## 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.

## 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.