## Change the Vowel

The following puzzle contains a clue within each clue. The answer, of course, fits the clue, but each answer is also one of the words within the clue with its vowel sound changed. For instance, the clue “the distance from the top of your hat to the sole of your shoes” contains the word *hat*, and if you change the vowel sound from a short *a* to a long *i*, you get *height*, which fits the description.

As always, there’s a catch. Every answer to the clues below is a mathy word.

- The number of permutations of three different colored socks.

- This value is the same for 3 + 1 and 2 + 2.
- Do this with two odd numbers and you’ll get an even number.
- Three hours before noon.
- You may fail to correctly expand (
*a*+*b*)(*c*+*d*) if you don’t remember this mnemonic. - One represents this fractional portion of toes on the hoof of a deer.
- The last element of a data set arranged in descending order.
- The measure of central tendency made from the most common data points.
- The number of contestants ahead of third place if there’s a tie for first place.
- If the leader is forced to drop out of a race, the runner-up takes over this place.
- The square root is needed to calculate the length of the hypotenuse in this type of triangle.
- The graph of the declination of the Sun during a year can be approximated by this type of curve.

Answers

- six

- sum
- add
- nine
- FOIL
- half
- least
- mode
- two
- first
- right
- sine

## 100 Problems for the 100th Day of School

In May 2020, I delivered a webinar titled *One-Hundred Problems Involving the Number 100*. Every problem included a problem that somehow used the number 100, maybe as the number of terms in a sequence, the length of a hypotenuse in inches, or the number of digits written on a whiteboard. At the end of the webinar, NCTM President Trena Wilkerson challenged me to create a collection of 100 problems for which the answer is always 100.

So, I did.

My process was simple. I just wrote problem after problem with little concern for topic or grade level. Some of the problems were good; others were not. Some of the problems were difficult; others were easy. Some of the problems required knowledge of esoteric math concepts; others required nothing more than the ability to add and subtract. But I wrote 100 problems, then I reviewed them and deleted those that weren’t good enough. Then I wrote some more, and cut some more, and so forth, until I finally had a collection of 100 problems that were worthy.

And I’m going to share all of them with you in just a minute. But first, a math problem for which the answer is** not** 100.

As I said, I wrote the problems as they came to me, not necessarily in the order that I’d want to present them. But to keep track of things, I numbered the problems 1‑100. Since they were in the wrong order, I had to rearrange them, meaning that Problem 92 in the draft version eventually became Problem 1 in the final collection; Problem 37 became Problem 2; Problem 1 became Problem 3; and so on. You get the idea. So, the question…

You have a collection of 100 items numbered 1‑100, but the items are out of order. When you arrange the items in the correct order, how many would you expect to be labeled correctly? (Less generically, how many of my problems had the same problem number in the draft version and the final collection?)

The solution to that problem is more beautiful than I would have initially guessed. Have fun with it.

Without further ado, here is the collection:

**Problems with 100 as the Answer**

My goal was to release these problems in time for the 100th day of school, which most schools celebrate in late January or early February. I hope this collection reaches you in time. And I present the problems one per page, so you can decide which one(s) you’d like to use with your students. If you teach algebra, then perhaps you’ll print and share Problems 46 and 53; if you teach third grade, perhaps Problem 2 will be more appropriate. But the problems cover a wide range of topics and difficulty levels, so feel free to use whichever ones you like. (Be forewarned, though. The answer to **every** problem is 100, so unless your students are absolutely terrible at identifying patterns, you probably won’t want to share every problem with them. At least, not at the same time. I’m sharing this collection in time for the 100th day of school, but feel free to use any problem at any time.)

My favorite problem in the collection? I like Problem 47:

Above the bottom row, each number in a square is the sum of the two numbers below it. What value should replace the question mark?

Feel free to let me know if you or your students have a favorite.

p.s. – Bonus points if you can identify the origin of the 100 in the image at the top of this post.

## My 0.04 Seconds of Fame

In 2017, I attended the International KenKen Championship and filmed the final round, which I posted previously on this blog. But filmmakers Louis Cancel, Chris Flaherty, and Daniel Sullivan were there that day, too, and their cameras were significantly more sophisticated than my Samsung S8. Their footage of the competition, coupled with myriad interviews of competitors, organizers, and the inventor of KenKen himself, Tetsuya Miyamoto, has resulted in a new documentary, * Miyamoto and the Machine*, recently released by

*The New Yorker*. It tells the story of KenKen’s origins and attempts to answer the question, “Can a computer make puzzles as beautiful as those created by humans?”

Many aspects of the film will appeal to kenthusiasts, but my favorite moment occurs at 17:14. Competitor Ellie Grueskin is competing in the finals, and just over Ellie’s left shoulder is a barely visible, occasionally funny, middle-aged math guy holding — wait for it — a Samsung S8!

Yep, that’s me. I’m a star!

You have to ask yourself, what kind of monster would author such a shamelessly self-promotional post and not even provide one KenKen puzzle for the reader to enjoy? Definitely not me, so here you go.

After you solve the puzzle, definitely watch * Miyamoto and the Machine*. It’s 25 minutes well spent.

## A Pattern Puzzle for the New Year

Over at the Visual Patterns site, the directions state that if you click on a pattern, you’ll get to see the number of objects in the 43rd step. Why 43? I assumed that it had to do with Fawn Nguyen being a fan of Troy Polamalu — which, as far as I’m concerned, would be just one more reason to have an infinite amount of respect for her — but when I asked about it, Fawn explained that 43 was chosen as…

…a random number that was farther down the step number to prevent students from finding the number of objects recursively, but not too far.

This explanation sits well with my beliefs. In my book *One-Hundred Problems Involving the Number 100*, I stated that it’s appropriate to ask students to find the 100th term in a sequence because 100 is “big enough to exhilarate, but not so big as to intimidate.” The same could be said about 43.

Following Fawn’s lead, here’s a problem to get you in the spirit for the new year. Feel free to share this problem with your students on or near January 1.

*How many squares would be in the 43rd element of this sequence?*

Coincidentally, I shared this sequence with Fawn, and it now appears as #392 on the Visual Patterns site.

Speaking of sequences, here’s my favorite infinite sequence joke.

Infinitely many mathematicians walk into a bar. The first says, “I’ll have a beer.” The second says, “I’ll have half a beer.” The third says, “I’ll have a quarter of a beer.” They continue like this, each one ordering half as much as the last. The barman stops them and pours two beers. One of the mathematicians says, “That’s it? That’s not enough for all of us!” The bartender replies, “C’mon, folks. Know your limits.”

For fun, figure out how much beer the 43rd mathematician asked for.

And as a little more fun, guess the value of all the coins in the glass below. As a hint, there are the same number of quarters, dimes, and nickels, but three times as many pennies as dimes. (Said another way, Q:D:N:P::1:1:1:3.)

If you think about it a little, you’ll realize the answer without doing any computation.

Happy New Year!

## Mathy Zoom Backgrounds

Do you seek the admiration of your colleagues or the respect of your students?

Do you wish to create the illusion that you’re funny and cool?

Do you long to be the envy of your virtual social circle?

Unfortunately, you’re reading a math jokes blog, which means there may not be much hope for you. But a possible start may be to download some of the math joke backgrounds below for your next online meeting. I’ve been using them for the past few weeks, and I don’t think it’d be an overstatement to say that I’m now the envy of the internet. I mean, I’ve got a face for radio, but you have to admit that I look pretty fantastic when there’s a math poem above my head and equations on either side of it:

And guess what? You can look that cool, too!

To use any of the images below, simply right click and “Save Image As…,” then install them as virtual backgrounds (Zoom, Google Meet). If you’d like a better look at any of them before deciding if they’re worth valuable memory on your laptop, just click on an image to open it full screen.

## KenKen 12 Puzzle for 12/12

Today is the twelfth day of the twelfth month, and in honor of the date, here’s a 4 × 4 KenKen puzzle that has 12 as the target number in each cage. The entire puzzle has only four cages, and it only uses addition and multiplication. Have at it!

But a post with just a KenKen puzzle isn’t much of a post, especially on a math jokes blog. So let’s consider some jokes that have to do with the association between 12 and a dozen. The following are some mathematical insults you can use if you’re playing the dozens.

Yo momma is so fat, she’s proof that the universe is expanding exponentially.

The shortest distance between two points is around yo momma’s ass.

Yo momma is so fat, her volume is an improper integral.

Yo momma is so crazy, when she received a can of Pepsi from the vending machine, she started jumping up and down, yelling, “I won! I won!”

Yo momma is so dumb, she thinks convex are inmates locked in a prism.

Yo momma is so infinitely fat, she can eat as much as she wants and not gain any weight.

Yo momma thinks cosine is what she does for a loan.

Yo momma is so dumb, she sleeps with a ruler to keep track of how long she sleeps.

Yo momma is so fat, she took geometry because she heard there was gonna be π.

Yo momma is so fat, the ratio of her circumference to diameter is 4.

Yo momma is so fat, in a love triangle she’d be the hypotenuse.

Yo momma thinks *coincide* is what you should do when it’s raining.

The integral of your mom is fat plus a constant, where the constant is equal to more fat.

Yo momma is so dumb, she doesn’t know the difference between a doughnut and a coffee cup.

Yo momma is so dumb, she thinks crossing a mosquito and a mountain climber yields |mosquito| × |mountain climber| × sin(θ).

The derivative of yo momma is strictly positive.

Yo momma is so dumb, she serves beer in Klein bottles.

Yo momma is so dumb, she thinks that if two people go into a hotel and three come out, the first two must have pro-created.

Yo momma is so dumb, she can’t even solve a second‑order non‑homogeneous differential equation.

Yo momma is so fat, her dress size requires an exponent.

The limit of yo momma’s ass tends to infinity.

Yo momma is so fat, when she steps on the scale, it displays π without a decimal point.

Yo momma’s muscle-to-fat ratio can only be explained by irrational complex numbers.

Yo momma is so ugly, Pythagoras wouldn’t touch her with a 3-4-5 triangle.

## Guess the Graph

The bar graph below was created because of a recent discussion with my wife. The title and axis labels have been removed. **Can you identify the data set used to create the graph?** I’ll give you some hints:

- The data set contains 32 elements.
- It’s based on a real-world phenomenon from this year.
- The middle five categories account for 81% of the data.
- The special points marked by A, B, and C won’t help you identify the data set, but they will be discussed below.

Got a guess?

No clue? Okay, one more hint:

- The vertical axis represents “Teams.”

Still not sure? Final hints:

- Point A represents the lowly J-E-T-S, who are currently winless.
- The region outlined by B shows that 26 teams have from 3 to 7 wins.
- Point C on the graph represents my Pittsburgh Steelers, whose record is a perfect 10‑0. (It’s my hope that I’ll still be able to gloat on Friday morning, after the Steelers host the Ravens on Thanksgiving night.)

This graph was generated while discussing the current standings in the NFL with my wife, who speculated that there seemed to be a lot of really good teams and a lot of really bad teams this year. The horizontal axis represents the number of wins. As it turns out, the distribution above is somewhat typical at this point in the season. At the end of most seasons, about 2/3 of the teams finish a 16-game season with 5 to 10 wins. It may be a little unusual that there are 8 teams with 7 wins, but it’s not statistically cray-cray.

If you’ve read this far, then you may enjoy these other math-related football trivia questions:

- Describe two ways in which an NFL game can end with a score of 2‑0.
- What’s the greatest score that cannot be attained by scoring only touchdowns (7 points) and field goals (3 points)?
- Express the ratio of width:length of a football field. For length, include the end zones.
- What are the only positions allowed to wear single-digit uniform numbers?
- During a typical broadcast of an NFL game, approximately what percent of the time is spent actually playing football (as opposed to commercials, half time, or just milling around between snaps)?

Happy Drinksgiving! And, go Stillers!

—

Answers

- A game can end 2‑0 if one team scores a safety and the other team doesn’t score at all. It can also end 2‑0 if one team forfeits before either team has scored, by league rule. (In high school and college, a forfeit is officially recorded as a 1‑0 loss.)
- 11 points. Any point total above that is (theoretically) possible. Below that, it’s not possible to score 1, 2, 4, 5, or 8 points.
- A field is 53 1/3 yards wide and 120 yards long. In feet, that’s 160:360, which can be reduced to 4:9.
- Quarterbacks and kickers.
- According to several analyses, 11 minutes of a three-hour broadcast is spent actually playing. That’s about 3%. Sheesh.

## Mathy Portmanteaux

The term *portmanteau* was first used by Humpty Dumpty in Lewis Carroll’s *Through the Looking Glass*:

Well, ‘slithy’ means “lithe and slimy” and ‘mimsy’ is “flimsy and miserable.” You see, it’s like a portmanteau — there are two meanings packed up into one word.

Interestingly, the word *portmanteau* itself is also a blend of two different words: *porter* (to carry) and *manteau* (a cloak).

Portmanteaux are extremely popular in modern-day English, and new word combinations are regularly popping up. Sometimes, perhaps, there are **too many** being coined. In fact, one author refers to these newcomers as *portmonsters*, a portmanteau of, well, *portmanteau* and *monster* that attempts to capture how grotesque some of these beasts are. An abridged list of portmonsters would include *sharknado*, *arachnoquake*, *blizzaster*, *snowpocalypse*, *Brangelina*, *Bennifer*, *Kimye*, *Javanka*, *fantabulous*, and *ridonkulous*.

Portmanteaux seem to proliferate most easily in B-movie titles, weather, and celebrity couples, but the world of math and science is not free from them. Here are a few mathy portmanteaux, presented, of course, as equations.

**ginormous** = *giant* + *enormous*, really big

**guesstimate** = *guess* + *estimate*, a reasonable speculation

**three-peat** = *three* + *repeat*, to win a championship thrice

**clopen set** = *closed* + *open set*, a topological space that is both open and closed

**b****it** = *binary* + *digit*, the smallest unit of measurement used to quantify computer data

**pixel** = *picture* + *element*, a small area on a display screen; many can combine to form an image

**voxel** = *volume* + *pixel*, the 3D analog to pixel

**fortnight** = *fourteen* + *night*, a period of two weeks

**parsec** = *parallax* + *second*, an astronomy unit equal to about 3.26 light years

**alphanumeric** = *alphabetical* + *numeric*, containing both letters and numerals

**sporabola** = *spore* + *parabola*, the trajectory of a basidiospore after it is discharged from a sterigma

**gerrymandering** = *Elbridge Gerry* + *salamander*, to draw districts in such a way as to gain political advantage (In the 1800’s, Governor Elbridge Gerry redrew districts in Massachusetts to his political benefit. One of the redrawn districts looked like a salamander.)

**megamanteau** = *mega* + *portmanteau*, a portmanteau containing more than two words, such as DelMarVa, a peninsula that separates the Chesapeake Bay from the Atlantic Ocean and includes parts of Delaware, Maryland, and Virginia

**meganegabar** = *mega* + *negative* + *bar*, the line used on a check so that someone can’t add “and one million” to increase the amount

(By the way, when Rutgers University invited *Jersey Shore* cast member Snooki Polizzi to speak to students on campus in 2011, they paid her $32,000, which is $2,000 more than they paid Nobel and Pulitzer Prize winning author Toni Morrison to deliver a commencement address six weeks later.)

## Getting Back to My Roots

For years, this blog represented the finest mathematical humor that the internet had to offer. That hasn’t been the case so much recently, so it’s time I got back to my roots — of course, for me, those would be cube roots…

I was inspired to craft this post of horrendously bad puns when my sister’s friend shared this photo with me:

And I figured if I have to suffer, you should, too.

How many math grad students does it take to change a light bulb? Just one, but it takes nine years.

What’s the best tool for math class? Multi-pliers!

Think outside the regular quadrilateral.

When asked how good she was at algebra, the student replied, “Very able.”

What’s the difference between the radius and the diameter? The radius.

Are you depressed when you think about how dumb the average person is? Well, I’ve got bad news for you… nearly half the population is even dumber.

How do you make one disappear? Add a

g, then it’sgone.

Writing haiku is

tough, because you have to count.

Writers don’t like math.

Light travels faster than sound. This is why some people appear bright until you hear them speak.

The grad student had trouble getting the pizza box into the recycling can. It was like trying to put a square peg in a round hole.

How is the moon like a dollar? Both have four quarters.

Don’t look now, but there’s a suspicious man over there with graph paper. I think he’s plotting something.

## Thinking > Symbols

As if having to wear a mask while flying isn’t scary enough, last week I had a harrowing experience.

It’s normally a three-hour flight from Yellowknife in the Northwest Territories to Portland, but about 20 minutes into a recent flight, there was an unsettling noise followed by an announcement from the pilot: “Ladies and gentlemen, this is your captain. Please don’t be alarmed by the noise you just heard near the left wing. Let me assure you that everything is okay. We’ve lost an engine, but the other three engines are working just fine. They’ll get us to Portland safely. But instead of taking 3 hours, it’s going to take 4 hours.”

She said this matter-of-factly, as if she were commenting on the color of my shirt or pointing to a squirrel that was enjoying an acorn. It was reassuring that she didn’t seem worried — but still, we had *lost an engine*! The cabin grew quiet as we waited to see what would happen next. But, nothing. We continued to fly as if nothing had happened, and the din of conversation slowly resumed.

And then there was another noise near the right wing, followed by the pilot: “Ladies and gentlemen, this is your captain again. Unfortunately, we’ve lost another engine. But the two remaining engines are functioning properly. We’re going to get you to Portland safely, but now it’s gonna take about 6 hours.”

Needless to say, you could hear a pin drop as we passengers waited with baited breath. And sure enough, there was another noise, and the pilot again: “Ladies and gentlemen, we’ve lost a third engine, but we’ll make it safely to Portland with the one engine we’ve got left. It’s just gonna take 12 hours.”

That was when the guy in the seat next to me lost it. “I sure hope we don’t lose that last engine,” he yelled, “or we’re gonna be up here **all day**!”

That story isn’t true. None of it. Not. One. Single. Fact. I mean, come on! Yellowknife? Why would I need to go to Yellowknife? To brush up on my Dogrib?

But the following story *is* true.

My honorary niece Ivy occasionally calls for help with math. Not surprisingly, the frequency of calls has increased with online learning during the pandemic. This is a modified version of a question about which she recently called:

A plane traveling against the wind traveled 2,460 miles in 6 hours. Traveling with the wind on the return trip, it only took 5 hours. What was the speed of the plane? What was the speed of the wind?

It’s assumed that the plane flew at the same average speed in both directions and that the wind was equally strong throughout. On a standardized test, those assumptions would need to be stated explicitly. But on this blog, I make the rules, and I refuse to add more words to explain reasonable assumptions without which the problem would be impossible to solve.

As presented to Ivy in an online homework assignment, the problem stated that *x* should represent the plane’s speed and *y* should represent the speed of the wind. My first question was, “Why? What’s wrong with *p* and *w* as the variables for plane and wind, respectively?” But my second, and perhaps more important, question was, “Why are they forcing algebra on a problem that is much easier without it?”

Somewhere, many years ago, someone decided that the methods of substitution and elimination were critically important, if not for success in life then at least for success in high school. Anecdotally, this seems to be true; I was a bad-ass systems-of-equations solver in high school, and I was also the captain of the track team that won the county title, so… yeah. Sadly, that someone seemed to put less stock in a slightly more critical skill: thinking.

Here’s the deal. Against the wind, the plane covered 2,460 miles in 6 hours, so the plane’s speed on the way out was 410 miles per hour. With the wind, the plane covered those same 2,460 miles in just 5 hours, so the plane’s speed on the way back was 492 miles per hour. Since the wind is constant in both directions, the unaided speed of the plane must have been 451 miles per hour, exactly halfway between the against-the-wind and with-the-wind speeds. Visually,

The top portion of that diagram represents the plane flying against the wind; the bottom portion represents the plane flying with the wind. This makes it obvious that the the unaided speed of the plane is equidistant from 410 miles per hour and 492 miles per hour, and that distance is the speed of the wind.

Algebraically, the problem might be solved like this…

which is no different than the logical approach above: *x* ‑ *y* is the speed of the plane against the wind, and *x* + *y* is the speed of the plane with the wind, so the numbers are obviously the same. But using algebra just seems so much more… *cumbersome*.

Students in elementary classrooms often explain their thinking with drawings and diagrams. Unfortunately, some of that great thought is abandoned when students take a course called Algebra, and the mechanical skill of symbolic manipulation replaces the mathematical skill of logical reasoning. That’s a tragedy. As Jim Rubillo, former executive director at NCTM, used to say, “We have to stop being so symbol‑minded.”