## Guess the Graph on Square Root Day

Today is **Opening Day** in Major League Baseball, and 13 games will be played today.

It’s also **Square Root Day**, because the date 4/4/16 transforms to 4 × 4 = 16.

With those two things in mind, here’s a trivia question that seems appropriate. **Identify the data set used to create the graph below.** I’ll give you some hints:

- The data set contains 4,906 elements.
- It’s based on a real-world phenomenon from 2015.
- 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?

- The horizontal axis represents “Distance (Feet)” and the vertical axis represents “Frequency.”

Still not sure? Final hints:

- Point A on the graph represents Ruben Tejada’s 231-foot inside-the-park home run on September 2, 2015.
- Point B on the graph represents the shortest distance to the wall in any Major League Baseball park — a mere 302 feet to the right field fence at Boston’s Fenway Park.
- Point C represents the longest distance to a Major League wall — a preposterous 436 feet to the deepest part of center field at Minute Maid Park in Houston.

Okay, you’ve probably guessed by now that the data underlying the graph is **the distance of all home runs hit in Major League Baseball during the 2015 season**. That’s right, there were 4,906 home runs last year, of which 11 were the inside-the-park variety. The distances ranged from 231 to 484 feet, with the average stretching the tape to 398 feet, and the most common distance being 412 feet (86 HRs traveled that far). The data set includes 105 outliers (based on the 1.5 × IQR convention), which explains why a box plot of the data looks so freaky:

The variety of shapes and sizes of MLB parks helps to explain the data. Like all math folks, I love a good graphic, and this one from Louis J. Spirito at the Thirty81Project.com is both awesome and enlightening:

Here are some more baseball-related trivia you can use to impress your friends at a cocktail party or math department mixer.

**Who holds the record for most inside-the-park home runs in MLB history?**

*Jesse Burkett, 55 (which is 20 more than he hit outside-the-park)*

**Which stadium has the tallest wall?**

*The left field fence at Fenway Park (a.k.a., the “Green Monster”) is 37 feet tall.***Which stadium has the shortest wall?**

*This honor also belongs to Fenway Park, whose right field wall is only 3 feet tall.***Although only 1 in 446 home runs was an inside-the-park home run in 2015, throughout all of MLB history, inside-the-park home runs have represented 1 in ____ home runs.**

158**Name all the ways to get on first base without getting a hit.**

*This is a topic of much debate, and conversations about it have taken me and my friends at the local pub well into the wee hours of the morning. I have variously heard that there are 8, 9, 11, and 23 different ways to get on base without getting a hit. I think there are 8; below is my list.*

*(1) Walk*

*(2) Hit by Pitch*

*(3) Error*

*(4) Fielder’s Choice*

*(5) Interference*

*(6) Obstruction*

*(7) Uncaught Third Strike*

*(8) Pinch Runner*

**What is the fewest games a team can win and still make the playoffs?**

*39. The five teams in a division play 19 games against each of the other four teams in their division. Assume that each of those teams lose all of the 86 games against teams***not**in their division. Then they could finish with 39, 38, 38, 38, and 37 wins, respectively, and the team with 39 wins would make the playoffs by winning the division.**Bases loaded in the bottom of the ninth of a scoreless game, and the batter hits a triple. What’s the final score of the game?**

*1-0. By rule, the game ends when the first player touches home plate.***In a 9-inning game, the visiting team scores 1 run per inning, and the home team scores 2 runs per inning. What is the final score?**

*16-9. The home team would not bat in the bottom of the ninth, since they were leading.*

## Mathegories

In case you missed it, the following mathy challenge was presented by Will Shortz as the NPR Sunday Puzzle on February 28:

Find two eight-letter terms from math that are anagrams of one another. One is a term from geometry; the other is from calculus. What are the two words?

The irony of this puzzle (for me) appearing on that particular Sunday is that five days later, I delivered the keynote presentation for the Virginia Council of Teachers of Mathematics annual conference, and I had included the answer to this puzzle as one of my slides. I wasn’t trying to present an answer to the NPR Puzzle; I was merely showing the two words as an example of an anagram. The following week, Will Shortz presented the answer as part of the NPR Sunday Puzzle on March 6.

SPOILER:

**Slide from My Presentation**

(aka, the answer)

What I particularly enjoyed about the March 6 segment was the on-air puzzle presented by Shortz. He’d give a category, and you’d then have to name something in the category starting with each of the letters W, I, N, D, and S.

I’ve always heard that **good teachers borrow, great teachers steal**. So I am going to blatantly pilfer Shortz’s idea, then give it a mathy twist.

I’ll give you a series of categories. For each one, name something in that category starting with each of the letters of M, A, T, and H. For instance, if the category were *State Capitals*, then you might answer Madison, Atlanta, Topeka, and Harrisburg. Any answer that works is fine. But for many of the categories, you’ll earn bonus points for mathy variations. For instance, if the bonus rule were “+1 for each state capital that has the same number of letters as its state,” then you’d get two points for Atlanta (Georgia) and Topeka (Kansas), but only one point for Madison (Wisconsin) and Harrisburg (Pennsylvania).

There are nine categories listed below, and the maximum possible score if all bonuses were earned would be 79 points. I’ve listed my best answers at the bottom of this post, which yielded a score of 62 points. **Can you beat it?** Post your score in the comments.

Want to play this game with friends or students.Download the PDF version. |

**Movie Titles**

(+1 for a math movie)

M _____

A _____

T _____

H _____

**Historical Figures**

(+1 if the person is a mathematician or scientist)

M _____

A _____

T _____

H _____

**Games**

(+1 if the game is mathematical)

M _____

A _____

T _____

H _____

**School Subjects**

(+1 for mathematical subjects)**
**M _____

A _____

T _____

H _____

**Words with One-Word Anagrams
**(+1 if it’s a math term)

**M _____**

A _____

T _____

H _____

**Words Containing the Letter “Q”
**(+1 if it’s a math term)

**M _____**

A _____

T _____

H _____

**Math Terms
** (+3 if all four terms are related, loosely defined as “could be found in the same chapter of a math book”)

**M _____**

A _____

T _____

H _____

**Words Containing the Letters M, A, T, and H**(+1 if the letters appear in order, though not necessarily consecutively; +2 if consecutive)

**M _____**

A _____

T _____

H _____

**Words with a Single-Digit Number Word Inside Them
**(such as

*asi*, but -1 if the number word is actually used numerically, such as

**nine***; +2 if the single-digit number is split across two or more syllables)*

**four**thsM _____

A _____

T _____

H _____

*The following are my answers for each category.*

**Movie Titles
**

*Moebius*,

*Antonia’s Line*,

*Travelling Salesman*,

*(A) Hill on the Dark Side of the Moon*

(8 points)

**Historical Figures
**Mandelbrot, Archimedes, Turing, Hypatia

(4 points)

**Games**

Mancala, Angels and Devils, Tic-Tac-Toe, Hex

(8 points)

**School Subjects**

Mathematics, Algebra, Trigonometry, History

(7 points)

**Words with One-Word Anagrams
**mode (dome), angle (glean), triangle (integral), heptagon (pathogen)

(8 points)

**Words Containing the Letter “Q”
**manque, aliquot, triquetrous, harlequin

(6 points)

**Math Terms**

median, altitude, triangle, hypotenuse

(7 points)

**Words Containing the Letters M, A, T, and H**

ma

**t**c

**h**, aro

**math**erapy,

**th**e

**ma**tic, ho

**m**eop

**ath**ic

**(8 points)**

**Words with a Single-Digit Number Word Inside Them
**mezza

**nine**, ar

**two**rk, t

**one**, h

**eight**

(6 points)

## How Dumb Are You?

I recently purchased the book *How Smart Are You? Test Your IQ *for the same reason that I always purchase books like this — often, there are one or two gems buried amid a pile of mundane, mind-numbing questions.

Having just finished the last quiz, here’s all you need to know about this book:

- I found it on the discount table at Barnes and Noble.
- There is a picture of a wise, all-knowing owl on the front cover. (Ooh, an owl! I feel smart already!)
- The tag line on the cover reads, “Calculate Your IQ in Minutes,” yet the Introduction states, “Your scores will not reflect your actual intelligence.”

When it comes to measuring your IQ using this book, the following scale will be more effective than anything you’ll find between the covers:

Did You Buy this Book? |
IQ Score |

Bought | < 75 |

Didn’t | > 125 |

The book contains 50 quizzes with 10 questions each. Each question is worth 16.5 points, so your IQ is found by multiplying the number correct on a given quiz by 16.5.

**I hate to deliver the bad news. **

**The results of your IQ test have come back negative.**

Sadly, there were no gems among the 500 questions in the book. (Honestly, I found it more difficult to calculate my score than to answer most of the questions.) Yet there were quite a few duds. And that’s where we’ll start today’s story.

One question asked the reader to identify the next number in the series:

5, 13, 21, 29, 37, 44, …

You may notice that 5 **+ 8** = 13, 13 **+ 8** = 21, and 21 **+ 8** = 29, so you might think that the rule is “add 8.” But 37 **+ 8** ≠ 44, so the pattern fails. You don’t even need to check the addition, though; since the first term is odd and the common difference is even, all terms must be odd. The number 44 should have stuck out like a sore thumb to any editor worth his salt. Yet that did not stop the author from listing 44 + 8 = 52 as the correct answer.

Similarly, another problem asked:

A high school has 40 students in its senior class. Forty percent of the seniors are taking physics, 30 percent are taking chemistry, and

10 percent are taking neither. How many seniors are takingneitherphysics or chemistry? (Ed. note: emphasis added.)

You might first think that 4 students are taking neither physics nor chemistry (nevermind that the problem used *or* instead of *nor*), since the problem says that 10% are taking neither, and 10% of 40 is 4. Upon seeing the correct answer listed as 16 students, you might then think, “What the f**k?” And that would be a justifiable reaction. I suspect that this was meant to be one of those questions where the numbers in the three groups adds to more than 100%, so the overlap becomes important, but this problem is an epic fail as presented.

**Some people should have to pass an IQ test
to drive or reproduce. Fail the test,
you get birth control and a bus pass. **

A little later, on a quiz titled “Unscramble the Letters I,” readers were directed to unscramble the letters

delif

to create an English word or name.

The Internet Anagram Server says that there are three: *field*, *filed*, *flied*. Finding one of them without the Internet seems like a reasonable challenge. But within the book, the problem is presented as a **multiple-choice question**:Oh, my. Anyone smart enough to read a book would see immediately that *fled* doesn’t have enough letters, *flies* has an *s* instead of the requisite *d*, and *delight* has too many letters. How many people have been misled by this quiz, scoring a 165 and then thinking that they were Harvard material?

My favorite in this section, though, was the scrambled-letter collection

lydarceptt

which I immediately recognized to be *pterodactyl*, but then thought, “No, wait, there’s no *o*.” Yet *pterodactyl* was the only reasonable option among the four answer choices (*Pericles*, *lethargic*, *pterodactyl*, and *pictogram*), so I ignored the omission and collected another perfect score of 165. (Yay, me!)

As I said above, there were no gems, but I’ll end with the only problem in the entire book that I even mildly enjoyed:

A car traveled 281 miles in 4 hours and 41 minutes. What was the car’s average speed in miles per hour?

This one was also presented as a multiple-choice question, but it’s more fun to solve without the options. Have at it.

## Angle of Opportunity

My wife and I noticed that one of our sons has been getting his pants wet while urinating. He’s 8; these things happen. But when it occurred twice on consecutive days, we had reason for concern. When we inquired, he explained, “Sometimes when I start to pee, I hit the back of the seat. So I push my penis down, but then I hit the front of the toilet, and the pee ricochets and gets my pants wet.”

My wife began to pursue a line of investigative questioning, but I stopped her. “This is just simple geometry,” I explained.

I could have predicted my wife’s reaction. She said:

Not

everythinghas to be a math problem.Especiallythis.

Even if that were true (it’s not), this situation still begs for some trigonometric analysis.

I’m just over 6 feet tall, so my fire hose is approximately 20″ above the toilet when I urinate. As shown in Figure A, when I stand a reasonable horizontal distance from the commode, my *angle of opportunity* is approximately 30°.

My son, on the other hand, barely clears 4 feet. His water gun is less than 6″ above the toilet when he urinates, so his angle of opportunity is a mere 20°, as illustrated in Figure B.

The images clearly indicate why mothers tell their sons (and husbands), “Stand closer to the toilet when you go!” Doing so increases the angle of opportunity and thus decreases the likelihood of a “clean-up in Aisle 3.”

But more importantly, the above images and some quick trig calculations show that an adult male — who probably has greater control than a young boy, anyway — also has a 50% greater range through which to aim when making a deposit.

Upon completing my explanation, I turned to my son. “Though it may be harder for you to hit the mark, that doesn’t excuse peeing on your pants. I think you need to be more careful.”

I then addressed my wife. “I also think we need to cut him a little slack on this one.”

“And *I* think,” she said, “that *you* are absolutely unbelievable.”

With that, she excused herself.

I’m not sure where she went, but I suspect it was to text one of her friends about how lucky she was: not only is her husband good at math, but he can apply it in extremely esoteric situations.

—

Rather remarkably, there has actually been serious scientific investigation into this phenomenon:

More importantly, there are a number of jokes at the intersection of math and urination:

Why do statisticians choose the last urinal?

Because there’s only a 50% change of being splashed by someone else.What’s in the toilet of the math department restroom?

A natural log.What does a mathematician call a toilet seat?

An ass-toroid.

## Mathematically Unconscious

Both of my sons sleepwalk. At least once a week, one of them will wake up an hour after bedtime, walk down the stairs, and start speaking gibberish. They have no idea what they’re saying, because they aren’t awake — even though their eyes are open. (Freaky!)

During a recent somnambulation, Alex stood at the top of the stairs. He appeared frustrated. Finally, he said:

I just need to find the numbers. It shouldn’t take long.

As you might well imagine, it’s a little scary to have your son walking and talking while asleep. The only solace is that his subconscious thoughts are about math.

I don’t sleepwalk. But I recently had a dream in which I attended a cocktail party and asked the other attendees a most unusual question:

I suspect that my 7 years as an editor and 4 years as a question writer for MathCounts are to blame, but that doesn’t make it any easier to accept.

I vividly remember a dream I had in college, on the night prior to my Linear Algebra midterm. Feeling unprepared for the exam, my nightmare consisted of two brackets pinching my head like a vice, while numbers floated past.

I awoke in a cold sweat at 5 a.m., and proceeded to a study carrel for more test prep.

I was happy to learn that other folks dream about math, too. While subscribed to a listserve for former instructors of the Center for Talented Youth, I received a message from Mark Jason Dominus that read, “I dreamt of the following problem while I was sleeping last night. When I woke up, I convinced myself that it was a good problem, so I’ve decided to share it.”

The volume of a 3 × 3 × 3 cube is 27 cubic units, and the volume of a 2 × 2 × 1 rectangular prism is 4 cubic units. Theoretically, six prisms should be able to fit inside the cube, with three cubic units empty. But can you arrange six 2 × 2 × 1 prisms so they fit inside a 3 × 3 × 3 cube?

Good luck, and sweet dreams!

## 4 Folds, 40 Theorems, and Chinese New Year

If you asked a mathematician how to make an origami coin purse for Chinese New Year, you might receive the following instructions:

- Rotate a square piece of paper 45°.
- Reflect an isosceles right triangle about a diagonal of the square so that congruent isosceles right triangles coincide.
- Reflect the lower left vertex of the isosceles right triangle so it coincides with the opposite leg, such that the base of the isosceles right triangle and one side of the newly formed isosceles acute triangle are parallel.
- Reflect the other vertex of the isosceles right triangle to coincide with a vertex of the isosceles triangle formed in the previous step.
- Fold down a smaller isosceles right triangle from the top.
- Fill the coin purse with various denominations of currency, such as a $2 bill, a $1 gold coin, a quarter, or a penny.

Don’t like those directions? Perhaps these pictures will help.

To be truly authentic for Chinese New Year, you’ll want to use red paper instead of white.

The Common Core State Standards for Mathematics (CCSSM) have added rigor to geometry by requiring students to prove no fewer than a dozen different theorems. Further, the geometry progressions document formulated by Dr. Hung-Hsi Wu suggests that a high school geometry course should use a transformational approach, basing a majority of the foundational axioms on rotations, translations, and reflections.

What struck me about the origami coin purse was the myriad geometry concepts that could be investigated during this simple paper-folding activity. Certainly, the idea to use origami in geometry class in nothing new, but this one seems to have a particularly high return on investment. Creating the coin purse requires only **four folds**, and though perhaps I lied by claiming **forty theorems** in the title, the number of axioms, definitions, and theorems that can be explored is fairly high. (Claiming “forty theorems” wasn’t so much untruthful as much as it was archaic. The word *forty* has been used to represent a large but unknown quantity, especially in religious stories or folk tales, as in *Ali Baba and the 40 Thieves*, it rained “40 days and 40 nights,” or Moses and the Israelites wandered the desert for 40 years.)

For instance, the fold in Step 3 requires parallel lines, and the resulting figures can be used to demonstrate and prove that parallel lines cut by a transversal result in alternate interior angles that are congruent.

No fewer than four similar isosceles right triangles are formed, and many congruent isosceles right triangles result.

The folded triangle in the image above is an isosceles acute triangle; prove that it’s isosceles. And if you unfold it from its current location, the triangle’s previous location and the unfolded triangle together form a parallelogram; prove it.

The final configuration contains a pair of congruent right triangles; you can use side-angle-side to prove that they’re congruent.

But maybe you’re not into proving things. That’s all right. The following are all of the geometric terms that can be used when describing some facet of this magnificent creation.

- Parallel
- Congruent
- Similar
- Alternate
- Angle
- Square
- Reflection
- Rotation
- Translation
- Transversal
- Bisect
- Coincide
- Right Angle
- Perpendicular
- Isosceles
- Vertex
- Triangle
- Trapezoid
- Parallelogram
- Pentagon

And the following theorems and axioms can be proven or exemplified with this activity.

- Sum of angles of a triangle is 180°.
- Alternate interior angles formed by a transversal are congruent.
- Any quadrilateral with four congruent sides is a parallelogram.
- Congruent angles are formed by an angle bisector.
- Triangles with two sides and the included angle congruent are congruent.

There is certainly more to be found. What do you see?

## Snowzilla Math for Back to School

I don’t know where you live, or how much snow you’ve gotten, or whether your kids have been out of school for multiple days. But here in Falls Church, VA, it’s Thursday, January 28 — five full days after snow stopped falling from Winter Storm Jonas — and our schools are **still** closed.

My sons lounge around in their pajamas all day, only getting off the couch to interrupt my work-from-home day and ask for macaroni and cheese. It’s starting to feel like we’ve had two eight-year-old brothers-in-law take up residence.

That’s why I’ve used data from Snowzilla to create a series of math activities. Today’s assignment is for them to complete the following and not bother me till they’re done. Feel free to use any of these with the youngsters in your life, whether they’re your biological offspring from whom you need a break at home or your charges in a classroom who might enjoy the challenge.

What do geometry teachers do in a blizzard?

Make snow angles.Schools have had a record number of snow days. At this rate, the only math kids are doing is how many glasses of wine their mom drinks before 2 p.m. – Jimmy Fallon

—

**1.** Chris Christopher, a macroeconomist at IHS Global Insight, estimated that Jonas’s economic impact would be somewhere between $500 million and $1 billion.

**a.** Write both of those numbers in the form 2^{m} × 5^{n}.

**b.** If all values within the range are equally likely, what is the probability that the impact will be greater than $800 million?

**c.** If all values within the range are normally distributed, what is the probability that the impact will be greater than $800 million?

**Bonus.** Can you think of another occupation where it’s appropriate to state a prediction in which the upper end of the range is double the lower end?

—

**2.** In the Washington, DC, area, the average weekly sales in a typical supermarket is about $10 per square foot. In the two days leading up to Jonas, traffic to brick-and-mortar stores was 7.5 percent higher than usual. The graph below, based on national averages, shows the percent of weekly shoppers at grocery stores each day of the week.

Putting all this information together, as well any other data that you can find online, draw a graph that approximates sales at a typical grocery store in Washington, DC, for the month of January.

—

**3.** Virginia Governor Terry McAuliffe estimated that snow removal costs the commonwealth $2 million to $3 million per hour. Estimate the total cost for Virginia to clean up Jonas’s mess.

—

**4.** According to City Comptroller Scott Stringer, the cost of snow removal in New York City is approximately $1.8 million per inch. Estimate the total cost for New York City to remove the snow from Jonas.

—

**5.** The estimate above is an average for 2003 to 2014. The two graphs below show the snowfall totals and snow removal costs for those 12 years. Which years had the highest and lowest cost-per-inch? (Click each image to enlarge.)

—