OneLetter 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 OneLetter 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, 26letter 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 tripledigit 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 looptail version of this lowercase letter in print, less than onethird of participants in a Johns Hopkins study could correctly pick it out of a fouroption 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 “looptail g“) can be recognized but not written by most adults, as recounted on the DBrief 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 endtoend 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, onequestion 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 ninetyfive percent liars and snobs, he replies, “In Chandler City it’s ninetynine. 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 parabola whose intercept was the exposed stripe of Charlie’s back
the pressure of the water made concentric circles behind my clenchedshut eyelids
the agricultural buildings were at discreet distances from 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.
The BeAll, EndAll of EndtoEnd Comparisons
It’s become quite fashionable to use “end to end” comparisons to visually demonstrate just how large or long or vast something is.
 If all the atoms in your body were laid end to end, they would be able to encircle the Earth over 41 billion times, a length of almost 56 light years. (Trove 42)
 If you laid the Knicks’ top six big men from end to end, you would get 41 feet and 4 inches of pain, and enough sore necks, feet, knees and ankles to fill a modest orthopedic ward. (New York Times, April 11, 2013)
 If all of the fiber optic cable in the world were laid end to end, it would encircle the Earth 25,000 times. (NPR)
 If all the rolls of toilet paper used in the United Kingdom in a year were laid end to end, they would reach further than Mars. (@ThomasCrapperCo)
 If you laid all the DNA from all your cells side by side, their combined length would be about twice the diameter of the Solar System. (Science Focus)
 If your blood vessels were laid end to end, they would be over 100,000 miles long for an adult and 60,000 for a child. (Franklin Institute)
 If you laid all the M&M’s produced in a year end to end, they would stretch for a million miles. (MF4MF)
Though ubiquitous, these comparisons are often unreliable:
I checked with Google to see just how long the [Aleppo] souk actually is, if all its streets were laid end to end, and found it to be, variously, seven, eight, ten, twelve, thirteen, sixteen, and “about 30” kilometres. (Jonathan Raban)
But accuracy be damned. The point of these comparisons is not to demonstrate precise computational ability. Instead, they are meant to provide a reference point for a statistic that would be otherwise difficult to interpret.
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.
(This reminds me, for no good reason, of an entry in the Washington Post Style Invitational from some years back, in which entrants were asked to submit bad similes and metaphors: “He was as tall as a 6‑foot, 3‑inch tree.”)
Such endtoend comparisons are not new, however. According to the Quote Investigator, this type of comparison was used in 1885 to describe the Vanderbilt family’s $200,000,000 fortune:
Enough to buy 40,000,000 barrels of flour at $5 each. If these barrels were placed end to end, they would reach around the Earth on the parallel of Boston, or they would fence in every State in the Union.
Alexander Wolcott, in his 1934 bestseller While Rome Burns, quoted Dorothy Parker as saying:
If all the girls attending [the Yale prom] were laid end to end, I wouldn’t be at all surprised.
The most famous of these comparisons, however, is probably the following:
If all economists were laid end to end, they would not reach a conclusion.
Who said it? Who knows. It’s most often attributed to George Bernard Shaw, but it seems that the quip existed a full decade before Shaw was ever credited. It has also been attributed to Isaac Marcosson, Farmer Brown, Stephen Leacock, and William Baumol. Regardless of its originator, it has been reiterated and modified a thousand times:
 If the nation’s economists were laid end to end, they would point in all directions. (Arthur H. Motley)
 If all economists were laid end to end, there would be an orgy of mathematics.
 If all the economists in the world were laid end to end, it would probably be a good thing.
My favorite endtoend comparisons, like the one attributed to Shaw, are usually gardenpath sentences. They begin with an astounding statistic, but just when you think some simpler comparison will be made, they smack you in the nose with a twist:
 Just to be clear, if you carefully removed, and laid end to end, all the veins, arteries, and capillaries of your body, you will die. (Neil deGrasse Tyson)
 If all the world’s managers were laid end to end, it would be an improvement.
 If you laid all our laws end to end, there would be no end. (Arthur “Bugs” Bae)
 If all the salmon caught in Canada in one year were laid end to end across the Sahara Desert, the smell would be absolutely awful.
 If all 206 bones were removed from your body and laid end to end… you’d be dead.
 If all the cars in the world were laid end to end, someone from California would be stupid enough to try to pass them.
 If all the joggers were laid end to end, it would be easier to drive to work in the morning. (Milton Berle)
 If all students who fell asleep in their seats during math class were laid end to end, they’d be a lot more comfortable.
Finally — and with absolutely no bias whatsoever (wink, wink) — I present my alltime favorite endtoend comparison, gloriously penned by my friend and colleague Gail Englert, and which appears on the back cover of More Jokes 4 Mathy Folks:
If you took all the people who fell on the floor laughing when they read this book and laid them endtoend, you’d have a very long line of people. It’d be a silly thing to do, but at least you’d know who to avoid at a cocktail party.
Do you have a favorite endtoend comparison? Have at it in the comments.
There Are 2 Things that Happened Yesterday…
Yesterday was a banner day.
Last night, I was finally able to carve out some time to bingewatch Season 2 of Trial & Error, and I was rewarded with a classic math joke in Episode 1. When lead investigator Dwayne Reed arrives at the house of accused murderer Lavinia PeckFoster, he says:
There are two things that Reeds don’t trust: doctors, Pecks, and math.
I love it!
Upon realizing that I might be able to get my sitcomwriting career off the ground by reformulating stale math jokes, I promptly submitted my resume to NBC.
But, wait… there’s more!
Earlier in the day, I received NCTM‘s email newsletter Summing Up, which contained an unexpected surprise. In the section titled “NCTM Store,” there was a blurb about my most recent book, More Jokes 4 Mathy Folks, under the headline Just Published!
I had no idea that NCTM decided to sell my book, let alone that they were going to publicize it. My ignorance not withstanding, I couldn’t be more delighted!
If you’re looking for some great, light summer reading — something that can be enjoyed poolside while sipping a mojito — then pick up a copy of More Jokes 4 Mathy Folks from NCTM today! Not only will your purchase support a great organization (and my sons’ college fund), you’ll also receive a 20% discount for being an NCTM member.
Following the lead of Dwayne Reed, here are jokes that begin, “There are n kinds…,” all of which appear in More Jokes 4 Mathy Folks:
 There are only 2 kinds of math books: those you cannot read beyond the first sentence, and those you cannot read beyond the first page. (C. N. Yang, Nobel Prize in Physics, 1957)
 There are 2 kinds of people in the world: those who don’t do math, and those who take care of them.
 There are 3 kinds of people in the world: positive, negative, and relative.
 There are 2 kinds of people in the world: those who are wise, and those who are otherwise.
 There are 2 kinds of statistics: the kind you look up, and the kind you make up.
 There are 2 kinds of experienced actuaries: those who say they have made significant forecasting errors, and liars.
 There are 10 kinds of people in the world: those who understand binary, and those who don’t.
 There are 10 kinds of people in the world: those who understand binary, and 9 others.
 There are 10 kinds of people in the world: those who understand ternary; those who don’t understand ternary; and, those who mistake it for binary.
 There are 11 kinds of people: those who understand binary, and those who don’t.
 There are 8 – 3 × 2 kinds of people in the world: those who correctly apply the order of operations, and those who don’t think that 6 ÷ 2 × (1 + 2) = 9.
 There are 2 kinds of people in the world: logicians and ~logicians.
 There are 2 kinds of people in the world: those who can extrapolate from incomplete data…
Nationals Win Probability, and Other Meaningless Statistics
The first pitch of last night’s NationalsPhillies game was 8:08 p.m. That’s pretty late for me on a school night, and when a 38minute rain delay interrupted the 4th inning, well, that made a late night even later.
The Phillies scored 4 runs in the top of the 5th to take a 6‑2 lead. When the Nationals failed to score in the bottom of the 5th, I asked my friends, “What are the chances that the Nationals come back?” With only grunts in response and 10:43 glowing from the scoreboard, we decided to leave.
On the drive home, we listened as the Nationals scored 3 runs to bring it to 6‑5. That’s where the score stood in the middle of the 8th inning when I arrived home, and with the Nats only down by 1, I thought it might be worth tuning in.
The Nats then scored 3 runs in the bottom of the 8th to take an 86 lead. And that’s when an awesome stat flashed on the television screen:
Nats Win Probability
 Down 62 in the 6th: 6%
 Up 86 in the 8th: 93%
Seeing that statistic reminded me of a Dilbert cartoon from a quartercentury ago:
I often share Dogbert’s reaction to statistics that I read in the newspaper or hear on TV or — egad! — are sent to me via email.
I had this kind of reaction to the stat about the Nationals win probability.
For a weather forecast, a 20% chance of rain means it will rain on 20% of the days with exactly the same atmospheric conditions. Does the Nats 6% win probability mean that any team has a 6% chance of winning when they trail 62 in the 6th inning?
Or does it more specifically mean that the Nationals trailing 62 in the 6th inning to the Phillies would only win 1 out of 17 times?
Or is it far more specific still, meaning that this particular lineup of Nationals players playing against this particular lineup of Phillies players, late on a Sunday night at Nationals Stadium, during the last week of June, with 29,314 fans in attendance, with a 38minute rain delay in the 4th inning during which I consumed a soft pretzel and a beer… are those the right “atmospheric conditions” such that the Nats have a 6% chance of winning?
As it turns out, the win probability actually includes lots of factors: whether a team is home or away, inning, number of outs, which bases are occupied, and the score difference. It does not, however, take into account the cost or caloric content of my midgame snack.
A few other stupid statistics I’ve heard:
 Fifty percent of all people are below average.
 Everyone who has ever died has breathed oxygen.
 Of all car accidents in Canada, 0.3% involve a moose.
 Any time Detroit scores more than 100 points and holds the other team below 100 points, they almost always win.
Have you heard a dumb stat recently? Let us know in the comments.
How Would You Answer These Questions?
The following is one of my alltime favorite assessment items:
Which of the following is the best approximation for the volume of an ordinary chicken egg?

The reason it’s one of my favorites is simple: it made me think. Upon first look, I didn’t immediately know the answer, nor did I even know what problemsolving strategy I should use to attack it.
I used estimation, first assuming that the egg was spherical — and, no, that is not the start of a math joke — and then by attempting to inscribe the egg in a rectangular prism. Both of those methods gave different answers, though, and not just numerically; each led to a different letter choice from above.
Not satisfied, I then borrowed a method from Thomas Edison — I filled a measuring cup with 200 mL of water, retrieved an ordinary egg from my refrigerator, and dropped it into the cup. The water level rose by 36 mL. This proved unsatisfying, however, because although choice B is numerically closer to this estimate than choice C — only 29 mL less, compared to 34 mL more — it was five times as much as B but only half as much as C. For determining which is closer, should I use the difference or the ratio?
It was at this point that I decided the answer doesn’t matter. I had been doing some really fun math and employing lots of grey matter. I was thinking outside the box, except when I attempted to inscribed the egg in a rectangular prism and was literally thinking inside the box. And, I was having fun. What more could a boy ask for?
On a different note, here’s one of the worst assessment questions I’ve ever seen:
What is the value of x?
3 : 27 :: 4 : x
I can’t remember if it was a selectedresponse (nee, multiplechoice) item, or if was a constructedresponse question. Either way, it has issues, because there are multiple possible values of x that could be justified.
On the other hand, it’s a great question for the classroom, because students can select a variety of correct responses, as long as they can justify their answer.
The intended answer, I’m fairly certain, is x = 36. The analogy is meant as a proportion, and 3/27 = 4/36. (Wolfram Alpha agrees with this solution.)
But given the format, it could be read as “3 is to 27 as 4 is to x,” which leaves room for interpretation. Because 27 = 3^{3}, then perhaps the correct answer is 64 = 4^{3}.
Or perhaps the answer is x = 28, because 3 + 24 = 27, and 4 + 24 = 28.
Don’t like those alternate answers? Consider the following from Math Analogies, Level 1, a software package from The Critical Thinking Company that was reviewed at One Mama’s Journey.
If this analogy represents a proportion, then the correct answer is $10.50, but that’s not one of the choices. Instead, the analogy represents the rule “add $1,” and the intended answer choice is $10.00.
What amazing assessment items have you seen, of either the good or bad variety?