math Words? It has 17 letters, and you’ll get a big hint if you check the time.

What two math words, both having the same number of letters, are equally appropriate to describe a triangle whose sides are congruent?

And what’s the funniest math word? Personally, I think it’s *syzygy*, but according to Tomas Engelthaler, it’s *logic*. In Humor Norms for 4,997 English Words, Engelthaler and Hills (2017) describe a method for determining which words are funniest. I emailed Engelthaler to ask which math word is funniest, and he responded as if it were a completely reasonable question. Without hesitation, he shared a list of math words and their humor rankings, and these five were at the top of the list:

- logic
- math
- theory
- science
- graph

The overall funniest English word, according to Engelthaler’s research? *Booty*. Go figure.

While you may not think that any of those words, mathy or otherwise, are laugh-out-loud funny, this isn’t debatable; it’s based on science.

If you take issue with this research, you’ll need to discuss it with Engelthaler and his colleagues. Please write to him directly to say that you’re bumfuzzled, that his research is malarkey, or that you think he’s a nincompoop.

## Mr. Consistency, Khris Davis

If you flipped four coins, the probability of getting exactly one head would be 0.25.

But the probability of doing that four times in a row is much lower, somewhere closer to 0.0039, or about 1 in 250.

Now, imagine flipping 100 coins four times, and getting the same number of heads each time. The odds of that happening are only slightly better than impossible. In fact, if every person *in the entire world* were to flip 100 coins four times, it would still be highly unlikely that this would ever happen.

That’s how rare it is, and it gives you some idea of what Major League Baseball player Khris Davis just pulled off. The Oakland Athletics outfielder just finished his fourth consecutive season with a batting average of .247. That’s right — the same average four seasons in a row.

Davis had some advantage over our coins, though. For starters, he wasn’t required to have the same number of at-bats every year. Moreover, batting averages are rounded to three decimal places, so his average wasn’t *exactly* the same during those four years; it was just really, really close:

**2015**: .24745 (97 hits in 392 at-bats)**2016**: .24685 (137 in 555)**2017**: .24735 (140 in 566)**2018**: .24653 (142 in 576)

How could something like this happen? According to Davis, “I guess it was meant to be.“

Perhaps it *was* predestination, but I prefer to put my faith in numbers.

Empirically, we can look at the data. From 1876 to present, there have been 19,103 players in the major leagues. The average length of an MLB career is about 5.6 years, which means that an average player would have about three chances to record the same batting average four seasons in a row. It’s then reasonable to say that there have been approximately 3 × 19,103 = 57,309 opportunities for this to happen, yet Khris Davis is the only one to accomplish this feat. So experimentally, the probability is about 1 in 60,000.

Theoretically, we can look at the number of ways a player could finish a season with a .247 batting average. In 2007, the Phillies’ Jimmy Rollins recorded an astounding 716 at-bats. That’s the most ever by a Major League Baseball player. So using a sample space from 1 to 716 at-bats, I determined the number of ways to achieve a .247 batting average:

- 18 hits, 73 at-bats
- 19 hits, 77 at-bats
- 20 hits, 81 at-bats
- 21 hits, 85 at-bats
- 22 hits, 89 at-bats
- 36 hits, 146 at-bats
- …
- 161 hits, 652 at-bats
- 161 hits, 653 at-bats
- …
- 177 hits, 716 at-bats

And, of course, there are the examples above from Davis’s last four seasons.

It’s interesting that it’s not possible to obtain a batting average of .247 if the number of at-bats is anywhere from 90 to 145; yet it’s possible to hit .247 with 161 hits for either 652 or 653 at-bats. I guess it’s like Ernie said: “That’s how the numbers go.“

All told, **there are 245 different ways to hit .247** if the number of at-bats is 716 or fewer.

That may sound like a lot, but consider the alternative: there are 256,441 ways to **not** hit .247 with 716 or fewer at-bats.

So, yeah. No matter how you look at it, what Davis did is pretty ridiculous. Almost as ridiculous as what happened to Saul…

Saul is working in his store when he hears a voice from above. “Saul, sell your business,” the voice says. He ignores it. His business is doing well, and he’s happy. “Saul, sell your business,” the voice repeats. The voice goes on like this for days, then weeks. “Saul, sell your business.” Finally, Saul can’t take it any more. He finds a buyer and sells his business for a nice profit.

“Saul, take your money, and go to Las Vegas,” the voice says.

“But why?” asks Saul. “I have enough to retire!”

“Saul, take your money to Las Vegas,” the voice repeats. It is incessant. Finally, Saul relents and heads to Vegas.

“Saul, go to the blackjack table and bet all your money on one hand.”

He hesitates for a moment, but he knows the voice won’t stop. So, he places his bet. He’s dealt 18, while the dealer has a 6 showing. “Saul, take a card.”

“What? The dealer has…”

“Saul, take a card!” the voice booms.

Saul hits. He gets an ace, 19. He sighs in relief.

“Saul, take another card.”

“You’ve got to be kidding me!” he pleads.

“Saul, take another card.”

He asks for another card. Another ace, 20.

“Saul, take another card,” the voice demands.

But I have 20!” Saul shouts.

“TAKE ANOTHER CARD, SAUL!”

“Hit me,” Saul says meekly. He gets another ace, 21.

And the voice says, “Un-fucking-believable!”

## One-Letter Quiz

The answer to each question below is a letter of the alphabet. Each letter is used exactly once. (Thanks for the idea, *Ask Me Another*.) Good luck!

**Want to amuse your friends, irritate your students, or annoy people you’ve just met? Download a PDF version of the One-Letter Quiz (without answers).**

- The letter used to represent the square root of -1.
- This letter is often added to indefinite integrals to show that any function with at least one antiderivative has an infinite number of them.
- The most frequently occurring letter in English words.
- The letter most recently added to the modern, 26-letter English alphabet.
- The letter represented by four dots in Morse Code.
- A type of road intersection with three arms.
- Although long out of use, this letter was used in the middle ages as the Roman numeral to represent 90.
- This letter is used for the temperature scale in which the boiling point is 212 degrees and the freezing point is 32 degrees.
- The most common blood type.
- The rating from the Motion Picture Association of America that requires children under 17 to be accompanied by an adult.
- The 43rd President of the United States.
- The only vowel that does not appear in the spelling of any single-, double-, or triple-digit numbers.
- Behind
*s*and*c*, the third most common letter with which English words begin. - With
*plan*, the letter used to refer to a typically less desirable alternative. - The Roman numeral for 500.
- The symbol for potassium on the periodic table.
- The most common variable in algebra.
- The Roman numeral for 5.
- The “score” used to indicate the number of standard deviations a data point is from the mean.
- The letter commonly used to refer to the vertical axis on a coordinate graph.
- Although every adult can recognize the loop-tail version of this lowercase letter in print, less than one-third of participants in a Johns Hopkins study could correctly pick it out of a four-option lineup.
- The clothing size that increases when preceded by an X.
- The shape of the “happiness curve,” which implies that most people are least happy in their 50’s.
- The shape of a logistic growth curve, which increases gradually at first, more rapidly in the middle, and slowly at the end, leveling off at a maximum value after some period of time.
- The only letter that does not appear in the name of any US state.
- The answer to the riddle, “It occurs once in a minute, twice in a moment, but never in a thousand years.”

**Answers (and Notes of Interest)**

- I
- C
- E
- J : in 1524, Gian Giorgio Trissino made a clear distinction between the sounds for
*i*and*j*, which were previously the same letter - H
- T
- N : see Wikipedia for a list of other Roman numerals used in medieval times
- F
- O
- R
- W : should probably be “Dubya” instead of “Double U,” but whatever
- A
- P : as you might expect, more English words start with S than any other letter; based on the ENABLE word list, P is the second most common initial letter, followed by C
- B
- D
- K : the symbol K comes from
*kalium*, the Medieval Latin for*potash*, from which the name*potassium*was derived - X
- V
- Z
- Y
- G : a lowercase
*g*can be written in two different ways, and the more common version in typesetting (known as the “loop-tail*g*“) can be recognized but not written by most adults, as recounted on the D-Brief blog - L
- U : see this article from
*The Economist*, especially this image - S
- Q
- M

## Which is Closest?

Not too long ago, I published a blog post about end-to-end comparisons, those silly feats of computational gymnastics that try to reduce an overwhelming statistic to something more tangible. Something like this:

If each piece of candy corn sold in a year by Brach’s — the top manufacturer of the waxy confection — were laid end to end, they would circle the Earth 4.25 times.

In writing that post, I inadvertently formulated a statistic that rather surprised me:

If all the players on an NFL team were laid end to end, they’d stretch from the back of one end zone to the opposite goal line.

That the players would almost line the entire field struck me as an amazing coincidence. And it got me to thinking — might this be true for other sports?

Not one to let sleeping dogs — or professional athletes — lie, I decided to investigate. Based on that research, here’s a simple, one-question quiz for you.

**Which of the following comparisons is the most accurate?**

- If all of the players on an
**NHL (hockey)**roster were laid end to end, they would reach from**one end of the rink to the other**. - If all of the players on an
**NBA (basketball)**roster were laid end to end, they would reach from**one end of the court to the other**. - If all of the players on an
**NFL (football)**roster were laid end to end, they would reach from**one end line to the other**. - If all of the players on an
**MLB (baseball)**roster were laid end to end, they would reach from**home plate to second base**. - If all of the players on an
**MLS (soccer)**roster were laid end to end, they would reach from**one end to the other**.

As you begin to think about that question, some notes:

- Every professional baseball stadium has different measurements. Fenway Park (Boston) is a mere 310′ from home plate to the right field wall, whereas Comerica Park (Chicago) extends 420′ from home plate to straightaway center. Consequently, the distance from second to home is used in the fourth answer choice, because it’s the same for every field.
- To my surprise, MLS stadiums are not uniform in length and width. Who knew? The length of the field must be at least 100 meters, at most 110 meters, and anywhere in between is fine. Assume an average length of 105 meters for the fifth answer choice.

Before you read much further, let me say how much fun I’ve had discussing this question around the dinner table and at the local pub. In spite of hard facts, there is resolute disagreement about player height, roster size, and field dimensions. And the shocking (or should I say predictable?) results raise an eyebrow every time. I only mention that to persuade you to think about the question, alone or with some friends, before continuing.

Okay, you’ve cogitated? Then let’s roll.

In researching the answer to the question, I was struck by how close the total length of all players on the roster is to the length of the field, court, or rink. Coincidence? Of course, a larger field requires more players, so perhaps this is the evolution of roster size that one would expect.

To answer the question, you need to know the height of an average player, the number of players on a roster, and the dimensions of professional venues. All of that data can be found in a matter of minutes with an online search, but I’ll save you the trouble.

League |
Average Height (in.) |
Players on a Roster |
Combined Height, Laid End to End (ft.) |
Dimensions |

NHL | 73 | 23 | 140 | 200 feet (from end to end) |

NFL | 74 | 53 | 327 | 120 yards (360 feet, from end to end) |

NBA |
79 |
14 |
92 |
94 feet (from end to end) |

MLB | 73 | 23 | 140 | 127 feet (from home to second) |

MLS | 71 | 28 | 166 | 105 meters (345 feet, from end to end) |

As it turns out, the MLS comparison is the least accurate. The combined heights of soccer players is only 48% of the length of their field. The NHL comparison is a little better, with players’ heights extending 70% of the length of the field. But the NFL and MLB are both very close, with the players’ heights equalling 91% of the field length and 110% of the distance from home to second, respectively. Astoundingly, if the players on an NBA team were laid end to end, they’d come just 22 inches short of covering the entire court, accounting for a miraculous 98% of the length!

So there you have it. **D**, final answer.

One last thought about this. I play ultimate frisbee, a sport with a field that measures 120 yards (360 feet). For tournaments, our rosters are capped at 29 players, and I suspect my amateur teammates are, on average, shorter than most professional athletes. If we assume a height of 5’10” for a typical frisbee player, then the combined height is 172 feet. That puts us in the realm of soccer, with our combined length covering just 48% of the field.

If, like me, you play a sport that isn’t one of the Big 5 in the U.S., I’d love to hear about your sport’s field and roster size, and how it ranks with the comparisons above.

## Required Summer Reading: *The Grasshopper King*

If you’ve read *How Not to Be Wrong: The Power of Mathematical Thinking*, then you know that Jordan Ellenberg is extremely intelligent, well educated, and incredibly talented. In addition, he may be the best voice for mathematics in America today. (You may have come to the same conclusion by reading his “Do The Math” column in *Slate* or from any one of the articles he’s written for *The New York Times* or *The Wall Street Journal*.) But if you haven’t read *The Grasshopper King*, a nonfiction novel that Ellenberg wrote in 2003, then you are absolutely missing out on his gifts as a pure writer. It’s the tale of Stanley Higgs, an internationally acclaimed professor of Gravinics at Chandler State University; Samuel Grapearbor, a graduate student at CSU; and the silent relationship that forms when Grapearbor is assigned to watch Higgs after he decides — for no obvious reason — to stop talking.

Coffee House Press claims that the novel is about “treachery, death, academia, marriage, mythology, history, and truly horrible poetry.” I mean, what’s not to love?

I bought *The Grasshopper King* because of how much I enjoyed *How Not to Be Wrong*, but I had no intention of enjoying it nearly as much as I did. From the first page, though, I was enthralled with Ellenberg’s style. To amalgamate several of the Amazon reviews, “this is an unusual book,” but it is beautiful because of “the finely tuned precision of the writing itself.”

This is not a math book, but occasionally Ellenberg turns a phrase that reminds you he’s a mathematician. When Grapearbor’s girlfriend claims that New York is ninety-five percent liars and snobs, he replies, “In Chandler City it’s ninety-nine. Point nine repeating.” Other times, he’ll include mathematical terms that are, in fact, completely appropriate and economical, but not altogether necessary:

a grasshopper, stirred by some unguessable impulse, heaved itself out of the drench mess, rose and fell in a perfect, inevitable

parabolawhoseinterceptwas the exposed stripe of Charlie’s backthe pressure of the water made

concentric circlesbehind my clenched-shut eyelidsthe agricultural buildings were at

discreet distancesfrom one another

And, yes, I know that last one isn’t a math phrase… but I can’t help but read it as *discrete distances*.

If you like Pynchon or Wolfe or anything off the beaten path, then you’ll like this book. The characters are quirky and memorable, and the writing is unforgettable. I recommend spending a few hours with it during what you have left of this summer.