## Archive for June, 2012

### A Perfect State

A perfect number is a positive integer that is equal to the sum of its proper positive factors.

Perfect numbers, like perfect individuals, are very rare.

– Rene Descartes

They are very rare, indeed — there are only five perfect numbers less than 1,000,000,000. Because 6 and 28 are two of them, you might say that today is a perfect day.

Perfect Square: A nerd who never makes mistakes.

I recently coined the term *perfect state* to refer to a state for which the number of letters in its name is equal to the number of letters in the name of the state capital.

To pass the time on a recent car trip, I asked my sons to see how many perfect states they could find. During their search, they identified many states that were not perfect, and they giggled gleefully when I referred to them as *abundant* (more letters in the capital than in the state) and *deficient* (fewer letters in the capital than in the state).

They were extremely excited to learn that our home state, Virginia, is a perfect state. This did not surprise me — with beaches to the east, mountains in the west, urban living in the north, rural country in the south, and a whole lot of wine country in between, I’ve often argued that Virginia is the perfect state. (In their song *Old Dominion*, local band Eddie from Ohio describes Virginia as “just southeast of heaven to the surf and the hills.” Yeah, that’s about right.)

Anyway, my sons were able to find Virginia and **seven other perfect states** without the help of a map. Can you?

Need some help? Check out this map with a color-coded solution. The deficient states are white, the abundant states are dark blue, and the perfect states are light blue.

### Conversion Perversion

“Be there in a jiffy.”

If someone says that to you, then you know that that person should arrive soon. But did you know that *jiffy* is a technical term? Similarly, the expression “two shakes of a lamb’s tail” used to indicate a short period of time, but the unit of time known as a *shake* now has a specific designation.

- 1 sec = 100 jiffies = 100,000,000 shakes

I’m big into conversions. I often tell folks, if you need to convert between televangelists and expatriate poets, the following picture may be helpful to you:

That is, 1 Ezra Pound ≈ 454 Billy Grahams.

The following are some other fun conversions.

- π sec ≈ 1 nanocentury

It’s interesting that this is so accurate. It is within 0.5%.

- 1 furlong per fortnight (FPF) ≈ 1 cm/min

This one is even better. The error is less than 0.000025%.

- 1 m/s = 1 Hz/dpt (Hertz/dioptre)

This is what can happen when common units are replaced with uncommon units. *Hertz per dioptre* is an inside joke among physicists and yet another reason not to hang out with them. (*Dioptre* is a unit of measure for the optical power of a lens.)

- 1 square = 100 square feet

The term *square* is used in the construction industry, typically to measure a roof. For example, if a roof has an area of 1,000 square feet, then the contractor would order 10 squares of shingles. But you wouldn’t want to use this unit in regular conversation, because it leads to awkward phrases like a “one-square square,” which would be a square that measures 10 feet on a side.

- 1 gal ≈ 3 + π/4 L

This is one of my favorite conversions. It’s accurate to 0.00000003%.

- 1 Hubble-barn ≈ 13.1 L

A *Hubble length* is the length of the observable universe (a very, very big length), and a *barn* is 100 square femtometers (a very, very small area), so it’s neat that their product gives a very tangible volumetric result.

- 1 stone = 14 pounds

When asked for my weight, I usually respond, “About 13 stones.” Such a reply leaves room for interpretation, and it could be assumed that I weigh as little as 175 pounds or as much as 189 pounds. And I’m fine with that. What kind of rude bugger asks your weight, anyway?

On a related note, the following formula can be used to approximate the U.S. population for a given year. Let *x* = the last two digits of the year, and let *y* = the projected U.S. population for that year (in millions). Then,

*y*= π*x*+ 276

This result is based on projections from the Pew Research Center. This formula provides an accurate estimate (within 1%) of the actual population for every year since 2000, and it should give a reasonable projection for the next several decades, assuming there are no major catastrophes.

### No Respect for Mathy Folks

As we were watching my sons playing in the yard, my wife said to me, “They’re such sensitive children. Let’s wait till they’re older to tell them you’re a math guy.”

I get so little respect, I feel like Rodney Dangerfield. (“During sex, my wife always wants to talk. The other night, she called me from the hotel.”)

I’ve always heard that math folks aren’t boring. We just get excited by boring things.

Here are some one-liners that I hope you won’t find boring.

**Have you heard the one about the interesting mathematician?
**Nope, me neither.

**How do you drive a mathematician insane?**

Tie him to a chair, and force him to watch you fold a roadmap the wrong way.

**What is the Golden Rule for passing actuarial exams?**

Always leave yourself enough time to

**How does a mathematician liven up a party?**

Leave.

**How can you tell that a mathematician is having a mid-life crisis?**

He gets a faster calculator.

**What are the two rules for making sure that you know more than your students?**

(1) Don’t tell them everything you know.

**Where are geometers buried?**

The Symmetry.

**Which state has the largest population of mathy folks?**

Mathachusetts.

**Did you hear that a new largest prime number was found?**

It’s three times as big as the previous one.

### Great Gift for a Math Dad

Before school let out for the summer, every student in Eli’s class made a Father’s Day gift for their dads. When I arrived home today, I found my gift in a lunch bag with the following note stapled to it:

(Eli signed his name. The rest of the note was written by his teacher as Eli dictated the message.)

Truth be known, Eli and Alex never win because I let them win. Sure, I may occasionally misplay a turn, but I don’t just tank an entire game on purpose. (On the flip side, I never deliberately cheat just to beat them, either, even though I could totally get away with it.) Primarily, I think kids know when you’re letting them win, and I believe it sends the message that you think they’re not capable of winning on their own. I also agree with psychologist Sara Diemerman who says, “There’s nothing like winning fair and square to make a kid feel terrific.”

I recently did a Game Night for the Northern Virginia Math Teachers Circle. During that meeting, participants played the following game:

Player A chooses an integer from 2 to 9 inclusive. Then Player B multiplies Player A’s number by any integer from 2 to 9, then Player A multiplies the result by any integer from 2 to 9, and so on. The first player to get a result greater than 1000 wins.

Have fun figuring out the winning strategy for that game.

As part of our Father’s Day activities, I plan to teach this game to Eli and Alex. But they’re going to have to earn their victories.

### Yo Momma Is So Bad At Math…

The following insult about yo momma is funny, I don’t care who you are.

There are 3 types of people in the world: those who can count, and yo momma.

Of course, it may not be understood by people who don’t recognize the reference, but who cares? Throwing out a “yo momma” joke is mostly for the entertainment of the insulter, not the insultee. And besides, why would you associate with people who don’t understand the reference?

One of the common jokes using the format above is…

There are 10 kinds of people in the world: those who understand binary, and those who don’t.

And the follow-up to that one is…

There are 10 kinds of people in the world: those who understand binary, and 9 others.

While we’re on the subject of binary, here’s one of my original “yo momma” jokes:

Yo momma is so dumb, she thinks binary is a two-headed canary.

There are lots and lots of “yo momma” jokes out there. But did you know there was an entire genre of yo momma math and science jokes? Google can help you find many, many more, but the following are some of my favorites.

Yo momma is so fat, she is proof that the universe is expanding exponentially.

Yo momma is so fat, her volume is an improper integral.

Yo momma is so infinitely fat, she can eat as much as she wants and not gain any weight.

Yo momma is so fat, she took geometry because she heard there was gonna be π.

Yo momma is so fat, the ratio of her circumference to diameter is 4.

Yo momma is so fat, in a love triangle she’d be the hypotenuse.

Yo momma is so nasty, the shortest distance between her and any person is 50 cents.

Yo momma is so ugly, Pythagoras wouldn’t touch her with a 3-4-5 triangle.

Yo momma is so dumb, she doesn’t know the difference between a doughnut and a coffee cup.

Yo momma is so dumb, she thinks crossing a mosquito and a mountain climber yields |mosquito| × |mountain climber| × sin(θ).

Yo momma is so dumb, she serves beer in Klein bottles.

Yo momma is so dumb, she thinks that if two people go into a hotel and three come out, the first two must have pro-created.

Yo momma is so far behind the times, she thinks the best feature of her solar-powered calculator is the flashlight.

Yo momma is so dense, she refracts light.

### Therapeutic Numbers

I was lying on my left side, my right leg awkwardly bent so that my right foot was flat on the floor in front of me, and my left leg was extended straight out underneath my bent right leg. There was a weight strapped around my left ankle, and I was lifting my left leg as high as I could. “How many?” I asked.

“Thirty,” said my physical therapist.

I’m not sure if she heard me gulp. I had only done eight so far, and already my thigh was screaming.

But that was nothing compared to the guy next to me. He was lying face-down on a table, his head and arms hanging off of one end. In each hand was a dumbbell, and he had to rotate his shoulder joint until his arms were parallel to the ground. After his first few, he asked, “How many?” She told him 30, too.

He did a few more, and his grunts were getting louder. “How many?” he asked again, but now with an air of incredulity.

“One-hundred fifty-two,” our therapist said. “That’s *always* the answer the *second* time you ask.” She smiled, then she looked at me. “I’m not sure why 152 is the number I pick.” It seemed reasonable that she’d want to explain her choice to me. After all, I am a numbers nerd. (Not a dweeb, geek, or dork. See below.) But then I realized she doesn’t even know what I do.

“It’s a fine number to pick,” I said. “After all, it’s evenly divisible by the sum of its digits: 152 / (1 + 5 + 2) = 19.”

She squinted a bit, and she raised one eyebrow slightly. I was undeterred.

“But 153 might be a better choice,” I continued. “It’s the sum of the first 17 counting numbers: 1 + 2 + 3 + … + 17 = 153. And if you raise each of its digits to the third power and then add them, you get 153 again: 1^{3} + 5^{3} + 3^{3} = 153.”

She now raised both eyebrows. Her head shook a little as she asked, “You just *know* that?”

“Yes,” I said. “And I often wonder how much useful information I could keep in my head if it weren’t filled with all this trivia about numbers.”

Then there was a long, silent pause. It probably would have been uncomfortable to a less mathematical, more socially adept individual. But not me. However, I felt bad when my therapist started to squirm, so I continued.

“What’s exceptionally cool, though, is that if you take any three-digit multiple of 3, and then add the third power of its digits, and then add the third power of the digits of the result, and keep doing that, you’ll always get back to 153.”

There was another long, silent pause.

The shoulder guy next to me finished his exercises. “What next?” he asked.

“How about some dumbbell presses,” she suggested.

“How many?”

She looked at me. “15*3*,” she said, with a little extra emphasis on the three.

### Overpaid Whistle Blowers

According to a story in The Charlotte Observer, the NFL offered increased salaries and a pension plan to referees. Barry Wilner’s article from June 4 stated:

The NFL made a seven-year proposal that offered increases of between 5 percent and 11 percent in wages per year. First-year officials who made an average of $78,000 in 2011 would earn more than $165,000 by the end of the new agreement. A 10-year veteran in 2011 who made $139,000 would get more than $200,000 in 2018.

[League spokesman Greg] Aiello said the NFL also offered a retirement arrangement under which each official would receive annual contributions starting at $16,500 and increasing to almost $23,000, plus a wide range of investment opportunities and expanded reimbursement for medical insurance costs.

Remember that these guys only work 16 days a year.

According to the Bureau of Labor Statistics, the average K-12 teacher makes $52,000 to $55,000 a year. That’s for 180 days of work, not to mention late nights grading papers and weekends preparing lessons for the coming week.

If any teachers want to tell these guys that they already make too much, you can voice your opinion at www.canyoubelievetheypaymetoblowawhistle.com.

### Garrison Keillor Reads Math Poem

As Garrison Keillor said, “Here’s a poem for today by Mary Cornish, entitled *Numbers*.” (Or maybe you’d prefer to hear GK read the poem on The Writer’s Almanac.)

**Numbers**, by *Mary Cornish*

I like the generosity of numbers.

The way, for example,

they are willing to count

anything or anyone:

two pickles, one door to the room,

eight dancers dressed as swans.

I like the domesticity of addition—

*add two cups of milk and stir*—

the sense of plenty: six plums

on the ground, three more

falling from the tree.

And multiplication’s school

of fish times fish,

whose silver bodies breed

beneath the shadow

of a boat.

Even subtraction is never loss,

just addition somewhere else:

five sparrows take away two,

the two in someone else’s

garden now.

There’s an amplitude to long division,

as it opens Chinese take-out

box by paper box,

inside every folded cookie

a new fortune.

And I never fail to be surprised

by the gift of an odd remainder,

footloose at the end:

forty-seven divided by eleven equals four,

with three remaining.

Three boys beyond their mothers’ call,

two Italians off to the sea,

one sock that isn’t anywhere you look.

### P(Winning Lottery) > 0… but Just a Little

I know a fair bit about the probability of winning the lottery.

State-run lotteries are a tax on the mathematically challenged.

Given the odds of winning, then you might wonder why I occasionally buy scratch-off lottery tickets. Lord knows, my wife often wonders aloud about it. Believe it or not, there are three reasons that I buy these tickets:

- First, I’m from a rural town in the-middle-of-nowhere Pennsylvania. The rural poor are infamous consumers of lottery tickets. Consequently, I believe that buying lottery tickets is part of my genetic code.

- Second, it’s a guilty pleasure that is easier to indulge than buying PowerBall or Daily Number tickets. When you buy one of those, there is a human interaction, and I imagine that the clerk selling me the ticket is thinking, “Loser! Don’t you know how low your odds of winning are?” For the scratch-off tickets, you insert your money in a vending machine, and the tickets are dispensed. Sure, you may get a disapproving eye from a passer-by, but at least there’s no formal exchange with another human.

- Third, and most importantly, I know a bit about probability, but I also know a little about the intersection of math and psychology. As it relates to the lottery, the idea is fairly simple — make every third or fourth ticket a winner, and people who buy scratch-off tickets will win often enough that they’ll keep coming back for more. Truth is, the winning tickets usually have a prize equal to the price of the ticket or twice the price of the ticket. For instance, if the tickets cost $5, then the winning tickets usually have a pay-out of $5 or $10. When four tickets are sold for $5 each, the state collects $20 and only pays out $5 or $10. Good work if you can get it, eh?
This last point is actually the one that hooks me in. If I buy four tickets at a time, I can almost guarantee that one of them will be a winner. Consequently, I’ll only be giving $10 or $15 to the state instead of $20. (What a bargain, right? I walked into the store with $20, and I get to leave with $5 or $10. Who could pass that up?) But on the off chance that there are two winners in this group of four, or if one of the tickets is a big winner with a prize of more than double the price, well, then, this could work out all right for me.

Yes, I am fully aware that my argument is irrational and that I am slightly delusional. Recognizing my irrationality and delusion, I don’t buy scratch-off tickets very often; but, I do buy them occasionally.

So, why am I telling you all this? Because this morning, I bought four scratch-off tickets at the local supermarket.

**First Ticket:** It had a “5 Times” logo next to $10. That means I won $50.

“Wow!” I thought. “I’m already ahead $30.” And then, of course, I realized how unlikely it was that the other three would be winners.

**Second Ticket:** I matched not one, not two, but *three* of my numbers to the winning numbers for $5 each. That means $15 in winnings on the second ticket.

“Holy schnikeys!” I said out loud, though probably too soft for anyone else to hear. (I hope.)

**Third Ticket:** I matched two numbers for $5 each. That means another $10.

**Fourth Ticket:** Nada.

But, whatever. I was up $55, so who cares about that stupid fourth ticket?

I collected my winnings, and I walked across the street to Panera and ordered a chai tea latte and a bagel. I handed my MyPanera card to the clerk — indicentally, I hate the recent trend of naming something as MySomething, because then it’s really awkward when I want to refer to the MySomething that belongs to me by calling it my MySomething; but, I digress — and he told me that I had earned a free bagel. “You can have this one for free, if you want,” he said. Well, hell yeah!

A few hours later, I went to lunch with a new professional acquaintance. Even though *I had asked her* if she wanted to meet for lunch, *she* picked up the tab!

Can you believe it? Fifty-five dollars in lottery winnings, a free bagel, and a free lunch. Financially speaking, this could have been the luckiest day of my life. (Well, except for the day when I learned that an essay I’d written had won a honeymoon in Oaxaca for my wife and me. But that’s a story for another day.)

Yep, the chances of one good thing happening in a day are low. But the chances of three good things happening in a day? Infinitessimal! Guess I’m just blessed.

My former boss, Jim Rubillo, knows a thing or two about probability and statistics, too. Somehow, his favorite line seems appropriate for this post.

If you don’t believe in the power of random sampling, then the next time your doctor requests a blood sample, tell her to take it all!