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Answers (and Notes of Interest)
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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?
As you begin to think about that question, some notes:
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.
]]>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.
]]>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:
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:
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.
]]>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:
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:
Have you heard a dumb stat recently? Let us know in the comments.
]]>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?
]]>Seventyfive percent of Great Britain — Scotland, Northern Ireland, and the United Kingdom — does it. (Fucking Wales.)
The Academy of Motion Picture Arts and Sciences does it.
And now Maine, the first state to implement a onetoone laptop program for all students, does it, too.
The “it” is rankedchoice voting, a method that allows voters to rank the candidates in their order of preference. When used to elect a single candidate, rankedchoice voting helps to select a winner that reflects the support of a majority of voters.
Many cities and towns already use rankedchoice voting to elect mayors and members of council. But Maine is the first state to use it for state and federal elections.
Many people, including Jennifer Lawrence, support rankedchoice voting. (That should mean something, right? After all, she’s the only person born in the 1990’s who’s won as Oscar. So far, anyway. Maybe she supports rankedchoice voting because the Academy used rankedchoice voting to award her an Oscar? Who knows.)
This video gives a very simple example of how rankedchoice voting works.
But maybe there’s a better example. Imagine that a book club is trying to decide which book they should read next, and rather than just voting for their top choice, the group instead ranks each of three books:
The voting proceeds as follows:
Order  Number of Votes 
MJ4MF / Grapes / Riot  4 
MJ4MF / Riot / Grapes  5 
Grapes / MJ4MF / Riot 
6 
Grapes / Riot / MJ4MF 
2 
Riot / MJ4MF / Grapes 
2 
Riot / Grapes / MJ4MF 
1 
First, consider only the firstplace votes, and determine if any candidate received a majority. In this case, none of them received more than half of the firstplace votes: MJ4MF received 9 firstplace votes, Grapes received 8, and Riot received 3.
Since no one won, the candidate with the fewest firstplace votes is eliminated. Sorry, Colin Adams; we’ll have to say goodbye to Riot at the Calc Exam.
Now, the voters who had their first choice eliminated will have their votes counted for their second choice instead. So, 2 additional votes go to MJ4MF, while 1 vote goes to Grapes.
This means that MJ4MF, with 9 + 2 = 11 votes, is the winner, since Grapes received only 8 + 1 = 9 votes in the second round. Sorry, Greg Tang; you should’ve chosen a less worthy adversary.
C’mon, now… you didn’t really think MJ4MF was gonna lose, did ya?
On Tuesday, voters in Maine used rankedchoice voting to decide several political races, but they also got to vote on whether to use rankedchoice voting in future elections. Because rankedchoice voting takes time to tabulate, two of Tuesday’s contests were still undecided as of Thursday morning. So, voters literally were asked to decide if they should keep or reject a system that they had never seen used in an election. Doesn’t that seem just a bit odd?
You know what else is odd? Numbers that aren’t divisible by 2.
]]>I see a yellow light serpent… and a low APR.
What you may also find surprising is that this commercial reminds me of my fatherinlaw, Julian Block, a nationally known tax expert. Julian used to work for a company that connected advisors to people seeking help. An operator would ask a few questions of each caller and then connect the advisee with an appropriate consultant. Sometimes, however, the advisee would be connected to the wrong expert. And the problem? The company specialized in dispensing two types of advice: tax and psychic.
You may now see why the GEICO commercial reminds me of my fatherinlaw. When calls were incorrectly routed to him, he would become a de facto psychic tax advisor. I imagine conversations like the following:
Young Woman: My boyfriend just proposed. Should I marry him?
Julian: That depends. What’s his tax bracket?
These combinations — fortune teller and bank teller; psychic advisor and tax advisor — yield rather whimsical new professions. It made me wonder if there were others. Sadly, an hour of brainstorming yielded only a handful of satisfactory results:
Super Hero Intendent, who still has time to fight crime after 8 hours of dealing with stoppedup toilets
Dog Street Walker, who thinks the oldest profession is picking up Spot’s poo
Antique Debt Collector, who will accept payment in Ming vases and pocket watches
Switchboard Lottery Operator, who has a 1in500 chance of connecting you to the right person
Social Construction Worker, who can strike up a conversation with anyone while shingling a roof
Foreign Language Flight Instructor, who will teach you how to land safely after you give her the declension for agricola
Got any others you’d add to the list? Post them in the comments.
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1, 2, 4, 8, 16, __
and was dumbfounded to learn that the missing value was 31, not 32, because the pattern was not meant to represent the powers of 2, but rather, the number of pieces into which a circle is divided if n points on its circumference are joined by chords. Known as Moser’s circle problem, it represents the inherent danger in making assumptions from a limited set of data.
Last night, my sons told me about the following problem, which they encountered on a recent math competition:
What number should replace the question mark?
Well, what say you? What number do you think should appear in the middle stick figure’s head?
Hold on, let me give you a hint. This problem appeared on a multiplechoice test, and these were the answer choices:
Now that you know one of those four numbers is supposed to be correct, does that change your answer? If you thought about it in the same way that the test designers intended it, then seeing the choices probably didn’t change your answer. But if you didn’t think about it that way and you put a little more effort into it, and you came up with something a bit more complicated — like I did — well, then, the answer choices may have thrown you for a loop, too, and made you slap your head and say, “WTF?”
For me, it was Moser’s circle problem all over again.
So, here’s where I need your help: I’d like to identify various patterns that could make any of those answers seem reasonable.
In addition, I’d also love to find a few other patterns that could make some answers other than the four given choices seem reasonable.
For instance, if the numbers in the limbs are a, b, c, and d, like this…
then the formula 8a – 4d gives 8 for the first and third figures’ heads and yields 8 × 6 – 4 × 9 = 12 as the answer, which happens to be one of the four answer choices.
Oh, wait… you’d don’t like that I didn’t use all four variables? Okay, that’s fair. So how about this instead: ‑3a + b + c – 2d, which also gives ‑3 × 6 + 7 + 5 + 2 × 9 = 12.
Willing to help? Post your pattern(s) in the comments.
[UPDATE (3/9/18): I sent a note to the contest organizers about this problem, and I got the following response this afternoon: “Thanks for your overall evaluation comments on [our] problems, and specifically for your input on the Stick Figure Problem. After careful consideration, we decided to give credit to every student for this question. Therefore, scores will be adjusted automatically.”]
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