## Archive for July, 2012

### Golf Is a Good Walk Ruined

The title of this post is a quote attributed to Mark Twain. I would amend it as follows:

Golf may be a good walk ruined — but it’s a good time to think about math.

Not too long ago, I attended a conference at the Asilomar Conference Center on the Monterey Bay Peninsula in California. The property is adjacent to beautiful 17-Mile Drive and historic Pebble Beach. With some free time on the last afternoon, I decided to treat myself to a round of golf.

“Hello, this is Edie. Thanks for calling Pebble Beach Golf Club. How can I help you?”

“Hi, Edie. I’m wondering if I can get on the course to play 9 holes today.”

“You sure can,” she said, “but the price is the same for 9 or 18 holes.”

“Oh, okay. And what’s the price?”

“$495,” Edie said matter-of-factly.

I did a quick mental calculation, and I realized that the price would be $55 per hole.

What I said: “On second thought, Edie, I’m not sure I’ll have enough time today. Let me call back if my schedule opens up.”

What I thought: “Are you f**kin’ kiddin’ me?”

So instead of playing Pebble Beach, I found an executive course in Cupertino where I played 9 holes, rented clubs, and bought six golf balls, a bag of tees, and a new golf glove — all for less than $50.

Not only does this incident represent an exercise in fiscal responsibility, it also brought to mind divisibility rules. For instance, I realized that 495 was divisible by 9 because 4 + 9 + 5 = 18.

The rule for divisibility by 9 is rather easy to implement: just add the digits. If the result is divisible by 9, then the original number is divisible by 9, too.

A couple of weeks ago, a colleague told me that she and some friends had discovered an elegant rule for divisibility by 7. However, she was unwilling to share the method with me — she said I’d have more fun if I discovered it on my own. (I simultaneously loved and loathed her for that.)

I knew a rule for divisibility by 7, I discovered an inelegant method, and I discovered another method that was less inelegant. I’ll share my three methods below, but if you know (or discover) a different method, please share.

**The Method I Already Knew**

The rule for divisibility by 7 that I had heard before, and that can be found easily via Google, works as follows:

- Remove the units digit, and double it. (If the original number was 10
*m*+*n*, where*m*and*n*are positive integers, you’d calculate 2*n*.) - Subtract the result of Step 1 from the remaining number. (That is, you’d find
*m*‑ 2*n*.) - Repeat Steps 1 and 2 until you can determine if the result is a multiple of 7. If it is, then the original number is a multiple of 7, too.

This is best shown with an example. Let’s say you want to know if 8,603 is divisible by 7. Then first find 860 ‑ 2(3) = 854. Since it may not be obvious that 854 is a multiple of 7, repeat the procedure: 85 ‑ 2(4) = 77. Because 77 is a multiple of 7, then 8,603 must be a multiple of 7, too.

The problem with this procedure is that it takes too long for big numbers.

** The Highly Inelegant Method That I Devised**

Take the number, and continually subtract known multiples of 7. For instance, again using 8,603, first remove 7,000 to leave 1,603. Then remove 1,400 to leave 203. Then remove 210 to leave ‑7, which is a negative multiple of 7. So that means 8,603 must be a multiple of 7, too.

Since any number of the form 7 × 10* ^{k}* is a multiple of 7, a time-saving step in this process is to reduce every digit in the original number by 7. For 8,603, that means reduce 8 to 1 to leave 1,603. That’s the same as subtracting 7,000. But for a bigger number like 973,865, that shortcut could be implemented three times to leave 203,165, which might help a little. Taking this even further, remove other multiples of 7; for instance, since 16 ‑ 14 = 2, you could further reduce the number to 203,025.

I rather like this method for its simplicity, but it also takes too long.

**The Less Inelegant Method That I Found **

The reason that the rule for divisibility by 9 works is that every power of 10 is 1 more than a multiple of 9. Consequently, when a number of the form *m* × 10* ^{k}* is divided by 9, the remainder will be

*m*. For instance, when 7,000 is divided by 9, the remainder is 7; when 300 is divided by 9, the remainder is 3; and, when 80 is divided by 9, the remainder is 8. So if you subtract a lot of 9’s from 7,380, you would be left with 7 + 3 + 8 = 18 as the remainder, and since that result is a multiple of 9, then 7,380 is a multiple of 9, too.

This same idea can be applied for divisibility by 7.

- When you divide a number of the form
*m*× 10^{6k}by 7, the remainder will be*m*. - When you divide a number of the form
*m*× 10^{6k + 1}by 7, the remainder will be 3*m*. - When you divide a number of the form
*m*× 10^{6k + 2}by 7, the remainder will be 2*m*. - When you divide a number of the form
*m*× 10^{6k + 3}by 7, the remainder will be 6*m*. - When you divide a number of the form
*m*× 10^{6k + 4}by 7, the remainder will be 4*m*. - When you divide a number of the form
*m*× 10^{6k + 5}by 7, the remainder will be 5*m*.

You can then determine if a number is divisible by 7 by multiplying the units digit by 1, the tens digit by 3, the hundreds digit by 2, and so on, following the sequence 1, 3, 2, 6, 4, 5, as given above. If the result is a multiple of 7, the original number is, too.

As an example, again consider 8,603. Then calculate

1(3) + 3(0) + 2(6) + 6(8) = 63.

Since the result is a multiple of 7, then 8,603 is a multiple of 7, too.

**What You Got?**

Those are my three methods. As I said above, the third is less inelegant that the first two, but I still wouldn’t call it elegant.

Have you got a rule for divisibility by 7 that’s better than any of these? Do tell!

### 15 Math Spoonerisms

When we walked into the mall, there was a display for kitchenware. The signature piece in the collection was a copper wok. My friend Justin looked at it and turned to me. “That’s a great spoonerism,” he said.

That was many years ago. At the time, I didn’t know what a spoonerism was, so the humor (as it were) was lost on me. But I learned that a *spoonerism *is a play on words in which corresponding sounds are switched, and I now consider myself an above average spoonerist.

A spoonerism of cube root, for example, is **cute rube**, which might refer to an attractive country bumpkin.

(Though truth be known, that is actually a forkerism, since the ending sounds have been interchanged. Douglas Hofstadter coined the terms *kniferism* and *forkerism* to refer to the exchange of the middle and ending sounds of words. He reserved spoonerism for the exchange of the beginning sounds).

And a **fenerating junction** is an intersection where you can get a loan.

The following image shows **sailor tearies**…

…which might be what comes out of a seaman’s ducts when he deals with an infinite sequence.

And a **tractor fee** is what you’d pay for a piece of farm equipment.

Here are eleven others:

**Rare Squoot** — Given that no one has actually ever seen a squoot, they must be rare, indeed.

**Trite Wry Angle** — A sharp-tongued intersection of two lines whose comments are trivial.

**Kine Serve** — Cows that wait tables. (See Holy Cow!)

**Rational Roots** — Well, duh.

**Spinnier Lace** — Secret desire of every lingerie lovin’ lady.

**Hone Kite** — To perfect a wind-flying instrument.

**Saw of Lines** — “Read between the lines,” et al.

**Faulty Marryable** — Elizabeth Taylor, Larry King, and Zsa Zsa Gabor, to name a few.

**Formal Nectar** — A drink to be drunk when dressed like a skunk.

**Meriadoc Potion** — A concoction of Frodo’s best friend.

**Lewd Skeins** — Naked fowl.

Spoonerisms have had their place in pop culture. The *Kenny Everett Television Show* featured a character with the spooneristic name Cupid Stunt. As the story goes, the character was originally named Mary Hinge, but BBC vetoed the name for fear that announcers would mistakenly pronounce the spoonerism; rather bizarrely, they allowed Cupid Stunt despite the same risk.

Finally, this post started with a mention of a wok, so here’s a line from the song “High School Party” by Bo Burnham:

Let’s rob a Chinese restaurant or stroll around the block —

Either way, girl, we’re taking a wok.

### Guess Who’s the Best Mathematician of All Time?

Just as Anna Nicole Smith was the premier jeans model in the 1990’s, Carl Friedrich Gauss was the preeminent mathematician of the 1800’s. (Wow, did I really just compare a model known primarily for taking off her clothes and meeting an early demise with a mathematician who made significant contributions in number theory and statistics, as well as astronomy and optics? But the analogy isn’t completely absurd — after all, Carl Friedrich was one of the five hottest mathematicians.)

Recognizing the similarities between these two giants in their fields, it was impossible to resist the urge to create the following poster. Enjoy.

Here’s an up-close look at the Gauss^{®} symbol:

### Celebrity Sighting at Math Meeting

I have the pleasure of serving on the advisory committee for the Math Midway 2 Go, a traveling exhibit of the Museum of Mathematics. I get to see a lot of cool stuff.

When it opens on December 15, MoMath will be the only museum of mathematics in North America. If you happen to find yourself in Manhattan, check it out. The exhibits are really fun.

One of the exhibits in the Math Midway 2 Go is a number line with ornaments hanging from each number. For instance, a square ornament hangs from the numbers 1, 4, 9, 16, …, and a symbol that looks like an atom hangs from 2, 3, 5, 7, 11, 13, … (the atom symbol was used because “prime numbers are the building blocks of the number system”). However, I was not able to identify the symbol that hangs from the following numbers:

3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30,

32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96

I’ve been told that the symbol is a compass (the kind for drawing circles, not for orienteering). Unfortunately, that hint didn’t help me to identify the sequence of numbers. Do you know what the sequence is? **

A recent meeting of the advisory committee was held at a private school in NYC, and a number of parents were waiting in the hallway when our meeting ended. As I walked by, one of the parents stood up quickly, and I accidentally brushed against her. “Oh, I’m sorry, ma’am,” I said. She turned to look at me, and I looked back. First, I noticed how tall she was. Then I noticed something else. “Oh,” I said, “you’re Brooke Shields.” Turns out her kids go to this school. She smiled politely at my recognition.

She was dressed in casual clothes, and she was just there to pick up her kids. I didn’t want to be a nuisance, so I just said, “Have a great day.”

How cool is that? Go to a math meeting, meet a celebrity! And one I had a crush on when I was 13, no less!

** The sequence is the number of sides for *constructible polygons*, which are regular polygons that can be drawn with a straightedge and compass.

### What Helps You Remember?

The following joke is terrible, and I must confess that it’s an MJ4MF original. (One wonders why I’d admit that.)

Professor: The polynomial we’ll use will be

x^{3}+ 17x^{2}+ 51x+ 12, but many people find this expression difficult to remember.Student: That’s too bad. Those people should find a new monic.

That joke came to me (the way posing like Tim Tebow just came to Paul Pierce) tonight, when my wife informed me that she’d be returning from London on United Flight 919. “That’ll be easy to remember,” I said. “It’s a palindromic prime, and it’s the eleventh cuban prime.”

A mathematician walks into a tobacco shop and asks for a cigar. The tobacconist pulls one from the case and says, “I think you’ll really like this cigar. It’s from Havana, and I’ve sold 37 of them today.”

“Fantastic!” exclaims the mathematician. “I just love a good cuban!”

(Don’t you feel lucky? That joke is also an MJ4MF original, and this marks the first time that a post on MJ4MF contains more than one original joke. Congratulations for being part of history.)

The problem, of course, is that when I arrive at the airport, I won’t remember the airline, and I’ll remember “eleventh cuban prime” instead of the actual flight number, and it will take me several minutes to realize that (17 + 1)^{3} ‑ 17^{3} = 919 is the eleventh cuban prime. (Note that cuban primes can be expressed in the

form (*x* + 1)^{3} ‑ *x*^{3}, but not every number of the form (*x* + 1)^{3} ‑ *x*^{3} is a prime number.)

While it’s cool that the flight number is the eleventh cuban prime, it may be a completely useless mnemonic.

On the other hand, I used to remember the phone number for my favorite pizza shop (271‑8000) as the concatenation of 3^{3}, 1^{3}, and 20^{3}.

The house in which I grew up had house number 1331, which I remembered as the third row of Pascal’s triangle. (It’s also 11^{3}.)

And I’ll never forget the phone number at my apartment in college: 867‑5309. That’s certainly a memorable number, but what Tommy Tutone may not know is that 8,675,309 is a twin prime (8,675,311 is its partner), and it’s also the hypotenuse of a primitive Pythagorean triple: 8,675,309^{2} = 2,460,260^{2} + 8,319,141^{2}.

There are lots of mnemonics for remembering myriad things, both mathy and not:

- PEMDAS (order of operations)
- SOHCAHTOA (trig relationships)
- HOMES, or Super Man Helps Every One (Great Lakes)
- Every Good Boy Deserves Fudge (notes on the musical scale)
- ROY G. BIV (colors of the spectrum)
- Spring Forward, Fall Back (setting your clock for daylight savings)
- Lo d(Hi) Less Hi d(Lo), Draw the Line and Square Below (derivative of a quotient)

I think that mnemonics used to remember numbers are more interesting than those used to remember other things, though, primarily because they are often individually created and very personal; that is, they’re used to remember a specific address, phone number, or other piece of information that’s only pertinent to you.

So, tell me — **what’s your best mnemonic**, and what number does it help you remember?

### Live Longer: Have More Birthdays

Satchel Paige asked, “How old would you be if you didn’t know how old you were?”

There are lots of quotes about aging. Age is an issue of mind over matter, they say — if you don’t mind, it doesn’t matter. Or, you’re only as old as you feel. But the following is my favorite quote about age:

Anyone who stops learning is old, whether at twenty or eighty. Anyone who keeps learning stays young. The greatest thing in life is to keep your mind young.

With each passing year, I learn more and more things. But I also learn more about how much I’ll never know.

The number of years on Earth is not an accurate measurement of a life. Many people fill their entire lives with trivial matters.

A graduate student saw a professor working on a proof of the Riemann hypothesis. On the professor’s desk were thousands of papers with various notes about the problem.

“My goodness,” said the student. “Have you been working on this problem your whole life?”

“Not yet,” said the professor.

And what is age, anyway? It’s just a number. For instance, Paul Erdös claimed to be two and a half billion years old.

“When I was a child, the Earth was said to be two billion years old,” he said. “Now scientists say it’s four and a half billion. So that makes me two and a half billion.”

Now in the computer age, it seems that no matter how much we know, machines may know more than we do.

A computer manufacturer unveils a new computer that supposedly knows everything.

A skeptical man asks, “How old is my father?”

The computer thinks, then says, “Your father is 57 years old.”

“See?” says the man. “This is nonsense. My father has been dead for 20 years, and if he were alive, he’d be 71.”

“No,” replies the computer. “Your mother’s husband has been dead for 20 years. Your father is only 57, he’s currently fishing on Lake Michigan, and he just landed a three-pound trout.”