## Posts filed under ‘Uncategorized’

### What an Amazing Date!

There are lots of good dates — meeting at a bookstore coffee shop for, well, perusing books and sipping coffee; spending hours playing Super Mario Bros. and Pac-Man at a retro video game arcade; and, of course, going to a open-mic comedy show where one of the performers tells nothing but math jokes.

But great dates? Well, those are pretty rare. My first date with my wife — where I took her to a hotel and “whispered” to her from the couch across the lobby — is an example, though the stimulating conversation and her perfect laugh may have contributed more than the elliptical ceiling. (Maybe.)

Few dates, however, can compare to today’s date:

**12/3/21**

Look at that beautiful symmetry! Marvel at its palindromic magnificence! The way it rises then falls, like a Shostakovich melody.

But wait… there’s more! Consider the following pattern:

1 × 1 = 1

11 × 11 = 121

111 × 111 = ?

That’s right! The number 12,321 is a perfect square! And not only that, its square root contains only 1s.

Moreover, check this out:

1 + 2 + 3 + 2 + 1 = 9

That’s right! It’s a square number, and the sum of its digits is also a square number!

Finally, here’s a KenKen puzzle that makes use of the number, though it’s not unique unless one of the digits is already filled in:

No matter how you choose to celebrate, here’s hoping your day is as great as the date!

### Better Multiple-Choice

If I were a K-12 student right now, I’d want to live in San Diego. In 2020, San Diego Unified School District introduced a new district-wide math test that contained **no multiple-choice questions**. The district was allowed to use their own internal test instead of the state test last year, due to the pandemic, and they’re apparently allowed to use it again this year. Supposedly, the test moves away from a reliance on computational ability and instead measures three dimensions: students’ knowledge of mathematics (concepts and formulas); their application of that knowledge; and, their ability to communicate mathematically.

The optimist in me says, “It’s about time!” But the pessimist in me thinks, “Don’t they know that it’s a lot more expensive, and harder to ensure reliable and replicable results, when using humans to do the scoring instead of machines?” I’m old enough to remember the controversy and eventual dissolution of the Maryland State Performance Assessment Program (MSPAP) exams, which required extended answers and contained no multiple-choice questions. FairTest called the MSPAP test “perhaps the single best state exam,” but it was criticized for providing school-level but not individual student scores. Though generally agreed to have been a catalyst for improved teaching, it was replaced by an entirely multiple-choice test to meet the requirements of the Elementary and Secondary Education Act (ESEA).

Ah, the good old days. Reminiscing sure ain’t what it used to be. But, I digress.

To determine if multiple-choice questions are valid tools, I made a list of pros and cons regarding their use in educational assessment.

Pros | Cons |

They can be scored quickly. | The correct answer can be guessed. |

They can be scored objectively and without bias. | The correct answer can be found by process of elimination. |

They encourage students to think like the test creators instead of like themselves. | |

They provide no information about the student’s solution strategy. | |

They are required to have only one answer. | |

They exacerbate test anxiety. | |

They don’t prepare students for college or the work force. | |

Incorrect answer choices expose students to misinformation, which can influence future recall and thinking. |

Seems a bit lopsided.

In recent years, multiple-choice questions have gotten a bit of a makeover. Those in the educational assessment industry now call them “selected-response items” because, well, students get to select a response.

But this is just semantics. Referring to a pig as a mud wrestler may sound nicer, but the pig won’t be any less dirty.

It’s the advertising trope of…

New look, same great taste!

Or as a pretentious coffee brand said when they changed the label…

Innovative presentation, but consistent quality.

Truth is, selected-response items look like they’ve always looked, typically with a really boring prompt and even more boring answer choices. As one example, the following item is from a PACE (Packet of Accelerated Christian Education), which “integrate Godly character-building lessons into the academic content.”

**Mr. Louis Pasteur did experiments with milk.** Mr. Louis Pasteur was…

- a glass bottle
- an airplane
- a scientist

Despite your religious beliefs, you have to admit that this question is rather absurd. Would any student ever think that a glass bottle or an airplane would be referred to as “mister”? To be fair, this question appeared in a PACE packet in 2013, so it’s quite possible that it’s since been updated. Still, 2013 wasn’t that long ago, and there’s no time in history when those answer choices wouldn’t have been ridiculous.

And here’s one that was presented during a session at an NCTM regional conference:

To **convert to radians**, multiply by…

- π/180
- 180/π
- 225π/180
- π/40,500

Ignoring the fact that this question attempts to assess something that your calculator knows so you don’t have to, this question is fine. But in the reading passage directly above the question, it stated, “To convert an angle from degrees to radians, multiply by π/180.”

Well, that will just never do.

I’m not convinced that a great multiple-choice question actually exists. That said, some are better than others, so I offer you the following seven multiple-choice — or selected-response, or objective-response, or whatever-you-want-to-call-them — items.

What is the probability that you will **randomly choose the correct answer** to this question?

- 25%
- 50%
- 0%
- 25%

At any given time, **the number of people in the air** — that is, those who are flying in motorized aircraft, and not counting those who were recently launched by catapults or who have bounced on a trampoline — is closest to the population of…

- Flint, Michigan
- Seattle, Washington
- New York, New York

The approximate **volume of an average chicken egg** is…

- 7 cm
^{3} - 70 cm
^{3} - 700 cm
^{3} - 7,000 cm
^{3}

The **polar (north-to-south) diameter of the Earth** is about…

- 1,000,000 inches
- 20,000,000 inches
- 500,000,000 inches
- 1,000,000,000 inches

**One million one-dollar bills** weigh about as much as…

- a three-toed sloth
- a giant panda
- Chris Christie
- a grizzly bear
- a black rhinoceros

The total number of **calories in all the hot dogs consumed at Yankee Stadium** during one season of Major League Baseball is closest to…

- the number of five-card poker hands
- the number of possible license plates in Indiana
- the number of combinations in the Powerball lottery
- the number of humans on Earth
- the number of stars in the Milky Way

If **the residents of New Mexico joined hands and stood in a straight line**, they could reach from one side to the other of…

- New Mexico
- Rhode Island
- Texas
- Alaska

The answers to these questions will not be provided, though each question absolutely has a best answer among the choices. In lieu of an answer key, enjoy the following joke:

How do you keep a fool in suspense?

### Interview: Kerry Schultz, Saucon Valley High School

It sounds like the start of a math joke: Did you hear about the mathematical economist who became a teacher? The punch line is, “Her name is Kerry Schultz,” which, admittedly, isn’t very funny, but it’s absolutely true. Kerry used to work as an analyst for JPMorgan Chase but now teaches calculus and computer science at Saucon Valley High School in Hellertown, PA.

Seniors selected Kerry to be the faculty speaker at the 2021 SVHS commencement. During her speech, Kerry gave the graduates some sage advice. “When you get the choice to sit it out or dance, I hope you dance,” she told them, drawing from Lee Ann Womack’s 2000 hit. She also referenced one of my favorite publications:

After a difficult last two years, I promised to avoid the pandemic topic, and I wanted to be sure to keep this on the lighter side. So I brought my favorite book,

Math Jokes 4 Mathy Folks. I’m pretty sure this book is the reason I was chosen to speak tonight.

As it turns out, Kerry and I have more in common than just our love of math jokes. We both have twins. (In fact, she has twin 10-year-old daughters and a 9-year-old son. I’ve never been good with numbers, but I’m pretty sure that means that, at one point, she had three kids under age two in her house. My goodness!) Like my wife and me, Kerry and her husband both love math jokes, math memes, and all things numeric. The two of them used to play The Game of 24 on long car rides; my wife and I played Dollar Nim with our kids.

The comparison ends when it comes to exercise, though. I’m active, but Kerry runs at 4:30 a.m. most mornings, because she spends her afternoons taking kids to their various activities (soccer, baseball, football, swimming, cross country, and golf). She’s finished 10 marathons and hopes to run the Chicago Marathon in 2022. (I’ve also never been asked to speak at a graduation. Yet.) In her limited downtime, Kerry enjoys traveling or reading a good book on the beach.

I caught up with Kerry when a friend forwarded her picture from the Lehigh Valley Press. As it turns out, Kerry has a fascinating story about her path to education.

**Can you tell us how you got to Saucon Valley?**

I went to college with the hopes of becoming a math teacher, but others convinced me that I was “too smart” for that. So I graduated from Colgate University in 2000 with a degree in mathematical economics, and I went to work as an analyst at JPMorgan Chase in midtown Manhattan. Some might say it was glamorous, with lots of fancy meals and car service home every night — but I hated pretty much everything about it.

I was working in midtown on September 11, 2001, and my brother was working on the 90th floor of the South Tower. He was extremely lucky to escape the attack on the World Trade Center, but many of his co-workers did not. This was a pivotal day for me. I realized life was way too short to spend it doing something I hated. In the following weeks, I began looking for graduate programs in mathematics education. In 2002, I enrolled at Lehigh University, and in 2004, I began teaching middle school math in the Saucon Valley School District.

**What is your current role?**

I taught middle school math for five years while obtaining my principal certification from Lehigh. I then became a Coordinator of Academic Services and later an Instructional Coach for Math, all in Saucon Valley. In 2015, I requested a return to the classroom and was thrilled to be asked to teach high school math. I have been in the high school for five years now, teaching Algebra 2, Pre-Calculus, Calculus, and AP Computer Science Principles. I absolutely love my job!

**And if you weren’t teaching math?**

I’d love to be a professional athlete or work in an athletic setting — maybe a statistician for the NY Mets!

**What’s your favorite thing about teaching?**

The kids! It’s important to get to know each and every one of my students as best I can. Nothing is better than knowing I have made a difference in the life of a student. Sometimes it’s by helping them solve a difficult problem, sometimes by building their confidence, sometimes by showing up to their lacrosse game, and sometimes it’s simply by being there when they’ve had a rough day. The relationships I’ve built with students over the years are by far the most important thing to me.

**What is your favorite math joke(s)?**

My oldest favorite has to be, “What did 0 say to 8?” Now that I teach computer science, I really like, “There are 10 kinds of people: those who understand binary, and those who don’t.”

**Which math joke(s) do your students like best?**

The jokes that poke fun at mathematicians tend to be class favorites. “What do you call a beautiful woman on the arm of a math graduate student? A tattoo.” And, “What’s the difference between a large pizza and a mathematician? A pizza can feed a family of four.”

**What’s the funniest thing you’ve ever said during class? Or maybe, what’s the funniest thing that’s ever happened in class?**

This is definitely a case of “you had to be there,” but one year I had an Honors Calculus student convince me, with the help of her classmates, that after high school she was going into the family business of designing chairs. I was skeptical at first, but they were so believable and had so many details, they had me convinced for days. They told me that her family designed chairs for Nicki Minaj, and to this day I can’t hear that name without dying of laughter.

**What is your favorite area of mathematics? Is that also your favorite thing to teach?**

I don’t have a personal favorite, but I definitely love teaching calculus. Calculus is a great challenge for many students, but most of them are willing and able to put forth the effort to succeed. I enjoy helping students work through the difficulties, and I’m just as excited as they are when it all starts to make sense. It is fantastic when you see the light bulb go on!

### A Great Day for a Pattern

When I first saw today’s date in mm/dd/yy format — 11/12/21 — I thought, “Well, that’s pretty cool. It’s all 1s and 2s.” And then I thought, as I’m sure you did, “In ternary, that’d be 376,” because everyone thinks in ternary, right?

But then I looked at the number again, and I thought, “Ah, hello, old friend. Good to see you again.”

Those six digits form the fifth term of a famous pattern:

1

11

21

1211

111221

So your first question is, what’s the next term?

If you’ve never seen this pattern before, it’s worth a little of your time to try to figure it out before reading more about it at MathWorld.

Your second question — if you’re still reading — is, what’s the greatest digit that will ever appear in this sequence? As you can see above, the first five terms only contain 1s and 2s. What digits are in the sixth term? What digits will appear beyond the sixth term? How do you know?

### There Are Two Types of People…

It’s estimated that there are 7.9 billion people in the world, and counting. But in many ways, it’s fairly easy to divide us all into two types.

There are two types of people:

- Those who think the world can be divided into two types of people.
- Those who don’t.

The earliest known usage of the two-types format was by Mark Twain:

There are basically two types of people: people who accomplish things, and people who claim to have accomplished things. The first group is less crowded.

The potential origin of the two-types meme, as we know it today, may have been this ubiquitous math and computer science joke:

There are 10 types of people in the world: those who understand binary, and those who don’t.

A modification of that joke has appeared more recently for the uber-geeks:

There are 10 types of people in the world: those who understand ternary, those who don’t, and those who mistake it for binary.

Physicist C. N. Yang, who won the Nobel Prize in 1957, is credited with this version:

There are two types of math books: those you cannot read beyond the first sentence, and those you cannot read beyond the first page.

Two of my favorites were included in **More Jokes 4 Mathy Folks**:

There are three types of people: positive, negative, and relative.

There are two types of people: those who are wise, and those who are otherwise.

The number of modifications to the format are nearly infinite. To create your own, choose the number of things you wish to compare; choose the type of things you wish to compare; describe that number of things, making sure that two of them are diametrically opposed, as to cause an incongruous and humorous result; if possible, be self-deprecating in one of the descriptions; and finally, determine if you want it in paragraph form or as a bulleted list. For instance,

There are two types of math jokes:

- Those that are funny.
- Those that have appeared on this blog.

See? It’s not hard. Now you try. The following mathy examples can serve as inspiration.

There are three things I hate:

- People who can’t do simple math.
- Irony.

There are three things I hate:

- Bulleted lists.
- Lazy people.

There are two kinds of statistics:

- Those you look up.
- Those you make up.

There are three kinds of lies:

- Lies.
- Damned lies.
- Statistics.

There are two kinds of people. Avoid both of them.

There are two kinds of people:

- Those you want to drink with.
- Those who make you want to drink.

On the web, you’ll find all manner of visual adaptations of the meme.

There are two types of people:

There are two types of people:

And finally, there are two types of bloggers:

- Those who would write a blog post about the world containing two types of people.
- Those who would Google it first to see that there about 24,000,000 results for “there are two types of people.”

### A Funny Thing Happened at the Periodontist

People in Portland are nice. Like, really nice. *Nice to a fault*, some would say. It’s the reason the term “Portland nice” exists, and it’s the impetus for the following scene from Portlandia in which two drivers at an intersection insist — politely, but with increasing determination — that the other one go first.

Ashley, one of the assistants at my periodontist’s office, is Portland nice. So at an appointment a while back, when I settled into the chair, it wasn’t surprising that her opening question was, “Got any plans for the weekend?”

As it turns out, I did. The Museum of Mathematics had invited me to host a webinar as part of their Family Fridays series, and I offered to deliver Punz and Puzzles, an hour or so of, well, math puns and math puzzles. I told Ashley about this, and there was a long pause before she responded. Finally, she said, “Do you know a lot of math jokes?” Before I could assure her that, indeed, I knew at least two volumes’ worth, Dr. Thanik entered the room, and our conversation was temporarily paused.

Dr. Thanik then did what periodontists do: he told me about the procedure that he was going to perform, and he injected several gallons of novocain into my gums. While it took effect, Ashley said, “Tell me a joke.”

“What?” asked Dr. Thanik.

“Not you,” she said, waving a hand at me. “Mr. Vennebush. Before you came in, he was telling me that he’s doing a webinar tonight that involves math jokes.”

“Do you know a lot of math jokes?” Dr. Thanik asked.

I explained that I had literally written the book on them.

“Well, then… let’s hear one!” he demanded.

Neither of them seemed to care that my mouth was numb and any joke would be delivered through an excessive amount of drool. Fortunately, I have very little self-respect or regard for etiquette, so I didn’t care, either. I launched in.

“Well, you probably know the world’s most ubiquitous math joke,” I began. “Why is 6 afraid of 7?”

They responded in unison. “Because **7 8 9**!”

“Yes!” I said. “But there’s a follow-up. Why is epsilon afraid of zeta?”

Raised eyebrows. Blank looks. Silence.

“Because **zeta eta theta**!” I exclaimed.

“Oh, I should’ve gotten that!” Dr. Thanik said, with the knowing look of someone bearing a Greek surname.

He then performed the procedure. As he finished the last suture, he said, “Okay, I don’t quite have the wording right, but what about this? Why couldn’t the **tangent** get a loan? Because his parents wouldn’t **cosine**.”

“Did you just make that up?” I asked.

“You seem surprised,” he said.

“I just didn’t expect my periodontist to make references to trigonometry,” I replied.

“Well,” he said, “I know a lot of things. After all, I spent 20 years in school.”

I continued, “Well, I guess I’m also a little surprised that you were trying to formulate a math joke while performing gum surgery.”

“Fair,” he said.

And then it occurred to me. “Oh, of course!” I said. “I’ve got the *perfect* joke for you. Did you hear about the middle school math teacher who became a dentist?” I asked.

“No,” they said.

“Her specialty is **square root canals**!”

They both laughed politely. Like I said, Portland nice.

### Mathy One-Liners

To keep my edge, I read joke books and watch comedians. I modify the jokes I read and hear to fit my particular needs and, sometimes, I just steal a joke outright. I’d feel bad about doing this if I profited from it, but there is little to be gained by dropping a one-liner at a neighborhood happy hour.

I just finished *1001 One‑Liners and Short Jokes* by Graham Cann. It compensates with quantity what it lacks in quality. Although most of the jokes are not good — and many rely on British English, and others reference British culture, so they’re lost on me — there are more than a few chestnuts in the mix. I used this modification of one of his jokes while having dinner with my in-laws recently:

I don’t like coffee. It’s just not my cup of tea.

It garnered guffaws from my mother-in-law and groans from my sons, so it had the intended effect.

Another joke from the book is mathematical:

When I was two, I was really anxious because my age had doubled in just one year. I thought, “If this keeps up, by the time I’m six, I’ll be 90!”

It’s a terrible joke, not least because I’m unaware of any toddler concerned about their age. But more importantly, it’s wrong. If your age doubled from one to two in a year, then it would double to four by age three, to eight by age four, to 16 by age five, and to 32 by age six. Graham Cann clearly hasn’t studied exponential growth.

The following are other mathy jokes from the book, most of which I’ve modified at least slightly.

- I took an algebra test at school yesterday. My kleptomania is getting out of hand.
- For the three o’clock race, I backed a horse at ten to one. It came in at a quarter past four.
- One of every four frogs is a leap frog.
- My gun is made from a dozen pigs. It’s a 12-boar.

- Thirty percent of car accidents in Sweden involve a moose. I say it’s time that we stop letting moose drive. (For the record, that statistic is likely fabricated. It’s estimated that there are 4,500 car accidents involving moose every year, but there are far more than 15,000 car accidents annually.)
- Did you hear about the constipated accountant? He tried to work it out with a pencil — but he couldn’t budget.
- To the man who invented zero: Thanks for nothing.
- Statistically, six of seven dwarfs are not Happy.
- I, for one, like Roman numerals.
- If every human in the world laid down end‑to‑end along the equator, most of them would drown.
- Ninety-nine percent of politicians give the rest of them a bad name.
- Light travels faster than sound, which is why some people appear bright until you hear them speak.
- I tried to change my password to “14 days,” but my computer said it was too week.

There were 288 others that I chose not to share, because they were two gross.

### Math Puzzles with Letters

This week on the NPR Sunday Puzzle, host Will Shortz offered the following challenge:

Name a famous city in ten letters that contains an

s. Drop thes. Then assign the remaining nine letters their standard value in the alphabet — A = 1, B = 2, C = 3, etc. The total value of the nine letters is only 25. What city is it?

It’s not much of a spoiler to note that the average value of those nine letters must be less than three, since their sum “is only 25.” Consequently, a lot of those letters must occur at the beginning of the alphabet and — if eight of them were *a*‘s — there would be no letters later than *q* in the name of the city. But that’s as much as I’ll say; you can solve the puzzle on your own. (When you do, you can submit your answer for a chance to play next week’s on-air puzzle live with Will Shortz.)

Mathematician Harold Reiter uses a similar problem with elementary school students. Using the same idea — that each letter has a value (in cents) equal to its position in the alphabet — he asks students to find a dollar word, that is, a word whose letters have a sum of 100. As it turns out, there are many. Based on a nonexhaustive search, there are at least 3,500 dollar words, and likely a whole lot more. In a quick perusal of the list, one word jumped out: **oxygon**. Nope, that’s not a typo. It’s an archaic term meaning “a triangle with three acute angles.”

All of this talk of letters reminds me of my favorite puzzle, which I call Product Values. Using the same scheme — that is, A = 1, B = 2, C = 3, etc. — find the product value of a word by *multiplying* the values of the letters. So, for instance, *cat* has a product value of 3 × 1 × 20 = 60. How many words can you find that have a product value of 100? Based on the ENABLE word list, there are nine. (If you need some help, you can use the Product Value Calculator at www.mathjokes4mathyfolks.com.)

To end this post, a few math jokes that involve letters:

And Satan sayeth, “Let’s put the alphabet in math.” Bwa-ha-ha-ha-ha.

Romans had no trouble with algebra, because X was always equal to 10.

### Brood X Anniversary

The Brood X cicadas have erupted again after 17 years, which can mean only one thing: it must be our 17th wedding anniversary.

You see, my wife and I were married in 2004, the last time the Brood X cicadas surfaced. Despite our poor choice of date — honestly, who plans an outdoor wedding during peak cicada time? — we were unaffected. Hundreds of cicadas were chirping away about a mile from the venue, but only two made an appearance at the event: one was seen crawling across the stage in front of the band, and the other was sunning itself on the concrete wall of a water fountain in the garden.

My best man and his wife recently sent us the following photo:

The accompanying text read, “They’re ba-a-a-a-ack! It must be your 17th anniversary!”

At dinner on Sunday night, my wife handed me a card with this image on the front:

She knows me well, and she knew a Venn diagram (no relation) would resonate. Of course, I couldn’t help but wonder why the area of overlap was so small! After 17 years, shouldn’t the image look more like this?

Was she trying to tell me something? Luckily, the message inside gave a clear indication that I had nothing to worry about. (No, I won’t elaborate as to what it said. Use your imagination. Nice boys don’t kiss and tell.)

Venn diagrams are useful for expressing relationships between two or more things. This one appeared on Twitter at least five years ago, but I only discovered it recently.

In my humble opinion, the king of Venn diagrams is Demetri Martin, from whom we get this wonderful comparison of ants, bears, and people:

My personal contribution to the genre stems from a realization about the disciplines that claim April as their national month: math, poetry, and humor:

Finally, a Venn diagram tautology.

### MathCounts Problems, Practice, and New Friends

Later today, my sons will represent Sellwood Middle School at the Oregon State MathCounts competition; and, if they do well enough, they’ll represent Oregon at the 2021 Raytheon Technologies MathCounts National Competition. They were invited to compete in the state competition today because they finished first and second in the local MathCounts competition:

The local MathCounts organizers hosted a virtual celebration for participants, and at the end, I asked if any students from other schools who qualified for the state competition would be interested in some joint practice sessions. As a result, the coach and two students from Access Academy joined Alex, Eli, and me for two 90-minute sessions this past week, during which we solved some previous state- and national-level problems. One with which we had great fun and lots of discussion was about cryptocodes:

A certain cryptocode must contain one letter from the set {X, K, M, Z} and three distinct letters from the set {W, X, Y, Z}. The four letters can be arranged in any order, and since X and Z are in both sets, these letters may each appear twice in an arrangement. How many cryptocodes are possible?

2017 MathCounts National Competition, Target Round, Problem 8

When asked for their answers, Alex suggested a number that was 24 too low, and Eli gave a number that was 24 too high. After discussing the solution, the other coach said, “Alex and Eli, I think it’s awesome that the average of your answers was the correct answer! Is that because you’re twins?” Now, that’s funny.

A video solution from Sjoberg Math is available on YouTube; my solution is below.

I’m occasionally asked how to prepare for MathCounts competitions. Our in-home preparation program involved several parts:

- Leave math and puzzle books — such as those written by Ben Orlin, Alex Bellos, and Martin Gardner — on the living room table for them to discover.
- Watch YouTube videos — such as those from Numberphile, Matt Parker, and 3Blue1Brown — to see that math can be fun (and that math people can be funny).
- Talk about math and solve problems at the dinner table. One of our favorites is determining the number of clinks that happen after a toast, when everyone at the table offers “Cheers!” and taps glasses with everyone else.
- Use the Art of Problem Solving‘s MathCounts Trainer. Of note, my sons discovered this on their own and started using it because they enjoy solving problems, not because they were training.
- Complete a few MathCounts competitions from previous years. This is, in fact, the only part of the regimen that was actual training, and all students in our math club did two practice competitions prior to the local competition. The main purposes were to expose them to the types of questions on MathCounts competitions; to prepare them for the intensity of the competition (they are presented with 38 questions to be attempted in about 90 minutes); and, most importantly, to prepare them for the reality that they likely won’t get all of the questions correct, which, for most math club students, stands in stark contrast to their performance on the assessments they complete in their regular math class.

I offer this list to anyone who is coaching or interested in coaching a MathCounts team. The purpose of MathCounts is to get students excited about math; the stated mission is “to build confidence and improve attitudes about math and problem solving.” Winning may be fun, but it’s not the goal. To quote Boris Becker, “I love the winning. I can take the losing. But most of all, I love to play.” MathCounts provides students a chance to play with math, and most of them won’t win. Still, it’s an amazing opportunity to show kids how much fun math can be.

—

There are 16 ways to choose the four letters for a cryptocode. The codes in blue text (eight combinations in the middle columns) can each be arranged in 4! = 24 ways.

XWXY XWXZXWYZ XXYZ | KWXY KWXZ KWYZ KXYZ | MWXY MWXZ MWYZ MXYZ | ZWXY ZWXZ ZWYZ ZXYZ |

The codes in green text (six combinations in the first and last column) have a repeating letter (either X or Z), so they can be arranged in 4!/2! = 12 ways each.

Finally, the codes in **red bold text** (one combination in the first and last column) can also be arranged in 4! = 24 ways — but watch out! They’re the same sets, both consisting of W, X, Y, and Z. So only count that set once, not twice.

In total, then, there are 9(24) + 6(12) cryptocodes. Alex explained that this could be computed by rewriting it as 18(12) + 6(12) = 24 × 12, and one of the students from Access Academy rewrote it as 9(24) + 3(24) = 12 × 24. Both obviously reveal the answer, 288 cryptocodes.

I’m 99% certain that that’s the correct answer. And I’m 100% certain that it’s **two gross**.