## As If Your Life Depends on It

It was early Wednesday morning (or late Tuesday night, depending on how you look at it) of Finals Week. Yes, I should have been studying — or sleeping; it was 3 a.m., after all — but I was young and in love, and wandering through the quads and into unlocked academic buildings on Penn State’s campus with my girlfriend held far more appeal than the problems and theorems in my linear algebra textbook. I remember a light snowfall and how beautiful she looked in the lamplight. I remember my surprise when I pushed on the main door to Sparks Building and it opened. But what I remember most from that night is a quote that a psychology professor had borrowed from a student’s paper and taped to her office door:

Many things depend on many things.

Indeed.

I don’t remember that girlfriend’s name. And I remember very little from my linear algebra course. But I’ll never forget that quote, and I’ve repeated it many times in business meetings.

de·pen·dentn.what hangs from de necklace

Dependence is a topic that rears its head frequently in mathematics, from algebra to probability, and it’s useful in a variety of contexts.

Football, for instance. Redskins safety David Bruton showed his understanding of dependence during a recent radio interview:

I’m between 225 and 230 [pounds], depending on what I had for lunch.

And measurement. Comedian Ron White understands dependence, too:

Now, I’m between 6’1″ and 6’6″, depending on which convenience store I’m leaving.

Some things aren’t really dependent at all…

The economy depends on economists in the same way that the weather depends on forecasters.

And some things are subjective…

Your true value depends entirely on what you are compared with.

Some things depend on whom you ask…

A teacher said to her student, “Billy, if both of your parents were born in 1967, how old are they now?”

After a few moments, Billy answered, “It depends.”

“On what?” the teacher asked.

“On whether you ask my mother or father.”

And other things on your perspective…

How long a minute feels depends on what side of the bathroom door you’re on.

The location of an animal?

Where can you find polar bears?

Depends on where you lost them!

But the better answer to that joke is, “Just check their polar coordinates!” (You’re welcome.)

This post wouldn’t be complete without an obligatory old-person joke…

An old man is flirting with a woman at the senior center. He asks her, “If I took you out for a night of wining, dining and dancing, what would you wear?”

The old woman replies shyly, “Depends.”

And finally, one last math joke…

How many math professors does it take to plaster a wall?

Depends how hard you throw them.

## Flag Day Math

Tuesday, June 14. Flag Day. It’s nearly impossible for mathy folks to **not** tell this joke today.

Several engineers were attempting to measure the height of a flag pole. They only had a measuring tape, and they were getting quite frustrated trying to slide the tape up the pole. They could get the tape no more than a third of the way up the pole before it would bend and fall down.

A mathematician asks what they’re doing, and they explain. The mathematician offers to help. She removes the pole from the ground, sets it down, and measures it easily. She then returns the measuring tape to the engineers, and walks off.

When she leaves, one engineer says to the others, “That’s just like a mathematician! We need to know the height, and she gives us the length!”

Those who know it will also tell this one, or a variant.

How do statisticians determine which banner to hoist?

They take a flag poll.

And then there are jokes about specific flags.

I’m about as motivated as the guy who designed the Japanese flag.

Honestly, I want to stop. But I can’t. Just one more…

What’s the best thing about Switzerland?

I don’t know, but the flag’s a big plus.

Okay, seriously… I didn’t invite you here today to listen to bad jokes. (Well, that’s not the **only** reason, anyway.)

I invited you here today to have a little Flag Day fun with math. The projectionist Shahee Ilya has **converted the flag of every country into a pie graph** based on its colors. For example, the Austrian flag has two red stripes and one white stripe, so it is converted to a pie graph as follows:

Pretty cool, huh?

What follows are pie graphs for ten flags. Even if you are geographically challenged, I assure you that you’ve heard of all ten countries represented below. **Can you name the country whose flag was used to create each pie graph?**

Stumped by the challenge? Here’s a hint: The countries whose flags are represented above are **the ten most populous countries on Earth**. (Admittedly, had someone asked me to name the ten most populous countries prior to writing this post, I would have been lucky to identify half of them.)

And just to put some space between the pie graphs above and the countries whose flag they represent below (i.e., the answers), I include for your enjoyment one of the most hideous puns you’ll ever see, modified from an even worse version at Six Puns:

During a recent heat

wave, apollrevealed that beads of sweat had amassed(mast) on the secretary’s forehead and a virus wasripplingthrough the officestaff. Although the boss knew that the secretary was very sick, he saw no reason toban herfrom the office. Instead, he wrote a note withpennant(pen and) paper, and heflagged the issue to be addressed with thestandardprotocol.

If you tolerated that, you certainly deserve the answers…

Nigeria | Pakistan |

Russia | United States |

Indonesia | Japan |

Bangladesh | Brazil |

China | India |

Click on over to **shaheeilyas.com/flags** to see the pie graph for every country in the world. Clicking on the pie graph will reveal the flag and country name.

## All 6’s and 7’s

Tech N9ne has said that the title of his album *All 6’s and 7’s* means “in a state of confusion and disarray.” Well, of course it does; that’s what it meant when Shakespeare (1595) used the phrase in Richard II…

But time will not permit: all is uneven,

And every thing is left at six and seven.

…that’s what it meant when Chaucer (1380s) used the phrase in *Troilus and Criseyde*…

But manly set the world on sixe and sevene;

And, if thou deye a martir, go to hevene.

…and that’s what it meant when Sirenia (2002) titled their debut album *At Sixes and Sevens*.

Today, we’re at sixes and sevens, in a sense. The date is 6/7, and this post is all about the many variations of the classic math joke, “Why is 6 afraid of 7?” Think you’ve heard them all? Think again. You’ll be overwhelmed by the sheer number of variations that have been collected from the farthest corners of the web, but hopefully it won’t throw you into a state of disarray and confusion.

(Original)

Why is 6 afraid of 7?

Because 7 8 9.

(Purist)

Why is 10 afraid of 7?

Because 7 8 9.

Some folks claim this makes more sense, since 10 would be next in line.

(Dice)

Why is 6 afraid of 7?

He’s playing craps and his point is 10.

(Turkish)Why is 5 (bes) afraid of 6 (alti)?

Because 6 (alti) 7 (yedi) 8 (sekiz).

In Turkish, the word for 7 (yedi) is also the word for “ate.”

(Greek)

Why is ε (epsilon) afraid of θ (theta)?

Because ζ (zeta) η (eta) θ (theta).

(Modern)

Why is 6 afraid of 7?

Because he’s a registered six offender.

(Anti-Joke)

Why is 6 afraid of 7?

It isn’t. Numbers are not sentient and therefore are incapable of feeling fear.

(Microsoft)

Why Windows 10?

Because Windows 7 8 9.

This was one of four jokes on the t-shirt worn by Joe Belfiore when presenting new features of Windows 10 at the Build 2015 conference. Microsoft never released a Windows 9 and skipped straight to Windows 10.

(Meta-Joke)

Why don’t jokes work in base 8?

Because 7 10 11.

(Canadian)

Why do Canadians prefer jokes in hexadecimal?

Because 7 8 9 A.

(Star Wars)

Why is Yoda afraid of 7?

Because 6 7 8.

Don’t get it? Say it out loud using your best Yoda voice, and pause briefly after the 6.

(PG-13)

Why did 6 break up with 7?

Because 7 8 9 out.

(Castiel fromSupernatural)

Why is 6 afraid of 7?

I assume it’s because 7 is a prime number, and prime numbers can be intimidating.

## Demitri Martin and Me

As I was watching *If I* by Demetri Martin, I realized something.

I

loveDemitri Martin, because IamDemetri Martin.

Not literally, of course. I didn’t inhabit his body and take over his soul. (Would if I could!) Nor is this blog a ruse that appears to be written by Patrick Vennebush when it is, in fact, written by Demitri Martin. I just mean that he and I are about as similar as two people can be without entering the world from the same womb. Check out this list:

Demitri Martin |
Patrick Vennebush |

He’s weird. (In a good way.) | I’m weird. (No disclaimer.) |

He did Mensa puzzles as a kid. | I did Mensa puzzles as a kid. |

He uses convoluted mnemonics to remember numbers. | I use convoluted mnemonics to remember numbers. |

He uses drawings and visual aids during stand-up performances. (See below.) | I use drawings and visual aids during math presentations. (See below.) |

He was influenced by Steven Wright, Emo Philips, Eddie Izzard, and Mitch Hedberg. | I watched every Steven Wright performance on cable television when I was a teenager; my favorite joke is from Emo Philips; I own every Eddie Izzard CD; and one of my great regrets is that I never saw Mitch Hedberg perform live. |

He was slated to play Paul de Podesta in Moneyball but was replaced by Jonah Hill. |
I wasn’t in Moneyball, either. |

He was born in a prime number year (1973). | I was born in a prime number year (1971). |

He won a Perrier Comedy Award. | I sometimes drink Perrier while watching Comedy Central. |

He once attended class wearing a gorilla suit. | I had no fashion sense in college. |

He is extremely allergic to nuts. | I’m not allergic to them, but I really don’t like crazy people. |

One of Demetri’s drawings:

Oh, sure, I could list hundreds of other similarities between Demitri and me, but I think the list above is enough to see that the coincidence is uncanny. I mean, we practically live parallel lives.

Demetri used to sneak Mensa puzzle books — not muscle mags or girlie mags — into school to read during class. One of the puzzles purportedly from his *Mensa Presents Mighty Mindbusters* book:

If a crab-and-a-half weigh a pound-and-a-half, but the half-crab weighs as much again as the whole crab, what do half the whole crab and the whole of the half-crab weigh?

He said that solving problems from those books was validating.

When I got one right, I’d be like, “Yes! I

amsmart! These other idiots don’t know how much the crabs weigh.” But I do. Because I just spent Saturday working it out.

I solved puzzles like this, too. I don’t know if they made me feel smart, but I enjoyed the way I felt when I figured out a particularly tough one.

From the way he describes it, such puzzles may have had the same effect on both of us.

Whatever the reason, I spent a lot of time as a kid doing these puzzle books. And it came to shape the way I see the world. So now, as an adult, I see the world in those terms. For example, to me a phone number is always a sentence or an equation. Like my friend Becky…

He goes on to say that he remembers Becky’s phone number using a convoluted, mathematical mnemonic:

That is, he converts the first three digits into an expression that is equal to an expression formed by the last four digits. He concludes that it’s “much simpler,” but it’s unclear how.

Now that’s some crazy, messed-up sh*t.

And I’d probably think it even weirder… if I didn’t do it, too.

One night many years ago, my roommate Adam asked for the number of the local pizza shop. I replied, “3^{3}, 1^{3}, 20^{3},” because that’s how I saw it. Adam looked at me like I was nuts, and he was probably onto something.

My friend AJ’s street address is 6236, which I remember as 6^{2} = 36.

My street address growing up was 1331, which I associated with the third row of Pascal’s triangle. (It also happens to be 11^{3}, but I didn’t know that at the time.)

I chose the four digits of my PIN because… no, wait, that wouldn’t be prudent.

My co-worker Julia’s extension is 2691. I used to remember this as 2 + 6 = 9 – 1, until I recognized a more elegant geometric mnemonic: the sequence 2, 6, 9, 1 forms an isosceles trapezoid on my office phone’s keypad — or it would, were the buttons equally spaced.

I can’t explain why I do this. Perhaps, as Demetri says, it’s the influence of all those puzzle books. Or maybe it’s just that the mental conversion to an equation gives the number meaning, making it more memorable. Or perhaps it’s that I’m wired to see the world through a mathematical lens, despite not wearing glasses.

Larry McCleary, author of *The Brain Trust Program,* claims that numbers are difficult to remember because “most of us don’t have any emotional attachment to particular numbers.” Mr. McCleary, I’d like you to meet my friend Demetri…

Demitri and I are both into anagrams.

Even when I walk down the street, things look a little different. The signs… the letters dance around. It becomes a little puzzle for me. So, say MOBIL, the gas station — that becomes LIMBO. STARBUCKS becomes RACKS BUST. CAR PHONE WAREHOUSE… AH, ONE SOUR CRAP — WHEE!”

Yeah, I do that, too…

My first car was a CHEVROLET IMPALA, which transforms to COMPARATIVE HELL. Our neighbor’s son is CARSON, whom I jokingly call ACORNS. And I can’t see a STOP sign without also thinking of OPTS, POST, POTS, and TOPS.

If you’re reading this, you likely have some things in common with Demetri, too. **What number mnemonics do you use, or what anagrams to do you see?**

## XII Puzzle

Yesterday, my wife and I celebrated our 12th anniversary. We celebrated at home, with the boys and a home-cooked meal. I created the following puzzle to fill the time between dinner and dessert.

Each of the **12 answers** in this puzzle is a **12-letter word** that contains the **letters X, I, and I**, a reference to the Roman numeral XII. Those three letters appear in the proper order, though they may be separated by other letters.

For example, if you were given the clue, “Of or relating to the study of flags,” you would say, “VE**XI**LLOLOG**I**C,” which consists of 12 letters and has X, I, and I as the third, fourth, and eleventh letters, respectively.

Below are the clues, each presented in two parts. The first part is the real clue, and the second part *in italics* is a fun addendum specifically for our anniversary.

Enjoy, and good luck!

**Device for putting out a fire**,*like the one I needed when mommy set my heart ablaze*.**Feeling of excitement or elation**,*like the feeling I had when mommy said, “I do!” (possibly arising from the trepidation that she might not)*.**Insufficient oxygen due to abnormal breathing**,*which I experience regularly when mommy kisses me*.**Lowest part of the sternum**,*which holds in the abdominal diaphragm and prevents me from experiencing asphyxia when mommy is nearby*.**Someone who takes money or other things through force or threats**,*which you might call mommy for stealing my heart*.**State of being so happy (or drunk) as to lose control of your faculties or behavior**,*which is the state I’ve been in since I fell in love with mommy*.**Someone who loves and studies words**,*like mommy and daddy*.**Serving as an example**,*like how I serve as a warning to women about why they shouldn’t get married*.**Someone who studies the adverse effects of chemicals on humans**,*like the scientist who told me that mommy’s love is as addictive as Vicodin*.**Torturous, intensely painful, or mentally agonizing**,*which are three ways that mommy has occasionally described living with me*.**To increase as much as possible, or the process of trying to find the best option**,*like the one I used to find the best wife in the world*.**In the US, a 1 followed by 51 zeroes; in the UK, a 1 followed by 96 zeroes**,*or how much I love mommy on a scale of 1 to 10*.

Having trouble figuring out the answers? Well, I won’t give them to you, but if you search ***x*i*i*** at www.morewords.com, it’ll return all 666 words that contain X, I, and I in the proper order. That should significantly limit your search. You’ll then need to do a little work to figure out which 12-letter words fit the clues above.

## 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.