## Covering 100 Squares

There’s an old math joke that says math books are sad because they have too many problems. But I disagree; I believe that most math books — and, in particular, textbooks — are sad because they have too many *exercises*.

To try to release more true problems into the wild, I recently wrote a book called *One-Hundred Problems Involving the Number 100*. Published by NCTM, the book contains problems, suggestions for classroom use, and solutions. Some of the problems are old chestnuts, such as, “What is 1 + 2 + 3 + ⋯ + 100?” (Problem 21: Gauss and Check). Others are complete originals, and occasionally a little silly, such as, “If all the positive integers from 1 to 100 were spelled out, how many letters would be used?” (Problem 1: Spell It Out). But my favorite problem in the book, which I’ve shared twice during NCTM’s 100 Days of Professional Development (**May 14**, **Oct 7**; login required), is Problem 100: Covering with Squares.

**As shown below, a square grid with 100 smaller squares can be covered by 100 squares (each measuring 1 × 1), by 25 squares (each measuring 2 × 2), or by 13 squares (one 6 × 6, two 4 × 4, two 3 × 3, two 2 × 2, and six 1 × 1).**

**Find all values of 0 < n < 100 for which it is impossible to cover a 10 × 10 grid with n squares of integer side length.**

I’ll admit, I’m not ecstatic about the phrasing of the question at the end. The symbolic representation makes it succinct, sure, but this problem can be investigated by students who may be too young to understand that notation. It might be better to just ask, “Can you cover it with two squares? Can you cover it with 58 squares? For how many different numbers of squares can you find a covering?”

You can, of course, explore this problem using graph paper. But for a digital experience, click on the link below and explore using Google Slides:

When you click on that link, you’ll be required to “Make a copy” for your own Google Drive. This is done for two reasons: first, it’ll prohibit any editing to the original file, so that other folks who use that link will receive a similarly pristine copy; and second, it places a copy in your Google Drive that you can play with, modify, or share with your students. If you do that, be sure to change **/edit** and anything that follows at the end of the URL to **/copy**, which will require anyone you share it with to make a copy, too. Of course, you could also just share the link above with your students, and they can have a copy of the original file.

When using this problem in the classroom, start by showing a few examples — such as the ones above — and then have all students find a covering for the grid that’s different from your examples. Because every student can find at least one covering, this breeds confidence. In addition, it ensures that all students understand the problem. When all students have found a covering, have them share it with a partner. From then on, allow students to work in pairs or small groups to find other coverings.

To facilitate the discussion, I draw a blank hundreds grid on the whiteboard. As unique coverings are found, students are allowed to enter the number in the hundreds grid. Of course, I require them to first verify with a partner that they have counted the number of squares correctly; once confirmed, they can show the covering to me; and if that number hasn’t been entered in the hundreds grid yet, they get to enter it. This can be motivating for students, and it’s a good opportunity to get less participatory students engaged.

When using this problem recently with a group of students, this is how the hundreds grid looked after about 20 minutes:

You’ll notice that numbers appear in different colors; each group received a different color dry erase marker. What you might also notice is that the numbers 17, 20, 23, 26, …, 65 all appear in red. This seemed more than coincidental, so I asked about it. The group members explained:

*We realized we could cover the grid with a one 6 × 6 square and sixteen 2 × 2 squares:*

*We then divided one of the 2 × 2 squares into four 1 × 1 squares. That meant that one 2 × 2 square was replaced by four 1 × 1 squares, increasing the number of squares by three. So, the grid was now covered with 17 + 3 = 20 squares. *

*If we kept doing that — if we kept dividing the 2 × 2 squares — then we’d keep adding three more squares, so we could get 23, 26, 29, and so on, all the way up to 65.*

The realization that one configuration could be transformed into many others allowed students to find coverings for myriad numbers. For what values of *n* could the grid not be covered? That’s left as a question for you. Have fun!

## When a Half Is More Than a Half (and When It Ain’t)

Tonight, the dreadful Philadelphia Eagles defeated the pathetic New York Giants 22‑21 in a match-up of horrendous one-win teams. But not all one-win teams are created equal: in late September, the Eagles played the Bengals to a 23‑23 tie in a game that might have featured the all-time worst ending ever. As a result, the Eagles entered tonight’s game with a horrid 1‑4‑1 record, but not to be outdone, the Giants entered the game with a slightly more putrid 1‑5 record.

In football, a tie counts as a half-win (and a half-loss). But half-wins are sometimes worth more than half a win, sometimes they’re worth less than half a win, and sometimes they’re worth exactly half a win. Let me ‘splain.

After their win tonight, the Eagles record is 2‑4‑1. For the time being, that puts them atop the lowly NFC East:

Eagles | 2-4-1 |

Cowboys | 2-4 |

Washington | 1-5 |

Giants | 1-6 |

Philadelphia has played 7 games and won 2 1/2 of them. That is, they’ve won

of their games. That puts them ahead of Dallas, who has won

of their games. So, the Eagles are currently in first place by 1/42 of a game.

But let’s say the Eagles had entered tonight with a 3‑2‑1 record and the Cowboys were 4‑2. After tonight’s win, the Eagles would be 4-2-1, and they would’ve won

of their games. The Cowboys, on the other hand, would have won

of their games, and the Cowboys would be leading the division by 1/42 of a game.

So that half-win tie? It’s worth more to the Eagles because they’re terrible. Were they at least mediocre, that tie wouldn’t be as valuable.

On the other hand, if the Eagles had entered tonight with a 2‑3‑1 record and the Cowboys were 3‑3, then the Eagles would have been 3-3-1 after tonight’s win, and they would’ve won

of their games. Similarly, the Cowboys would have won

of their games, and the teams would’ve been tied for first in the pitiful, talentless, miserable NFC East.

(Yes, I’m being hard on the NFC East, but it isn’t unwarranted. The average power ranking of the four teams is 28, when the lowest possible is 30.5. The four starting quarterbacks have thrown nearly as many interceptions as touchdowns (24 TDs, 22 Ints), and the four teams’ top running backs have more fumbles than touchdowns (11 TDs, 12 Fum). Seriously, this division may be all-time bad.)

All that said, it’s highly unlikely that the season will end with the Eagles having played more games than the Cowboys. Then again, with COVID‑19, who knows what might happen?

It’s often been said that football is “a game of inches.” But given the importance of half-wins, isn’t it time we started saying that football is “a game of fractions”?

## 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:

**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.

**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

.

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?

Finally, I’ll leave you with the best advice I’ve ever received:

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:

- ONIG
- HQ
- RAOR
- 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.