## Problems at the 2016 MathCounts National Competition

Yesterday, **Edward Wan** (WA) became the 2016 MathCounts National Champion. He defeated Luke Robitaille (TX) in the finals of the Countdown Round, 4-3. In the Countdown Round, questions are presented one at a time, and the first student to answer four correctly claims the title.

“This one is officially a nail-biter,” declared Lou DiGioia, MathCounts Executive Director and the moderator of the Countdown Round. Three times, Wan took a one-question lead; and three times, Robitaille tied the score on the following question. The tie was broken for good when Wan answered the following question:

What is the remainder when 999,999,999 is divided by 32?

This year’s winning question was relatively easy. What makes me say that? Well, for starters, when an odd number is divided by an even number, the remainder will be odd; and because 32 is the divisor, the remainder has to be less than 32. Consequently, the remainder is in the set {1, 3, 5, …, 31}, so there are only 16 possible answers.

But more importantly, most MathCounts competitors will be well trained for a problem of this type. It relies on divisibility rules that they should know, and it requires minimal insight to arrive at the correct answer.

I suspect that the following explanation of the solution is the likely thought process that Wan used to solve this problem; of course, all of this occurred **in his head in less than 7 seconds**, which does make it rather impressive.

A fact that you probably know:

- A number is divisible by 2 if it’s even.

But said another way…

- A number is divisible by 2 if the last digit is divisible by 2.

There are then corollary rules for larger powers of 2:

- A number is divisible by 4 if the last
**two**digits are divisible by 4.- For example, we can conclude that 176,432,9
**28**is divisible by 4 because the last two digits form 28, which is divisible by 4. The digits in the hundreds, thousands, and higher place values are somewhat irrelevant, because they represent some multiple of 100 — for instance, the 7 in the ten millions place represents 70,000,000, which is 700,000 × 100 — and every multiple of 100 is divisible by 4.

- For example, we can conclude that 176,432,9
- A number is divisible by 8 if the last
**three**digits are divisible by 8.- For example, we can conclude that 176,432,
**376**is divisible by 8 because the last three digits form 376, which is divisible by 8 since 8 × 47 = 376.

- For example, we can conclude that 176,432,
- A number is divisible by 16 if the last
**four**digits are divisible by 16. - A number is divisible by 32 if the last
**five**digits are divisible by 32. - And so on.

These observations lead to a generalization…

- A number is divisible by 2
if the last^{n}*n*digits are divisible by 2.^{n}

I won’t take the time to prove that statement here, but you can trust me. (Or maybe you’d like to prove it on your own.) I will, however, explain why it’s relevant.

A number will be divisible by 32 if the last five digits are divisible by 32. Consequently, any number that ends in five 0’s will be divisible by 32, which means that 1,000,000,000 is a multiple of 32. Since 999,999,999 is 1 less than 1,000,000,000, then it must be 1 less than a multiple of 32. Therefore, when 999,999,999 is divided by 32, **the remainder will be 31**.

The hardest part of solving that problem is recognizing that 999,999,999 is 1 less than a multiple of 32. But for most MathCounts students, that step is not very difficult, hence my contention that this was a relatively easy winning problem.

My favorite problem of the Countdown Round? Now, that’s another story, and it epitomizes what I generally love about MathCounts problems.

If

a,b,c, anddare four distinct positive integers such thata=^{b}c, what is the least possible value of^{d}a+b+c+d?

This problem has several things going for it:

- It’s simply stated.
- It’s easily understood, even by students who don’t participate in MathCounts.
- It has an entry point for all students, since most kids can find at least one set of numbers that would work, even if they couldn’t find the set with the least possible sum.
- Finding the right answer requires convincing yourself that no lesser sum exists.

It’s that last point that I find so interesting. While I was able to find the correct answer, it took a while to convince myself that it was the least possible sum. But since I don’t want to deprive you of any fun, I’ll let you solve the problem on your own.

As a final point, I’ll show you a picture that I took at the event. Do you see the error? What can I say… it’s a math competition… you didn’t expect them to be good with numbers, did you?

*Full disclosure: The error was corrected halfway through the competition during a break.*

## Sound Smart with Math Words

When law professor Richard D. Friedman appeared in front of the Supreme Court, he stated that an issue was “entirely orthogonal” to the discussion. Chief Justice John G. Roberts Jr. stopped him, saying, “I’m sorry. Entirely *what*?”

“Orthogonal,” Friedman replied, and then explained that it meant *unrelated* or *irrelevant*.

Justice Antonin Scalia was so taken by the word that he let out an **ooh** and suggested that the word be used in the opinion.

In math class,* orthogonal* means “at a right angle,” but in common English, it means that two things are unrelated. Many mathematical terms have taken a similar path; moreover, there are many terms that had extracurricular meanings long before we ever used them in a math classroom. *Average* is used to mean “typical.” *Odd* is used to mean “strange” or “abnormal.” And *base* is used to mean “foundation.” To name a few.

The stats teacher said that I was average, but he was just being mean.

You know what’s odd to me? Numbers that aren’t divisible by 2.

An exponent’s favorite song is, “All About the Base.”

Even words for quantities can have multiple meanings. Plato used *number* to mean any quantity more than 2. And *forty* used to refer to any large quantity, which is why Ali Baba had forty thieves, and why the Bible says that it rained for forty days and forty nights. Nowadays, we use *thousands* or *millions* or *billions* or *gazillions* to refer to a large, unknown quantity. (That’s just grammatical inflation, I suspect. In a future millennium, we’ll talk of *sextillion* tourists waiting in line at Disneyland or of *googol* icicles hanging from the gutters.)

Zevenbergen (2001) provided a list of 36 such terms that have both math and non-math meanings, including:

- angle
- improper
- point
- rational
- table
- volume

The alternate meanings can lead to a significant amount of confusion. Ask a mathematician, “What’s your point?” and she may respond, “(2, 4).” Likewise, if you ask a student to determine the volume of a soup can, he may answer, “Uh… quiet?”

It can all be quite perplexing. But don’t be overwhelmed. Sarah Cooper has some suggestions for working mathy terms into business meetings and everyday speech. Like this…

For more suggestions, check out her blog post How to Use Math Words to Sound Smart.

If you really want to sound smart, though, be sure to heed the advice of columnist Dave Barry:

Don’t say:“I think Peruvians are underpaid.”

Say instead:“The average Peruvian’s salary in 1981 dollars adjusted for the revised tax base is $1452.81 per annum, which is $836.07 below the mean gross poverty level.”

NOTE: Always make up exact figures. If an opponent asks you where you got your information, makethatup, too.

This reminds me of several stats jokes:

- More than 83% of all statistics are made up on the spot.
- As many as one in four eggs contains salmonella, so you should only make three-egg omelettes, just to be safe.
- Even some failing students are in the top 90% of their class.
- An unprecedented 69.846743% of all statistics reflect an unjustified level of precision.

You can see the original version of “How to Win an Argument” at Dave Barry’s website, or you can check out a more readable version from the Cognitive Science Dept at Rensselaer.

Zevenbergen, R. (2001). Mathematical literacy in the middle years. *Literacy Learning: the Middle Years*, *9*(2), 21-28.

## Heavy Cookies, Undervalued Coins, and Misconceptions

Simple question to get us started…

Which is worth more?

And **of course** the answer is, “The quarters, because 50¢ is more than 20¢,” right? But not to a kindergarten student or a pre-schooler who hasn’t yet learned how much coins are worth. A young student might argue, “Four is more than two.”

Why didn’t the quarter follow the nickel when he rolled himself down the hill?

Because the quarter had more cents.

Recently, I was asked to review an educational video for kindergarten math that had a similar question.

The video stated, “Can you tell the green, yellow, and orange cookies are heavier? That makes sense, doesn’t it? Because there are **more** of them!”

Uh, no.

This is the same logic that would lead one to claim that the value of four nickels is greater than the value two quarters because there are more nickels. It’s a huge misconception for students to focus on **number** rather than **value**. So it’s very frustrating to see this video reinforce that misconception.

For example, if each green, yellow, or orange cookie weighs 3 ounces, but each blue or purple cookie weighs 5 ounces, then the left pile would weigh 6 × 3 = 18 ounces, and the right pile would weigh 4 × 5 = 20 ounces, so the right side would be heavier. (Then again, are there really 6 cookies on the left and 4 on the right, or are some cookies hidden? Hard to tell.)

As far as I’m concerned, the only acceptable answer is that the pile of green, yellow, and orange cookies must be heavier — assuming, of course, that the balance scale isn’t malfunctioning — because the pans are tipped in that direction.

All of this reminds me of the poem “Smart” by Shel Silverstein.

SMARTMy dad gave me one dollar bill

‘Cause I’m his smartest son,

And I swapped it for two shiny quarters

‘Cause two is more than one!And then I took the quarters

And traded them to Lou

For three dimes — I guess he don’t know

That three is more than two!Just then, along came old blind Bates

And just ’cause he can’t see

He gave me four nickels for my three dimes,

And four is more than three!And I took the nickels to Hiram Coombs

Down at the seed-feed store,

And the fool gave me five pennies for them,

And five is more than four!And then I went and showed my dad,

And he got red in the cheeks

And closed his eyes and shook his head–

Too proud of me to speak!

## All Systems Go

I noticed the boys having an intense conversation in front of this sign at our local pizza shop:

When I asked what they were doing, they said, “We’re trying to figure out how much one slice and a beer would cost.”

As you read that, there were likely two thoughts that crossed your mind:

- Why can’t these poor boys look at a pizza menu without perceiving it as a system of equations?
- Why are eight-year-olds concerned with the price of beer?

The answer to both, of course, is that I’m a terrible father, and both beer and math are prominent in our daily lives.

But you have to admit that it’s pretty cool that my sons recognized, and then solved, the following system:

They didn’t use substitution or elimination because they didn’t have to — and, perhaps, because they don’t know either of those methods yet. But mental math was sufficient. If two slices and a soda cost $6.00, and one slice and a soda cost $3.50, then one slice must be 6.00 – 3.50 = $2.50. Consequently, two slices cost $5.00, so a beer must be 8.00 – 5.00 = $3.00. A beer and a slice will set you back $5.50.

I remember once visiting a classroom in Somerville, MA, and the teacher was reviewing the substitution method. My memory is a bit fuzzy, but the problem she solved on the chalkboard was something like this:

Mrs. Butterworth’s math test has 10 questions and is worth 100 points. The test has some true/false questions worth 8 points each and some multiple-choice questions worth 12 points each. How many multiple-choice questions are on the test?

The teacher then used elimination to solve the resulting system:

The math chairperson was standing next to me as I watched. “Why is she doing that?” I asked. “You don’t need elimination. It’s clear there have to be 8 or fewer multiple-choice questions (8 × 12 = 96), so why not just guess-and-check?”

“Because on the MCAS [Massachusetts Comprehensive Assessment System], if they tell you to use elimination but you solve the problem a different way, it’ll be marked wrong.”

So much for CCSS.Math.Practice.MP1. Although most of us would like students to “plan a solution pathway rather than simply jumping into a solution attempt,” apparently students in Massachusetts need to blindly follow algorithms and not think for themselves.

The following is my favorite system of equations problem:

I counted 34 legs after dropping some insects into my spider tank. How many spiders and how many insects?

Why is this my favorite system of equations problem? Because there is a unique solution, even though it results in just one equation with two unknowns. Traditional methods won’t work, and students have to think to solve it. Blind algorithms lead nowhere.

Other things that lead nowhere are spending your leisure hours reading a math jokes blog. But since you’re here…

Why did the student put his homework in a fish bowl?

He was trying to dissolve an equation.An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn’t care.

## Mathiest Fortnight of 2016

Monday, April 4, 2016, was Square Root Day, because the date is abbreviated 4/4/16, and 4 × 4 = 16. But if you’re a faithful reader of this blog, then you already knew that, because you read all about it in Monday’s post, Guess the Graph on Square Root Day.

But it doesn’t end there. It ain’t just one day. Oh, no, friends… this is a banner week. Or, really, a banner *two weeks*.

Tomorrow, April 8, 2016, is a **geometric sequence day**, because the date is 4/8/16, and 4 × 2 = 8, and 8 × 2 = 16.

And Saturday, April 9, 2016, is a **consecutive square number day**, because the month, day, and year are consecutive square numbers. Square number days, in which each of the month, day, and year are all square numbers — not necessarily consecutive — are less rare; there are 15 of them this year. But among them is 1/4/16, which rocks the intersection of square number days *and* geometric sequence days. (That’s right — I said “rocks the intersection.”)

And then Sunday, April 10, 2016, is an **arithmetic sequence day**, because 4, 10, and 16 have a common difference of 6. Though honestly, arithmetic sequence days are a dime a half-dozen; there are six of them this year.

Next Monday, April 12, 2016, is a **sum day**, because 4 + 12 = 16. Again, ho-hum. There are a dozen sum days this year, and there will be a dozen sum days *every year* through 2031.

And just a little further in the future is Friday, April 16, 2016, whose abbreviation is 4/16/16, and if you remove those unsightly slashes, you get 41,616 = 204^{2}. I’m not sure what you’d call such a day, other than *awesome*.

Admittedly, some of those things are fairly common occurrences. But, still. That’s six calendar-related phenomena in a thirteen-day period, which may be enough mathematic-temporal mayhem to unseat the previously unrivaled Mathiest Week of 2013.

Partially, this blog post was meant to enlighten and entertain you. But mostly, it was meant to send numerologists off the deep end. Mission. Accomplished.

You’ve endured enough. Here are some calendar-related jokes for you…

Did you hear about the two grad students who stole a calendar?

They each got six months!I was going to look for my missing calendar, but I just couldn’t find the time.

What do calendars eat?

Dates.

## 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)