## 12 Math Games, Puzzles, and Problems for Your Holiday Car Trip

It doesn’t matter if you’re one of the 102.1 million people traveling by car, one of the 6.7 million people traveling by plane, or one of the 3.7 million people traveling by train, bus, or cruise ship this holiday season — the following collection of games, puzzles, and problems will help to pass the time, and you’ll be there long before anyone asks, “Are we there yet?”

**Games**

*1. Street Sign Bingo*

Use the numbers on street signs to create expressions with specific values. For instance, let’s say you see the following sign:

The two numbers on the sign are 12 and 3, from which you could make the following:

12 + 3 =

1512 – 3 =

912 × 3 =

3612 ÷ 3 =

4

Of course, you could just use the numbers directly as 3 or 12. Or if your passengers know some advanced math, they could use square roots, exponents, and more to create other values.

Can you split the digits within a number and use them separately? Can you concatenate two single-digit numbers to make a double-digit number? That’s for you and your traveling companions to decide.

To play as a competitive game, have each person in the car try to get every value from 1‑20. It’s easiest to keep track if you require that players go in order. To make it a cooperative game, have everyone in the car work collectively to make every value from 1 to 100.

As an alternative, you could use the numbers on license plates instead.

*2. Bizz Buzz Bang*

Yes, I’ve played this as a drinking game. No, I don’t condone drinking while driving. No, I don’t condone under-age drinking, either. Yes, I condone playing this game with minors while driving. (See what I did there?)

The idea is simple. You pick two single-digit numbers, *A* and *B*. Then you and your friends start counting, one number per person. But each time someone gets to a number that contains the digit *A* or is a multiple of *A*, she says, “Bizz!” Every time someone gets to a number that contains the digit *B* or is a multiple of *B*, she says, “Buzz!” And every time someone gets to a number that meets both criteria, he says, “Bang!”

For example, let’s say *A* = 2 and *B* = 3. Then the counting would go like this:

one, bizz, buzz, bizz, five, bang, seven, bizz, buzz, bizz, eleven, bang, buzz, …

When someone makes a mistake, that round ends. Start again, and see if you can beat your record.

You can use whatever numbers you like, or modify the rules in other ways. For instance, what if bizz is for prime numbers and buzz is for numbers of the form 4*n* + 3? Could be fun!

*3. Dollar Nim* (or any other variation)

Nim is a math strategy game in which players take turns removing coins from a pile. Different versions of the game are created by adjusting the number of coins that can be removed on each turn, the number of coins in the pile originally, how many piles there are, and whether you win (normal) or lose (misère) by taking the last coin.

A good aspect of Nim is that you don’t actually need coins. To play in the car, players just need to keep track of a running total in their heads.

A version of Nim dubbed **21 Flags** was played on *Survivor Thailand* several years ago. There were 21 flags, and each team could remove 1, 2, or 3 flags on each turn. The team to remove the last flag won. This is a good first version to play in the car, especially for young kids who have never played before. Then mix it up by changing the initial amount and the number that can be removed on each turn.

Our family’s favorite version, **Dollar Nim**, is played by starting with $1.00 and removing the value of a common coin (quarter, dime, nickel, penny) on each turn. (While discussing this post with my sons, they informed me that they much prefer **Euro Nim**, which begins with 1€, but then the coin values to be removed are 50c, 20c, 10c, 5c, 2c, and 1c. It’s essentially the same game, but they like the different coin amounts.)

**Three-Pile Nim** consists of not just one but three piles with 3, 5, and 7 coins, respectively. On each turn, a player must remove at least one coin, and may remove any number of coins, but all removed coins must be *from the same pile*.

Finally, **Doubling Nim** is played as the name implies. On the first turn, a player may remove any number of coins but not the entire pile. On every turn thereafter, a player may remove any number of coins up to double the number taken on the previous turn. For instance, if your opponent removes 7 coins, then you can remove up to 14 coins.

For every version of Nim, there is an optimal strategy. We’ve wasted hours on car trips discussing the strategy for just one variation. Discussing the strategies for all the versions above could occupy the entire drive from Paducah to Flint.

*4. Guess My Number
*

One person picks a number, others ask questions to try to guess the number.

The simplest version is using “greater than” and “less than” questions. Is it greater than 50? Is it less than 175? And so forth. Using this method, the guessers can reduce the number of possibilities by half with each question, so at most, it should take no more than *n* guesses if 2^{n} > *m*, where *m* is the maximum possible number that the picker may choose. For instance, if the picker is required to choose a number less than 100, then *n* = 7, because 2^{7} = 128 > 100. Truthfully, this version of the game gets boring quickly, but it’s worth playing once or twice, especially if there’s one picker and multiple guessers. And a conversation about the maximum number of guesses need can be a fun, mathy way to spend 15 minutes of your trip.

A more advanced version excludes “greater than” and “less than” questions. Instead, the guessers can ask other mathematical questions like, “Is it a prime number?” or “Do the digits of the number differ by 4?” Those questions imply, of course, that the answer must be yes or no, and that’s typical for these types of guessing games. If you remove that restriction, though, then guessers could ask questions that reveal a little more information, like, “What is the difference between the digits?” or “What is the remainder when the number is divided by 6?” With this variant, it’s often possible to identify the number with two strategic questions.

**Puzzles**

Here are three puzzles that can lead to hours of conversation — and frustration! — on a car trip. Before you offer one to your crew, though, put forth the disclaimer that anyone who’s heard the puzzle before must remain mum. No reason they should spoil the fun for the rest of you. (The puzzles are presented here without solution, because you’ll know when you get the right answer. You can find the answer to any of them online with a quick search… but don’t do that. You’ll feel much better if you solve it yourself.)

*5. Dangerous Crossing*

Four people come to a river in the middle of the night. There’s a narrow bridge, but it’s old and rickety and can only hold two people at a time. They have just one flashlight and, because it’s night and the bridge is in disrepair, the flashlight must be used when crossing the bridge. Aakash can cross the bridge in 1 minute, Britney in 2 minutes, Cedric in 5 minutes, and Deng in 8 minutes. When two people cross the bridge together, they must travel at the slower person’s pace. And they need to hurry, because zombies are approaching. (Oh, sorry, had I failed to mention the zombie apocalypse?) What is the least amount of time that all four people can cross the bridge?

And how can you be sure that your method is the fastest?

*6. Weight of Weights*

Marilyn has a simple balance scale and four small weights, each weighing a whole number of grams. With the balance scale and these weights, she is able to determine the weight of any object that weighs between 1 kg and 40 kg. How much does each of the four weights weigh?

*7.* *Product Values*

Assign each letter a value equal to its position in the alphabet, i.e., A = 1, B = 2, C = 3, …, Z = 26. Then for any common word, find its product value by multiplying the value of the letters in the word. For instance, the product value of CAT is 60, because 3 × 1 × 20 = 60.

- Find as many common English words as you can with a product value of 60.
- Find a common English word that has the same product value as your name. (A little tougher.)
- Find a common English word with a product value of 3,000,000. (Zoiks!)

**Problems**

*8. The Three of Life*

This one looks so innocent!

What’s the probability that a randomly chosen number will contain the digit 3?

But spend a little time with it. And prepare for. Mind. Blown.

*9. Hip to Be (Almost) Square*

No calculators for this one.

There are four positive numbers — 1, 3, 8, and

x— such that the product of any two of them is one less than a square number. What is the least possible value ofx?

No spreadsheets, either.

*10. Hip to Be Square (Roots)
*

Just some good, old-fashioned algebra and logic to tackle this one.

Which of the following expressions has a greater value?

or

Surprised?

*11. Coming and Going*

Potentially counterintuitive.

At the holidays, Leo drove to his grandma’s, and the traffic was awful! His average speed was 42 miles per hour. After the holidays, however, he drove home along the same route, and his average speed was 56 miles per hour. What was his average speed for the entire trip?

And, no, your first guess was most likely not correct.

*12. The Year in Numbers*

A moldy oldie, to be sure, but this puzzle is always a crowd-pleaser for those who haven’t seen it before.

Use the digits of the new year — 2, 0, 1, and 9 — and any mathematical operations to form the integers from 1 to 100. For instance, you can form 1 as follows: 2 × 0 × 9 + 1 = 1.

For an added challenge, add the restriction that you must use the four digits *in order*.

**Bonus: A Book**

My sons get sick when they read in the car. That’s why I love the two books *Without Words* and *More Without Words* by James Tanton. Literally, there are no words! Each puzzle is presented using a few examples, and then students must follow the same rules to solve a few similar, but more challenging, puzzles.

Wherever you’re headed during the holiday break — driving to your relatives’ house, flying to Fort Lauderdale, or just relaxing at home — I hope your holidays are filled with joy, happiness, and lots of math!

## 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.