## Where Would We Fit?

In the classic book *101 Puzzle Problems*, Nathaniel B. Bates and Sanderson M. Smith make an astounding claim:

*The volume of the 1970 world population is less than the volume of the Houston Astrodome.*

That is, if you packed all of the people on Earth who were alive in 1970 — all 4 billion of them — as tightly as possible, they would’ve actually fit in the stadium, and they could’ve seen Evel Knievel jump 13 cars or Billy Jean King defeat Bobby Riggs.

Before you read any farther, you might wish to do a Fermi estimate to verify the reasonableness of this claim.

It’s an amazing fact!

Unfortunately, it’s also wrong.

Bates and Smith argue that the average human body has a volume of approximately 2 cubic feet. Their argument? That water weighs 62.5 pounds per cubic foot, and “the human body consists mostly of water.” Fair enough, if we think that an average human weighs about 125 pounds, which they probably didn’t, even in the 1970s.) Personally, I would’ve attacked the estimate a little differently. An average human can fit in a box that is 6 feet tall with a base that measures approximately 0.5 square feet — I know; I’ve been to funerals — and such a box has a volume of 3 cubic feet. But there’s a fair amount of wasted space in that box so, yeah, 2 cubic feet seems like a reasonable estimate.

Therefore, the world population in 1970 would have had a combined volume of 8,000,000,000 cubic feet.

So far, so good.

But Bates and Smith then make their statement about the Astrodome, with no calculations or estimates about the volume of the stadium to justify the claim. Please understand, I don’t fault Bates and Smith; as I understand it, the claim about everyone fitting in the Astrodome was a ubiquitous factoid during the decade of bellbottoms and disco. Still, you’d think that two mathematicians would have checked the stat before including it in a book.

The Astrodome has an outside diameter of 710 feet and a height of 208 feet. A cylinder with those dimensions would have a volume of

*V* = π × 355^{2} × 208 ≈ 82,000,000 cubic feet

Although this is surely an overestimate, it’s still an order of magnitude too small to fit the world’s population in 1970. There’s not nearly enough space to store all those buggers.

Bates and Smith state that the volume of the 1970 population “takes up about one-eighteenth of a cubic mile.” And I think therein lies the error. The height of the Astrodome is just shy of one-eighteenth of a mile, and the diameter is a whole lot more than one-eighteenth of a mile, so someone (incorrectly) assumed that a container with those dimensions would easily hold 1/18 cubic mile. But that’s not right. As any geometry student will tell you, shrinking each dimension to a fraction of its original length results in a volume that is the *cube* of that fraction. Oops.

The diameter of the Astrodome is about 1/7 of a mile, and the height is about 1/22 of a mile, so its volume would have been less than 1/1000 of a cubic mile.

So, what container would have been big enough to hold the 1970 population?

Honestly, who cares?

Things have changed. Today, we are bumping up against 8 billion people worldwide, the Astrodome was declared “unfit for occupancy” in 2016, and instead of calculating the volume of stadia, we can simply use Google to find the volume of some really large places.

In fact, Google claims that the volume of the Astrodome is 42,000,000 cubic feet. If we can believe what we read, then the calculations above were, sadly, unnecessary.

So, you may be wondering, **what places in the world could hold all of us**? That is, what has a volume of 16,000,000,000 cubic feet?

The building with the greatest volume in the world is the Boeing Everett Factory. It has a volume of almost 500,000,000 cubic feet — which means you’d need 32 of them to fit today’s world population.

The manmade structure with the greatest volume is the Great Wall of China, tipping the scales at just under 20,000,000,000 cubic feet. That would be large enough, though how would you get all of those people inside? For easier packing, you could turn to nature and just dump everyone into Sydney Harbour, which has a volume of 562,000 megaliters, or about 19,500,000,000 cubic feet.

## What Number Do You Hate?

In 2014, Alex Bellos conducted a poll to find out people’s favorite number. Based on those results, Maddy Fry wrote an article for *Time* in which she stated,

The least favorite number turned out to be 110, which was the lowest number to receive no votes.

That’s not quite true. It would be correct to say that 110 was the *least common* favorite number, but calling it the “least favorite number” makes it sound like it’s the number that folks like least. In a poll where folks were asked to choose just one favorite number, a number that gets no votes doesn’t make it the least liked number. It just means that no one picked it as their favorite. That’s a subtle but important distinction.

It could be the case — however unlikely — that even though no one picked 110 as their favorite number, it could be everyone’s *second-favorite* number.

On the other hand, I **do** have a least favorite number.

More than two decades ago, I heard a local Maryland band called Dead City Radio (not to be confused with the song *Dead City Radio* by Rob Zombie), and I bought their debut album. Although the band is now defunct, the image from that album cover holds a permanent spot in my psyche:

The cover includes disturbing imagery of a doll, a gun, graffiti, an atomic bomb explosion, and the number 219 on the door. Why 219? I spoke with DCR’s lead singer after the show, and he told me that it was serial killer Jeffrey Dahmer‘s apartment number. *Disturbing.*

As it turns out, that’s not true. Dahmer’s apartment number in Milwaukee was actually 213, though he did meet several of his victims at Club 219. I’m not sure if the DCR guys had it wrong, or if I misheard because my ears were still ringing after the concert, or if there’s some other explanation.

Regardless, I had no reason to question the statement when I first heard it, and I now have a fear and abnormal hatred of the number 219. It’s been my least favorite number for years. I obviously avoid room 219 when I stay at hotels. (And now that I know the truth, I avoid room 213, too. In fact, I try to avoid the second floor entirely. I don’t even want to walk past those rooms.)

But my hatred is deeper than just avoiding hotel rooms. When I score 219 points playing Dots, or when I receive $2.19 in change at the grocery store, or when my GPS tells me to turn right onto Route 219, a slight shiver runs down my spine.

I’m not the only person who despises a particular number. At The Top Tens, many people say they hate many numbers for a variety of reasons:

- 6 looks weird.
- 16 is so obnoxious. I can’t stand this stupid swagger of an integer. It should burn in hell.
- 12 will lead to endless controversies.
- 18 sucks because it’s when you have to say goodbye to your childhood.
- 39 is a multiple of 13… plus it’s so annoying.

**So, what number do you hate?** Complete the poll below. (And if it isn’t working for you, jump over to **this Google poll**.) Once I get a reasonable number of responses, I’ll clean the data and share the results. Check back in early 2019.

## Our Weigh or the Highway

On the outskirts of Paris, in a triple-sealed chamber, sits a golf-ball sized cylinder made of platinum and iridium. It’s officially known as the International Prototype of the Kilogram — or IPK, for short — but locals refer to it as *Le Grand K*.

Cast in 1879, the IPK serves as the international standard for mass. For a century-and-a-half, the accuracy of any weight measurement was linked to this very precisely hewn hunk of metal.

The kilogram is the only unit still defined in terms of a manufactured object, and that’s shaky and solitary ground on which to rest. In 1960, the only other SI unit still based on a physical artifact — the meter — was redefined in terms of the wavelength of light from a specified source, linking it to natural phenomena.

But no longer.

Today, Le Grand K will be retired when representatives from 60 countries will meet in Versailles to approve a new definition of the kilogram. Officially, the change won’t take place until May 20, 2019, which means that for a short time, the IPK, Pete Sessions, and Claire McCaskill will occupy a similar state of lame-duck limbo.

And what is the new definition for the kilogram? It’s pretty straightforward…

The kilogram (kg) will be defined by taking the fixed value of the Planck constant

hto be 6.626 070 15 × 10^{−34}J⋅s. The unit J⋅s is joule-seconds, which is equal to kg⋅m^{2}⋅s^{−1}, where the meter and second are defined in terms ofc, the speed of light, which is 299,792,458 meters per second (m⋅s^{−1}), and Δν_{Cs}, the ground state hyperfine splitting frequency of cesium-133, which is 9,192,631,770 Hz.

See? Easy peezy.

Over time, the IPK has, surprisingly, lost weight. And scientists can’t really explain why. The cylinder now weighs about 50 micrograms, or roughly the weight of an eyelash, less than it weighed in 1879. The irony, though, is that a kilogram, by definition, is equal to the weight of the IPK. So, technically, it isn’t that the kilogram has lost weight; in truth, the rest of the world has been getting a little heavier. (Keep this factoid in your back pocket for a few days. It is perhaps the best and most scientific excuse you’ll be able to offer if you don’t like the reading on your scale around the holidays.)

All this talk of Le Système International makes me think about the many important benchmark conversions that should be part of every science curriculum:

2000 mockingbirds = 2 kilomockingbirds

10^{12} microphones = 1 megaphone

10 cards = 1 decacards

1,000 grams of wet socks = 1 literhosen

And the most important benchmark, for when you need to convert between expatriate poets and televangelists:

And finally, if you’ve read this far, a PG-13 passage from *Wild Thing* by Josh Bazell:

In metric, one milliliter of water occupies one cubic centimeter, weighs one gram, and requires one calorie of energy to heat up by one degree centigrade — which is one percent of the difference between its freezing point and its boiling point. An amount of hydrogen weighing the same amount has exactly one mole of atoms in it. Whereas in the American system, the answer to “How much energy does it take to boil a room-temperature gallon of water?” is “Go fuck yourself,” because you can’t directly relate any of those quantities.

## 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!”