## Words No Longer Used

I’ve been listening to the audiobook of At Home by Bill Bryson, and there’s a segment where he talks about words previously used to refer to the bathroom. My favorite is

necessarium

with its Latin meaning of “necessity,” implying that a room dedicated to urinating and defecating may not be something we really want in our house but very much need.

This made me wonder about mathematical words that are no longer in use. Many have gone the way of necessarium, but I think they deserve consideration for reintroduction. Well, maybe not all of them. Let’s have a look…

octothorpe, n. : another name for the pound sign (#); the hashtag. Wouldn’t it be great if #worldoctopusday were read as “octothorpe world octopus day”?

surd, n. : a square root that cannot be reduced further. This word comes from its meaning in phonetics of “mute” or “voiceless” for an unvoiced consonant; in math, it refers to an expression that cannot be expressed (spoken) as a rational number. The following radical would be ab‑surd:

$\sqrt{ab}$

vinculum, n. : a horizontal line drawn over a group of terms in a mathematical expression that serve as a grouping, such as the line on top of a radical that indicates the number for which the root is to be taken, or the fraction bar, which appears over the entire denominator. Still used occasionally, but rarely.

Logo of Vinculum, a global software company.

solidus, n. : the diagonal slash “/” used as the bar between numerator and denominator of an in-line fraction. Also, a famous Roman bodybuilder.
synonym diagonal

virgule, n. : a diagonal slash resembling the solidus, but with slightly less slant, used to denote division for in-line equations. This is also the name for the line used to indicate a choice between two terms in writing, e.g., and/or or pass/fail.

lattermath, n. : aftermath. Okay, not really a math term, but on the list since it contains “math.”

porism, n. : an archaic type of mathematical proposition whose historical purpose is not entirely known. It is used instead of “theorem” by some authors for a small number of results for historical reasons.

Jacob’s staff, n. : a mathematical instrument used for measuring heights and distances; typically, a pole with length markings on it.

anthyphairetic ratio, n. : a continued fraction, such as

$\displaystyle 1 + \frac{2}{3 + \frac{4}{5}}$.

Same number of syllables as parallelogram and inequality, but cooler than either of those. If you looked at anthyphairetic and thought, “that’s Greek to me,” you’d be entirely correct.

## One-Hundred Problems Involving the Number 100

Although the following joke appears in Math Jokes 4 Mathy Folks —

Why was the math book sad?
Because it had so many problems.

— I’ve often contended that it isn’t true. Math books aren’t sad because they have too many problems. They’re sad because they have too many exercises.

But my forthcoming book isn’t the least bit melancholy, because it contains a multitude of honest-to-goodness, classroom-tested, student-approved, 100% legit math problems — a century of them, in fact, as implied by the title.

Disclaimer: The title is a lie. The book actually contains 101 problems. I was so excited, I just couldn’t stop myself when I got to 100. But don’t you worry; there’s no charge for that extra 1%.

As a sample, here are four problems from the book. To experience a fifth problem, register for an NCTM Author Panel Talk on Wednesday, October 7, 7:00 p.m. ET, when Marian Small, Roger Day, and I will be discussing rich tasks and sharing samples from each of our new books. The webinar will be moderated by NCTM Board Member Beth Kobett. Hope to see you there!

## Grid with 100 Paths

Due to current circumstances, the National Council of Teachers of Mathematics (NCTM) had to cancel their Centennial Annual Meeting and Exposition, which was to be held April 1-4 in Chicago. As a replacement, though, they presented an amazing gift to the math education community — 100 free webinars led by selected speakers from the Chicago program. Dubbed 100 Days of Professional Learning, these webinars are to be held on select days from April through October.

As part of the 100 Days, I presented “100 Problems Involving the Number 100” on May 14. To celebrate NCTM’s 100th Anniversary, I collected or created 100 problems, each of which included the number 100. One of my favorites, Dog Days, looked like this:

At the end of the webinar, the conversation continued “backstage” with several members of NCTM staff, NCTM President Trena Wilkerson, and me. During that conversation, Trena made the outlandish suggestion,

Now we need a collection of 100 problems for which the answer is always 100.

I had just finished preparing a webinar with 100 problems, and now she was asking for another 100 problems. But never one to shy away from a challenge, I began to think about what kinds of problems might be included in such a collection. One type that came to mind was path-counting problems, like this one:

Moving only north or east on the segments in the diagram, how many distinct paths are possible from A to B?

That particular grid, measuring just 3 × 4, has fewer than 100 distinct paths from A to B. (How many paths, exactly? That’s left as an exercise for the reader.) What got me excited, though, was wondering if there were any grids that have exactly 100 paths — and hence providing 1% of the content for the collection that Trena requested.

As it turns out, there are no unmodified m × n grids that have 100 distinct paths. But what if some segments were removed? For instance, what if one of the middle vertical segments were discarded from a 4 × 6 grid, as shown below? How many distinct paths from A to B would there be?

As it turns out, a lot more than 100. (Again, finding the exact number is an exercise I’ll leave for you. If you need help, Richard Rusczyk from Art of Problem Solving has a video showing how to count paths on a grid.)

So, this is where I leave you:

Can you create a grid with some segments removed that will have exactly 100 distinct paths?

Have fun! Good luck!

As for the webinar, which lasted only 60 minutes, there wasn’t nearly enough time to cover 100 problems, but we had fun with 5 of them.

If you missed the webinar, you can hear the discussion about the Dog Days problem and 4 others, as well as get a PDF of all 100 problems, via the links below.

Enjoy!

## Coronavirus and Mathematical Modeling

In Oregon, Governor Kate Brown banned gatherings of more than 250 people. Similar restrictions have been imposed in other states, too, and the Center for Disease Control and Prevention (CDC) recommends that organizers cancel or postpone any event that consists of 50 or more people. Moreover, the CDC recommends that you “put distance between yourself and other people,” because social distancing is believed to inhibit the spread of coronavirus. The virus is thought to spread between “people who are in close contact with one another (within about 6 feet).”

All of this information leads, of course, to an incredible opportunity for students to engage in mathematical modeling.

What size space would be appropriate for a large gathering to ensure that all attendees could maintain an adequate distance from one another?

This is a variation of a classic packing problem, a mathematical optimization problem that involves packing objects (in this case, people) into containers (concert halls, restaurants, or some other social gathering spaces).

To create a reasonable model, some assumptions must be made. For instance, one assumption might be that each person is treated as the center of a circle with radius 3 feet, and circles are not allowed to overlap when packed into the container. Consequently, no two people will ever be within 6 feet of one another.

Statistician George Box noted, “All models are wrong, but some of them are useful.” It’s reasonable to assume that each person is surrounded by a protective cylinder, but how could these cylinders fit together? What about this model could be improved? What aspects of this model are appropriate for analysis but don’t quite work in the real world?

One configuration that could work is arranging the 250 people into 10 rows of 25 people each. With 3 feet above and below, to the right and to the left, of every person, that arrangement could fit into a rectangle that measures 30 feet × 150 feet, which has an area of 4,500 square feet.

Is a better arrangement possible?

A corollary problem, of course, is considering the maximum number of attendees that a particular space could handle to maintain social distancing. For instance, our local synagogue suggested that congregants not attend Friday services, but they will accommodate those who feel strongly about attending. Their website states, “All services will be held in the main sanctuary, and we will encourage any participants to sit at a distance from others.” How many congregants could be seated in the sanctuary and still maintain safe distance?

## Talking Math and Coronavirus With Your Kids #tmwyk

Nothing like a global pandemic to spark a good math conversation.

If you’re a parent from Alabama, Florida, Illinois, Kentucky, Louisiana, Maryland, Michigan, New Mexico, North Carolina, Ohio, Oregon, Pennsylvania, Rhode Island, South Dakota, Virginia, Washington, West Virginia, and Wisconsin — and by the time this post is published, probably many other states — then you’ve got several weeks of quality time with your kids ahead of you. You may be wondering what you can do to fill their time in meaningful and productive ways. Well, my recommendation is to talk math any time you’re with your kids, but while COVID-19 is in the news, that suggestion may be more important than ever.

It won’t be long before you tire of questions from your kids about why they have to spend the next two to four weeks at home, about why you won’t let them go to the mall, about why their friends can’t come over, about why they shouldn’t play tag or duck, duck, goose. But don’t get frustrated by their questions. That curiosity is an opportunity to talk about the math of the pandemic while reinforcing the reasons for staying home.

The spread of any disease is dependent on four factors:

• the population of opportunity;
• the number of days an infected person remains contagious;
• the number of people with whom an infected person comes in close contact; and,
• the likelihood of contraction when close contact occurs.

Simulations based on these four factors can be conducted with the NCTM Pandemics app (which, unfortunately, requires Flash). The page on which that app resides talks about swine flu, because the app was developed in 2006. But the lessons to be learned from the app are as relevant today — maybe even moreso — as they were 14 years ago.

You can explore on your own, or you can watch the screencast below to see how the spread of coronavirus can be controlled if we all do our part to limit close contact with others.

With your kids, research and discuss appropriate numbers for each factor.

• For display purposes, the app limits the “population of opportunity” to 400. This number falls significantly short of the nearly 8 billion people worldwide who might be infected with coronavirus, but it’s enough to make a point.
• The number of days an infected person remains contagious is unknown, but healthline says that “people who have the virus are most contagious when they’re showing symptoms” and the infection starts with mild symptoms that “gradually get worse over a few days.” It’s reasonable to estimate that an infected person might be contagious for three to five days.
• The number of contacts is the only factor over which we have control. If you go to work or a shopping center, you may have contact with 20 people a day; if your child goes to school, she may interact with 50 other students. But if you follow CDC guidelines, stay home from work or school, and avoid public gatherings, you can reduce the number of contacts to just a handful.
• Finally, the chance of contraction is unknown. What is known is that an infected person is likely to transmit COVID-19 to between 2.0 and 2.5 other people if some type of quarantine does not occur. The corresponding chance of contraction would be in the range of 2-4%.

To convince your kids that staying home is a good idea, run the simulation with a large number of contacts. Even if the number of days contagious and chance of contraction are low, most of the population will become infected if the number of contacts is high. But then reduce the number of contacts and run the simulation again. As the number of contacts decreases, so, too, will the percent of the population that gets infected as well as the number of days before the pandemic burns itself out.

Of note, most of the population will be infected if the days contagious and chance of contraction are both high, regardless of the number of contacts. For instance, if days contagious and chance of contraction are both set to 10, then more than 80% of the population will be infected in the vast majority of simulations, even if the number of contacts is set to 2. However, there are very few diseases for which a person remains contagious for 10 days and the chance of contraction is 10%; and, those numbers are certainly higher than the data would suggest for COVID-19.

## If Jack Handey Were a Math Guy

In our old neighborhood, we had the Heidelberg Bakery, which we loved for cupcakes, Bavarian pretzels, and challah. But I really wish it were named the Heisenberg Bakery instead, so that one of the employees could have said to me:

Sorry, I can tell you the status of your order, or I can tell you the location of your order — but not both!

I went to a geometry lecture last night on circles that was fascinating. But it lasted two hours longer than expected, because the speaker kept going off on a tangent.

Math is everywhere, even English class, where there are add‑verbs, add‑jectives, and conjunctions.

But math really is in English class; you can use proportions to find the past tense of flew:

Sure, they say that the moon is made of cheese, but I prefer to think that it’s made of crust and filling. Then it’d be π in the sky!

To get from point A to point B, a mathematician takes a rhom‑bus.

Math for the Office:
1/2 hour of productivity + 7 1/2 hours on the internet = 1 good day at work!

The Math of Diets:
2 cheeseburgers + 46 fries + 1 diet soda = 1 totally healthy meal!

Square box. Round pizza. Triangular slices. WTF?

Today’s Special: Buy one cheeseburger for the price of two, and receive a second cheeseburger absolutely free!

I’m worried about that man over there drawing on graph paper. I think he’s plotting something.

Why is 6 afraid of 7?
Because math is terrifying.

If I had a dozen strips of bacon, and you took four of them, what would you have?
That’s right. You’d have a black eye.

## This is a Blog Post

One of my favorite warm-ups to use in presentations is the following:

This sine has threee errors.

It’s a bit of a joke grenade… pull the pin, wait five seconds, eventually some folks will start to chuckle. In addition to inciting laughter, it also works well as a formative assessment.

One of my favorite books is by Demetri Martin:

One of my favorite jokes is from Steven Wright:

I went to a bookstore and asked the woman, “Where’s the self-help section?” She said that if she told me, it would defeat the purpose.

One of my favorite comics is from Randall Munroe:

One of my favorite experiences happened at a Chinese restaurant:

And one of my favorite puzzles is from Gödel, Escher, Bach:

There are __ 0s, __ 1s, __ 2s, __ 3s, __ 4s, __ 5s, __ 6s, __ 7s, __ 8s, and __ 9s in this sentence.

I love that this puzzle can be solved with iteration: put in some numbers, see how that affects things and adjust, see how that affects things and adjust, ad nauseam, until you either find a solution, or until you run into an infuriating cycle and have to start over with new seed values.  For instance,

 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 → 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 → 1, 1, 10, 1, 1, 1, 1, 1, 1, 1 → 2, 10, 1, 1, 1, 1, 1, 1, 1, 1 → 2, 9, 2, 1, 1, 1, 1, 1, 1, 1 → …

If you haven’t figured it out by now, my favorite things often include self‑reference. I speak in self-referential sentences when I go to job interviews…

At the end of my job interview, the interviewer asked, “Finally, what is the question you’d least like to be asked during this interview?” I replied, “That was it.”

And when visiting my therapist…

I’m trying to be less self-deprecating, but I really suck at it.

Perhaps the best self-referential (and self-deprecating) line in history comes from Groucho Marx:

I would never join a club that would have me as a member.

But there are no shortage of self-referential jokes in the world.

I never make predictions, and I never will. (Paul Gascoigne)

What would the value of 190 in hexadecimal be?

A student asked, “What is the best question to ask, and what is the best answer to that question?” The teacher responded, “The best question is the one you just asked, and the best answer is the one I just gave.”

I am the square root of -1. Who am i?

No! No! No! I am not in denial!

When you’re right 90% of the time, you needn’t worry about the other 5%.

The reciprocal of the square root of 2 is half of what number?

It’s bad luck to be superstitious.

Twenty-nine is a prime example of what kind of number?

Break every rule.

## More HIPE

Nearly five years ago, I wrote about HIPE, a parlor game in which one person gives a particular string of letters, and the other people in the parlor try to guess a word with that same string of letters (consecutively, and in the same order).

Well, I recently rediscovered Can You Solve My Problems? by Alex Bellos, and I was pleasantly surprised to find that he included four HIPEs in that book:

1. ONIG
2. HQ
3. RAOR
4. TANTAN

The fourth is one that I had included in my previous post, Don’t Believe the HIPE, and all are good enough that they deserve wide distribution.

Just for fun, here’s a new list of HIPEs that might prove interesting.

• SSP
• LWE
• NUSCU
• CUU
• CTW
• KGA
• UIU
• XII

In an effort to collect a bunch of excellent HIPEs, I’m asking for your help. If you play the game with friends and discover a particularly delectable combination of letters, please share below or at https://forms.gle/otddCw1uLeDALrMo7.

## Truth, Lies, and Math in Portland

The city that I now call home — Portland, OR — is the most beautiful city in the country. With views of Mt. St. Helens to the north, Mt. Adams to the northeast, Mt. Hood to the east, and powerful rivers through the middle of town, it’s hard to look in any direction without having your breath stolen.

As it turns out, Portland is also the smartest city in the country. This fact is irrefutable, per the following data.

Although the data suggests that Portland might only be the second-smartest city in the country — Portland lags slightly behind Boston in per capita sales of Math Jokes 4 Mathy Folks — Stumptown leapfrogs Beantown because not a single person in Portland deigns to root for the Patriots.

The exemplary intelligence of Portlandians is only one of the many things I’ve discovered since moving to the Pacific Northwest. I’ve also learned that Portland is a beer Mecca; that, despite its reputation, the weather in Portland is far from terrible and, in fact, quite to my liking; it has some cool parks; and, Portland has a lot of bikes and a lot of bridges.

Beer

As a beer lover, I was ecstatic to hear that Portland had the most craft breweries of any city in the world.

Unfortunately, that was an old statistic, and Portland, OR, currently ranks #8 nationally in terms of craft breweries per capita:

1. Portland, ME
2. Asheville, NC
3. Bend, OR
4. Boulder, CO
5. Kalamazoo, MI
6. Vista, CA
7. Greenville, SC
8. Portland, OR
9. Pensacola, FL
10. Missoula, MT

The “other Portland” garners the top spot on the list. But it seems to me that if the list were culled to show only those cities where people actually want to live, the real Portland would again be near the top. (Asheville and Boulder absolutely give Portland a run for their money. But Kalamazoo and Vista? C’mon, now!)

Weather

Portland is known for gray skies and rain. But compare Portland to my previous hometown, Washington, DC.

The graphs below show that DC is warmer and wetter in the summer, but colder and drier in the winter.

But let’s dig into those numbers a little.

The average temperature in the two cities is remarkably similar, with Portland averaging 54.5°F and Washington, DC, averaging 55.7°F. But the hottest days are hotter in DC, and the coldest days are colder in DC. The temperate oceanic climate in Portland explains the cooler summers, the warmer winters, and the incredibly high number of homeless people.

Admittedly, Portland has more days of rain than Washington, DC — 156 to 115, in fact — but it receives a significantly smaller amount of rainfall — 36.0″ to 40.8″, a difference of nearly five inches.

Portland trails in hours of sunshine by roughly 10%, with 2,341 hours compared to DC’s 2,528. But Portland also has fewer days of snow per year, just 2.2 to DC’s 8.0, and much less accumulation — 3.0″ in Portland to a whopping 14.5″ in the nation’s capital.*

But rain, snow, sun, and temperature aside, there may be one statistic that is more important than all the others: Washington, DC, has significantly more days of Donald Trump, averaging over 300 per year since 2016; but since becoming President, Trump has spent nary a minute in Oregon.

Parks

Portland boasts Mill Ends Park, which holds the Guiness World Record for smallest park on the planet.

With a diameter of just 24″, the total area of Mill Ends Park is exactly π square feet, or approximately 0.000 072 acres.

Bikes

Portland has 94 miles of neighborhood greenways, 162 miles of bike lanes, and 85 miles of bike paths. That’s 341 biker‑friendly miles, which explains why more than 22,000 people ride their bikes to work every day. Over six percent of Portland’s commuters bike to work, which is twelve times the national average.

The joke in Portland is that, when you step off an airplane at PDX, they hand you a rain jacket and a dog. But if they really want folks to fit in, they better start doling out bikes, too.

Bridges

The Willamette (pronounced wuh-LAM-it, not WILL-uh-met) River separates the east and west sides of Portland, and it’s spanned by twelve bridges. When the Hawthorne Bridge was built in 1910, it was one of the first vertical-lift bridges anywhere in the country; now, it’s the last one still in operation. The Tilikum Crossing Bridge was the country’s first ever multi-modal bridge that accommodated light rail, streetcar, buses, and pedestrians — but not private automobiles. And the St. John’s Bridge, known for its 400-feet high, twin Gothic-style arches, previously held the records for the world’s longest pre-stressed twisted rope cables as well as the tallest reinforced concrete pier in the world.

Every morning as I cross the Sellwood Bridge, I look north to the smartest, drunkest, rainiest, most beautiful city in the country, and there’s no place I’d rather be.

* Every Portland resident who has relocated from some other part of the country will make a similar comparison between the weather in Portland and the weather in the city where they used to live. This is nothing more than rationalizing the decision to move to a city that only gets 144 days of sunshine a year.**

** Every Portland resident will also tell folks in other cities how bad the weather is, in an attempt to discourage others from moving to this amazing city. In short, they don’t want you here. I suspect, in fact, that they didn’t (and maybe still don’t) want me here. But too late, I’m staying. You, on the other hand, shouldn’t even think of coming here. I promise, you’ll hate it.

## 20 Math Problems for 2020

Happy New Year!

• It’s a leap year — yay!
• It’s the year of the rat — squeak!
• It’s an election year — okay, maybe it’s not that great of a year after all.

To get your mind thinking about something other than the associated realities of that last bullet, here are 20 problems to prepare you for the next 366 days.

1. What’s the difference between the number of positive integer factors of 2020 and the number of positive integer factors of 2020?
2. If all of the numbers from 1 to 2,020 were written down, how many digits would be used?
3. If all of the numbers from 1 to 2,020 were spelled out, how many letters would be used?
4. The numbers 1 through 2,020 are written on a whiteboard. At every stage, two numbers (say, a and b) are erased from the whiteboard and replaced with the sum a + b + 1. For example, if 187 and 2,013 were erased, then 187 + 2,013 + 1 = 2,201 would be written on the whiteboard. This process is repeated until a single number remains on the whiteboard. What is the number?
5. How many positive integers are less than the square root of 2,020?
6. How many different rectangles with integer side lengths and an area of 2,020 square units are possible?
7. A lighthouse is perched on the cliff of a rocky beach. Standing in the lantern room of the lighthouse, you are approximately 2,020 feet above the surface of the ocean below. How far can you see to the horizon?
8. A rectangular prism with integer edge lengths has a volume of 2,020 cubic centimeters. What is the maximum possible surface area of this solid?
9. Mike made two initial bank deposits on Monday and Tuesday. Then every day for the rest of the week, he deposited an amount equal to the sum of the previous two days’ deposits. If he deposited $2,020 on Saturday, and his deposit on Monday was less than$10, what was the amounts of his deposit on Tuesday?
10. What’s the sum of 101 + 202 + 303 + … + 2,020?
11. How many postive, four-digit numbers can be formed with the digits 2, 0, 2, and 0?
12. Saying that someone has 20/20 vision means they have normal visual acuity. That is, if you have 20/20 vision, then from 20 feet away, you can see what a normal person would see clearly from 20 feet away. In general, if you have 20/n vision, then from 20 feet away, you can see what normal people would see from n feet away. A person with 20/5 vision is looking at a billboard that is a mile away, and she can clearly see the letters on the billboard. How much closer would a person with 20/80 vision need to stand to clearly see the letters on the billboard? (Assume both people are standing at the boundary of where they can read the sign clearly.)
13. A triangle has two sides of length 20. What is the maximum possible area of this triangle?
14. Using only the digit 2 and the addition symbol, you can create expressions with many different values. For instance, you could use three 2’s and one addition symbol to make 22 + 2 = 24, or you could use  eight 2’s and three additional symbols to make 222 + 22 + 22 + 2 = 268. Using this process, what is the minimum number of 2’s needed to make an expression with a value of 2,020?
15. In how many zeroes does the number 2020 end?
16. Find the smallest possible string of consecutive positive integers that have a sum of 2,020.
17. One integer is removed from the set {1, 2, 3, …, n} so that the sum of the remaining numbers is 2,020. What integer was removed?
18. What is the product of the prime divisors of 2,020?
19. What is the maximum possible product for a set of positive integers that have a sum of 2,020?
20. Which of the following chains — each consisting of regular polygons with side length 1 unit — could be extended to have a perimeter of exactly 2,020 units?

1. 849
2. 6,973 digits
3. 47,123 letters
4. 2,043,229
5. 44 integers
6. 6 rectangles
7. approximately 52.5 miles
8. 8,082 square centimeters
9. $401 on Tuesday (and$5 on Monday)
10. 21,210
11. 3 numbers
12. the person would need to be 1/16 mile = 330 feet from the billboard, which is 4,950 feet closer
13. 200 square units
14. 29 2’s
15. 20 zeroes
16. 402, 403, 404, 405, 406
17. 60
18. 1,010
19. 3672 × 22
20. only the squares; for n squares, the perimeter is 2n + 2, and n = 1,009 yields a perimeter of 2 × 1,009 + 2 = 2,020

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

## MJ4MF (offline version)

Math Jokes 4 Mathy Folks is available from Amazon, Borders, Barnes & Noble, NCTM, Robert D. Reed Publishers, and other purveyors of exceptional literature.