Out of Sequence

If you came here looking for a good math joke and were thinking, “My God, I sure hope he didn’t post another story about his sons,” well, now would be a good time to close your browser.

Still with me? Good. Because I’ve just got to tell you about two things that have happened recently with Alex and Eli.

As we were walking through the neighborhood today, I was pointing to things that I thought the boys would enjoy. “Look,” I said as I pointed to the sky. Alex looked up. “An airplane!” he shouted.

I pointed to an automobile on the right. “A red car!” Eli exclaimed.

Then I pointed to a house on our left. Alex announced, “A square Fibonacci number!”

I admit, I didn’t see that coming. I had pointed to a house on our street, and there was a cat on the front porch that I wanted the boys to see. Instead, Alex saw the house number — 144 — and identified it as part of two different sequences. (I don’t think he recognized the lovely coincidence that 144 is the 12th term in both sequences, though I can’t be sure.)

My boys’ love of sequences is a result of teeth brushing. (No, really.) To make sure they brush their teeth long enough, I count while they brush — 15 seconds on the left side, 15 seconds on the right side, 15 seconds for the front teeth, and finally 15 seconds for “all around,” during which they’re supposed to brush the inside parts of their teeth. As you might well imagine, though, I was getting bored counting 1, 2, 3, …, 15 four times every night. I started to mix it up.

• I counted 1‑15 for the left side, then 16‑30 for the right, 31‑45 for the front, and 46‑60 for all around. But that got boring rather quickly, too.
• I switched to counting respectively by 1, 2, 3, and 4; that is, I’d count 1‑15 for the left, 17‑45 (by 2’s) for the right, 48‑90 (by 3’s) for the front, and 94‑150 (by 4’s) for all around. That was unsettling, though, because the boys started to think that 94, 98, 102, …, 150 were multiples of 4. While they recognized that counting by 2’s from 17‑45 gave the odd numbers, they weren’t able to discern that 94‑150 by 4’s analogously gave numbers n ≡ 2 mod 4. Nor did I expect them to — they’re only 3½.
• I therefore switched it up again and counted by 1’s, 2’s, 3’s, and 6’s, which meant that “all around” was now 96, 102, 108, …, 150, which provided the more satisfying pattern of multiples of 6.

Then one night, Eli shocked me. He said, “Daddy, I figured out another way to count by 5’s.” I wasn’t really sure what he meant, but I assumed that “counting by 5’s” was a general term he was using to refer to skip counting. I asked him to explain. He said, “Like this: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55.”

Upon hearing this, I thought what any geeky American father would think. “Holy sh*t. Did my three-year-old son really figure out the triangular numbers by himself?”

So I asked him, “What are those numbers, Eli?”

“Well,” he began, “there’s 1,” and then 1 + 2 = 3, and 1 + 2 + 3 = 6, and 1 + 2 + 3 + 4 = 10, and…

Through a few more questions, I realized that he had ascertained these sums using my old TI‑83 calculator. I am proud to share a picture of the boys happily playing with technology:

The boys had been playing with some of my graphing calculators because they liked typing letters to spell words. Little did I know they were entering expressions and learning some math, too. (Take that, all you stodgy opponents of calculators in the classroom!)

After that, there were no holds barred; during teeth brushing, I started busting out all kinds of sequences:

• Triangular numbers: 1, 3, 6, 10, …, where T(n) = ½(n)(n + 1).
• Square numbers: 1, 4, 9, 16, …, where S(n) = n².
• Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, …, where F(n) = F(n – 1) + F(n – 2).
• Feeby numbers: 11, 22, 33, 44, 55, …, where Fb(n) = 11n.
  (These are just the multiples of 11, but Eli named them the Feeby numbers, I think because it sounded a little like the first part of Fibonacci numbers, and two of the numbers in the list (55, 66) are elements of the Fibonacci sequence.)
• Perrin sequence: 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, …, where P(n) = P(n – 2) + P(n – 3).

I’ve been thinking about sharing Moser’s Circle Problem with them and showing them the sequence 1, 2, 4, 8, 16, 31, 57, 99, …. Then again, maybe not. 

Okay, so if you’re one of the folks who came for a math joke and tolerated that entire story, then by gosh you deserve a math joke. I don’t have a joke about triangular numbers (note to self: create a joke about triangular numbers), but I do have a joke that involves the word triangular.

What is small, green, and triangular?
A small, green triangle.

Yeah, I know… it’s lame.

Let me try to make amends with a cool triangular number problem.

Append the digit 1 to the end of every triangular number. For instance, from 3 you’d get 31, and from 666 you’d get 6,661. Now take a look at all of the divisors of the numbers you’ve created. What are the units digits of the divisors for every number created in this way? Can you prove that this result always holds?

I have a proof, but you’ll have more fun solving it on your own than reading my solution.

Just for Fun

Just a few jokes without a theme…

Sign hanging on the IT department wall:
“Theory — you know everything, but nothing works. Practice — everything works, but nobody knows why. In our department, we merge theory with practice: nothing works, and nobody knows why.”

A doctor tells a woman, “I have some bad news. You only have six weeks left to live.”
“That’s terrible!” exclaims the woman. “Doc, what should I do?”
“Are you married?” he asks.
“No.”
“Then find yourself an actuary, and marry him.”
“Will that help me live longer?” she asks.
“Well, no,” the doctor says. “But it’ll feel longer.”

How is that I can remember π to 100 digits and that the curl of the curl equals the gradient of the divergence minus the Laplacian… but I can’t remember where I park my car at the mall?

Top 6 Math Songs

On cold days, I look for creative ways for my sons to burn energy indoors. If I were forced to give it a title, today’s game would be called, “Up and Down the Stairs with a Song.” Eli ran up the stairs to the second floor while Alex ran down the stairs to the basement; then they both returned to the main level where they sang a song; then each boy ran up or down the other set of stairs; and, finally, they returned to the main level and rang the “dinger,” a bell included with one of their toddler games.

During the game, the song that Eli sang was mathematical:

1, 2, 3,
4, 5, 6, …
That’s how the numbers go.

7, 8, 9,
10, 11, 12, …
That’s all you really need to know.

13, 14, 15, 16, …
On and on they go.

17, 18, 19, 20, …
Gotta line ’em up just so!

The song is from “1-2-3, Count With Me,” a Sesame Street video starring Ernie (sans Bert). My favorite song from the video is Martian Beauty, but sadly, it just doesn’t hold the same appeal for Eli and Alex.

There have been lots of math songs through the ages, and the number has risen exponentially with YouTube. Generally, math songs are humorous. (Maybe because no one would listen to a math song that wasn’t funny?) Below are my top five six.

[update]
6. 7 8 9 – Barenaked Ladies

Thanks to Joshua Zucker for reminding me of this gem! How could I have forgotten? One of the moldiest of all oldie math jokes turned into a song. Shame on me for not including this in the original “Top 5 Math Songs” list.

5. That’s How the Numbers Go – Ernie (Sesame Street)

This is the song from the Sesame Street video “1-2-3, Count With Me.” I worried that it wasn’t sophisticated enough for readers of MJ4MF, but if Steven Strogatz can reference the video in a column for the New York Times, well, that’s credibility enough for me.

4. Lateralus – Tool

Even if you don’t like the genre, Lateralus by Tool gets big props for its intricate use of the Fibonacci sequence. The time signatures of the chorus change from 9/8 to 8/8 to 7/8. Drummer Danny Carey said the song “was originally titled 9-8-7 for the time signatures. Then it turned out that 987 was the 17th number of the Fibonacci sequence. So that was cool.” They exploited this relationship in several ways.

• The number of syllables in the verses follow the pattern ‎1, 1, 2, 3, 5, 8, 5, 3, 2, 1, 1, 2, 3, 5, 8, 13, 8, 5, 3, which rise and fall with the Fibonacci sequence.
• The song mentions spirals several times.
• The first word is sung at 1:37 into the song, and 97 seconds ≈ 1.618 minutes, which just happens to be the golden ratio, a number strongly associated with the Fibonacci sequence.

As it turns out, Tool has several other mathy songs, including Parabola, Forty Six and 2, and Cesaro Summability.

3. What You Know About Math? – Ethan Gilbert and Aaron Flack

Ethan Gilbert and Aaron Flack created a numerical sensation with What You Know About Math. With almost 4 million visits on YouTube, the song deserves a place on this list.

2. New Math – Tom Lehrer

This list could have easily been composed entirely of Tom Lehrer songs, since his titles include New Math, The Derivative Song, and Lobachevsky, but that seems unfair to the other artists who have contributed so much to the math music genre. So I tried to pick just one song that represents his body of work. (Personally, I think his best song is The Elements, but it’s not very mathy. And my apologies for linking to that particular video… I beg your forgiveness for including a link to a video that spells Lehrer’s name wrong and mistates the song title, but I chose it because it nicely displays the lyrics while the song plays.)

1. Finte Simple Group of Order Two – The Klein Four

The lyrics of Finite Simple Group of Order Two contain enough bad math puns to keep an undergraduate math major chuckling an entire semester. Doesn’t hurt that these guys are pretty good singers, too.

Tickling the Ivies

So tonight, I was supposed to attend an event called Dinner With Interesting People at the University of Pennsylvania. And guess what? I was going to be the interesting person!

I would have been speaking to students at Ware College House about possible careers in math education. For those of you who are interested in this subject, the Guide to Career Education is a great tool for finding courses and training programs when pursuing a math-related career. Of course, my talk was going to include more than a few jokes, including this one:

3 miles of medical tubing at the University
of Pennsylvania hospital = 1 I.V. League

Though my talk would surely have kept the audience at the edge of its seat, sadly, it had to be cancelled. That bummed me out. While the talk wouldn’t have earned me an honorary doctorate, I had planned to acquire a University of Pennsylvania sweatshirt while on campus. Which reminds me of a story.

In Possible Side Effects, writer Augusten Burroughs tells about a confrontation he had with a Harvard grad. Wearing a Harvard t‑shirt, he stepped into the elevator in his apartment building, and a woman stepped into the elevator with him. He wrote:

“Did you go to Harvard?” some bitch I’d never seen before in the elevator of my building asked. Something in her tone almost made it sound like, “You didn’t go to Harvard.” That tiniest note of accusation put me on the defensive. But I also loathed the idea of speaking to her just because of her smug haircut.

“Uh, no,” I admitted. “I just…” and let the sentence die in my mouth, hoping I wouldn’t have to explain why.

“Just what?” she said, now crossing her arms across her chest. She was smiling, but it was clearly a hostile smile.

“Well, I just like college t‑shirts,” I said. And I smiled, too. But I did so in a way that I hoped make me look friendly and not terribly intelligent. Thus, harmless.

[…]

“Don’t you think that’s a little deceptive?” she asked.

“Deceptive?” I said.

“Like telling someone you’re a doctor or a police officer when you’re nothing of the kind.”

Like many people in New York City, she was bossy and had a raging sense of entitlement. “Well,” I said, darker now. “I don’t think it’s any more deceptive than wearing four‑inch come‑f**k‑me pumps when one has no intention of ever f**king anybody.” I smirked and looked down at her pumps, then at her pinched, tight little spinster mouth.

That shut her up, and she walked out of the elevator. I could feel the heat of resentment wafting off her flesh.

And I felt guilty. Because the truth is, not only did I not go to Harvard, I didn’t even go to high school.

Truth is, if people saw me in a Penn sweatshirt and assumed that I was a Penn grad, I’d probably have felt a little guilty, too. “No,” I’d have to explain. “I’m actually a graduate of Penn State. We may not be as smart, but we’ve got a much better football team.” So maybe it’s for the best that the talk was cancelled…

Hail to the Chief

The following quote comes from Hugo Rossi, Professor Emeritus of Mathematics, University of Utah:

In the fall of 1972, President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.

That’s fantastic — and quite appropriate for President’s Day.

Presidents and mathematics have a long history together. Of course, everyone knows that James Garfield created a novel proof of the Pythagorean theorem using a trapezoid. But did you know that Andrew Johnson had no formal education and learned math (as well as reading and writing) from his wife? [update] And after graduating from the Naval Academy, Jimmy Carter, who tutored his midshipman classmates in mathematics, did graduate work in nuclear physics at Union College.

In honor of President’s Day, here are some of my favorite math jokes about presidents. (These jokes happen to reference George W. Bush and Barack Obama, but feel free to tell them about Bill Clinton, Ronald Reagan, Jimmy Carter, Millard Fillmore, or any other president whom you hold in high disdain.)

At his morning security briefing, President Bush was informed that there had been a tragedy. “Sir,” said his security advisor, “three Brazilian soldiers were shot last night.”
“Oh, my God,” says Dubya, his head dropping into his hands. After a moment, he collected himself and asked, “How many is a brazillion, anyway?”

“It is my understanding,” President Obama said at a news conference about education, “that math educators are frequently teaching algebra classes in which their students learn how to solve equations with the help of radicals. I can’t say that I approve of that…”

Who succeeded the first President of the United States?
The second one!

I Like My Women Like I Like My Math

My father had a predilection for embarassing my mother. For instance, when a waitress would ask how he’d like his coffee, he’d say:

I like my coffee like I like my women — hot, sweet, and light.

My father was the original bigot chauvinist. Had I been taking notes, I could have written Sh*t My Dad Says twenty years ago.  Here are a few other similes that I think my father would have appreciated:

I like my women like I like my mathematics — pure and beautiful, not complex and irrational.

I like my women like I like my math tests — full of problems, and easy to cheat on.

I like my women like I like my math problems — simple and easy.

I like my women like I like my calculus textbooks — full of curves.

I like my women like I like my research papers — interesting, intelligent, and covered in ink.

I like my women like I like my data — average, and within my range. 

I like my women like I like my equations — well-balanced.

I like my women like I like the tenth positive odd number — prime, and over 18.

I like my women like I like my ellipses whose major and minor axes are almost equal — just a little eccentric.

I like my women like I like the roots of x² + 2x + 1 = 0 — degenerate, and easy to find.

I like my women like I like my calculator-dependent students — with no interest in multiplying.

Places You Wouldn’t Want to Live on Valentine’s Day

Be thankful that you don’t live in the Gem State today:

Idaho law makes it illegal to give your sweetheart a box of candy weighing less than fifty pounds.

That’s a direct quote from You Can’t Eat Peanuts in Church and Other Little-Known Laws, a book by Barbara Seuling, originally published in 1975. The book is now out-of-print, but I was able to find it at a local used bookstore.

I never verified that the Idaho law is still on the books. My rationale — who cares if it’s still in effect? The fact that it was ever a public law is satisfying enough for me. Turns out, there are other places where it would be bad to live on Valentine’s Day. Maryland law puts a time limit on affection:

A kiss can last no longer than one second in Halethorpe, Maryland.

And shaving is an apparent necessity for every romantic Hoosier:

In Indiana, a mustache is illegal on anyone who “habitually kisses human beings.”

(As best I can tell, there is no law against sporting a ‘stache if you habitually kiss other species.)

It’s good that those three laws were not intended for the same locale. Can you imagine having to shave and then lug a 50-pound box of chocolate to your sweetheart’s, just for a peck on the cheek that lasts less than a second? No, thank you.

Mathy folks are often accused of being less romantic than the average person, but I don’t think that’s true. Case in point — I once knew a mathematician who loved his wife so much that he almost told her!

My college roommate was a set theorist. On Valentine’s Day, he gave his girlfriend the following note:

I ∈ U

She was an engineering major and knew a fair amount of math, but he had to translate for her: “I belong to U.”

Mean and Standard Deviation

As a follow-up to yesterday’s post, here’s a poem titled Mean and SD by Norman Chansky, professor emeritus at Temple University. Ostensibly, the poem first appeared in the Journal of Irreproducible Results, though I was unable to find an exact citation.

The mean is a measure of location,
The center of a population.
If at random a score you drew,
The mean’s the most likely score you’d view.

You can compute the mean in your slumber:
Sum the scores, and divide by the number.  
At the mean, sample scores converge;       
From the mean, these scores diverge.       
Near the mean, the scores are many.
In the tails, there are hardly any.         

But to measure a distribution’s variation,
From the mean, find each score’s deviation.
Each difference of D score, now you square.
Sum all D scores, all scores’ share.
Now this sum, divide by N.
That’s V, the variance, then.

The square root of V is called SD,
The gauge of a trait’s variability.
We’ve found two moments of a distribution,
Developed from each score’s contribution.

Picturing a universe, try to see:
Its center, the mean; its orbit, SD. 

Statistically Speaking…

My favorite quote from Lewis Carroll happens to be one of my favorite quotes:

If you want to inspire confidence, give plenty of statistics. It does not matter that they should be accurate, or even intelligible, as long as there is enough of them.

Here are a few statistical facts worth noting:

One of every four mathy folks suffers from mental illness. Now, think of three calculating friends. If they’re okay, then it’s you.

Fifty percent of Americans have an understanding of statistics that is below average.

69.8724% of all statistics reflect an unjustified level of precision, and 83.85% of all statistics are made up on the spot.

There are two kinds of statistics — the kind you look up, and the kind you make up.

There are three kinds of statisticians — normal, deviant, and skew.

Edison’s Birthday

Today’s date is kind of cool — 2/11/2011. It also happens to mark the 174th anniversary of the birth of inventor Thomas Alva Edison.

To honor his birthday, Google posted the following image this morning:

As it turns out, Edison was not particularly fond of mathy folks.

Edison once hired a math graduate to assist him. One day, he gave the assistant a light bulb and asked him to calculate its volume. The assistant determined the equation of a curve approximating the cross section of the bulb, then used calculus and his slide rule to find the volume of revolution. After many hours of calculating and checking his work, he presented his results to Edison.

“You’re off by at least 30%,” said Edison. “I just calculated the volume myself by taking the top off and filling the bulb with water.”

Edison’s thirst for knowledge was never quenched. He once visited a science institution and was asked to sign a guest book. The first column asked for the visitor’s name, under which he wrote, “Thomas A. Edison.” Another column asked, “Interested in?” under which he wrote, “Everything.”

Edison had a sense of humor. In honor of telegraphy, he bestowed the nicknames Dot and Dash on his first two children, Marion and Thomas Jr.

He was also an incredibly efficient man. At his summer residence, there was a heavy turnstile through which visitors had to pass to reach the main path to the house. A guest once asked Edison why, with all his clever inventions, he couldn’t have installed a less laborious turnstile. Edison explained that everyone who operated the turnstile helped to pump eight gallons of water into the tank on his roof.