## Archive for March, 2011

### MJ4MF Review

I was delighted to open the April issue of the *Mathematics Teacher* journal and discover a review of *Math Jokes 4 Mathy Folks*. Reviewer Leah Evans had some nice things to say:

Math Jokes 4 Mathy Folksis a delightful read. The author has compiled a vast array of puns, quips, and jokes meant for people of varied ages and mathematical expertise.[…]

I highly recommend this book as a diversion from the rigor of mathematics. It allows us to have a good laugh (or a long groan) at a joke that only “math nerds” would get, and it points out that humor can be found in all mathematical applications, even a telescoping series.

I love the cliffhanger at the end! The reference to a “telescoping series” is — I think — in regards to this joke (p. 89):

If you’re interested, here’s a copy of the entire review. (Click on the image to view a full-size version.)

### Lead by Counterexample

A wonderful, handwritten sign hangs on the door to my friend’s office:

There is no way of falsifying, “Unicorns exist.”

This is a good example of the nature of mathematics. Although providing just one counterexample is proof that something is not true, the inability to provide a counterexample is not proof that something is true. Understanding this concept is fundamental to understanding mathematics. Perhaps this guy could use a refresher…

At a conference, a mathematician proves a theorem. A young woman in the audience interrupts him. “Excuse me, sir, but I believe that your proof must be wrong. I have found a counterexample to your theorem.”

“Not a problem,” the speaker replies. “I have another proof.”

### Mitch Hedberg’s Numerology

Comedian Mitch Hedberg died six years ago today, on March 29, 2005. He was just 37 years old.

He was known for one-liners, and one of my favorites involves data analysis (sort of):

I went to a pizzeria, I ordered a slice of pizza, and the [guy] gave me the smallest slice possible. If the pizza was a pie chart for what people would do if they found $1,000,000, [then he] gave me the “donate it to charity” slice. I would like to exchange this for the “keep it” slice, please!

Here’s an MJ4MF original, based on one of Hedberg’s lines:

Sometimes in the middle of the night, I’ll wake with a profound result or an elegant proof, so I keep a pen by my bed to write such things down. But sometimes, if the pen’s been moved, I might lie awake for hours trying to convince myself that my thoughts weren’t really that profound or the proof wasn’t really that elegant.

Here are a few other Hedberg lines that are slightly mathematical:

My lucky number is 4,000,000,000. Unfortunately, it doesn’t come in real handy when I’m gambling. “Come on, 4,000,000,000! Aw, f**k! Seven. Not even close. I need some more dice. Four billion divided by 6, at least.”

I angered the clerk in a clothing shop today. She asked me what size I was, and I said, “Actual.” Because I am not to scale.

I hope the next time I move, I get a real easy phone number, something that’s real easy to remember. Something like 222‑2222. I would say, “Sweet.” People would say, “Mitch, how do I get a hold of you?” I’d say, “Just press 2 for a while. And when I answer, you’ll know you’ve pressed 2 enough.”

That last one reminds me of a classic math joke:

We’re sorry. The number you have dialed is imaginary. Please rotate your phone 90° and try again.

### Plainly Stated

One of my favorite applets at Illuminations is the State Data Map, which allowed me to create the following map depicting the number of U.S. Presidents born in each state:

Note that the states are color‑coded. Those states in which the greatest number of Presidents were born are the darkest shade of red; those in which no Presidents were born are white. In addition to allowing you to enter data, there are also pre‑loaded data sets. My favorite is the “Letters in State Name” set, from which I concocted the following trivia questions:

- Which state names have the most letters?
- Which state names have the fewest letters?

Feel free to think about it a few seconds before reading the next paragraph.

As it turns out, there are three states whose names contain 13 letters, and there are three states whose names have 4 letters. For what it’s worth, the mean number of letters is 8.24, and the median is 8.

My sons have a collection of foam letters for the bath tub. When the letters get wet, they stick to the side of the tub, and Alex and Eli love to use the letters to spell the names of states. Tonight, Eli spelled WYOMING. We then played a game where I’d give them the name of a state, and they’d try to spell it — but they couldn’t spell many of the state names because the set contains only one copy of each letter of the alphabet. This led to the following trivia question:

- Which states have names that can be spelled with bath tub letters, i.e., the state name contains no repeated letters?

Feel free to cogitate on that a while, too, then read on.

There are nine states with no repeated letters in their names. (Don’t feel bad if you weren’t able to identify all of them. I had to look at a map.)

Finally, here is a state trivia question a pro pos of absolutely nothing. For each pair of states below, identify the only state that borders both of them. (Each question has a unique answer.)

- North Carolina, South Carolina
- South Dakota, Illinois
- New Mexico, Missouri
- Oregon, Wyoming
- Missouri, West Virginia
- Wisconsin, Ohio

For the answers to all questions, check a map.

### Good Math in a Bad Economy

Times are tough. In February, the unemployment rate was 9.5%, and the U.S. cost of living hit an all‑time high. Some folks will take any job just to make ends meet…

A young man reported to the supermarket for his first day of work. The manager greeted him with a smile and a warm handshake, then promptly handed him a broom and said, “Your first task is to sweep the entire store.”

“But, I have a PhD in mathematics,” said the young man indignantly.

“Oh, I’m sorry,” said the manager. “Here, give me the broom — I’ll show you how.”

According to the Mortgage Bankers Association, at least 8 million Americans are at least one month behind on their mortgage payments. Still, some folks are trying to re‑pay their debts…

A mathematician and an engineer are walking down the street. A mugger approaches, pulls out a gun, and demands their money. “Just a second,” says the engineer. He takes out his wallet, turns to the mathematician, hands him a bill and says, “Here’s the $20 I owe you.”

### What is Your Favorite Number?

The WordPress Post-A-Week Challenge sends me a daily topic idea to consider for blog posts. Often, the prompts are not appropriate for a math jokes blog. For instance, some recent prompts have been:

- Grab the nearest book (or website) to you right now. Jump to paragraph 3, second sentence. Write it in a post.
- How do you find your muse?
- If you could bring one fictional character to life for a day, who would you choose?

But today’s prompt landed in my wheelhouse:

What is your favorite number, and why?

When Art Benjamin appeared on the Colbert Report, he said that 2,520 was his favorite number when he was a kid. When Stephen Colbert asked him why, he replied, “It was the smallest number that was divisible by all the numbers from 1 through 10.”

Tonight, I asked my twin sons Alex and Eli what their favorite numbers are.

Eli: 5, 15, 55, because my favorite number is really 5, but 15 and 55 are triangular numbers that have 5’s in them.

Alex: 21, because my favorite numbers used to be 1 and 2, and because it’s the number of cards you deal when we play Uno (3 players, 7 cards each).

My favorite number is 153, for lots of reasons:

- It is the smallest non-trivial Armstrong (or narcissistic) number — that is, it is an
*n*‑digit number that is equal to the sum of the*n*th powers of its digits: 1^{3}+ 5^{3}+ 3^{3}= 153. - Its prime factors are 3 and 17, and my birthday is 3/17.
- It is a triangular number. (Consequently, it’s the sum of 1 + 2 + 3 + … + 17.) As 351 is also a triangular number, 153 is also a reversible triangular number.
- It is the sum of the first five factorials: 1! + 2! + 3! + 4! + 5! = 153.
- The sum of its digits is 9, and the sum of its proper divisors is 9
^{2}. - It is one of only six known truncated triangular numbers, which means that 1, 15 and 153 are all triangular numbers.

Mathematician John Baez claims that his favorite numbers are 5, 8, and 24.

Got a favorite number? Share it, as well as the reason it’s your favorite, in the comments.

### Why Metric is 10 Times Better

Asking why the U.S. has not switched to the metric system is almost as pointless as asking why we still observe Daylight Savings Time.

Some folks still have trouble converting between the two systems, but there are celebrity mnemonics that can be used to help remember typical conversions. For instance, 1 Ezra Pound ≈ 454 Billy Grahams.

Here’s a trivia question for you: Besides the U.S., what other countries have not officially adopted the metric system? Are you ready for the answer? According to the U.S. Metric Association, Liberia and Myanmar are the only two additional hold‑outs of significance.

Truth be known, the U.S. has gone metric. The yard, the pound, and the gallon are now officially defined by reference to metric units. In 1975, the federal government adopted metric as the nation’s “preferred measurement system.” Though the United States Metric Board was created to manage the transition, the only noticeable change by the early 1980s was that liquor and wine were labeled in liters.

But seriously, there are plenty of great reasons why we haven’t switch to *Systeme Internationale*:

- Referring to football as a “game of millimeters” just doesn’t have the same ring to it.
- Inchworms will become centipedes.
- Meter sticks are very stubborn — they won’t give an inch!
- Currently, an ounce of prevention is worth a pound of cure. The conversion to metric would mean that a gram of prevention is worth approximately one‑sixtieth a kilogram of cure. Yuck.
- Nobody wants to go traipsing all over Hell’s half‑hectare.
- Cemetery workers would strike if they were asked to bury the dead “six meters under.”
- You could no longer love someone a bushel and peck. You would have to love them 37.9 liters.
- The famous barroom reprimand, “Mind your p’s and q’s” (pints and quarts), would become, “Mind your h’s and l’s” (half‑liters and liters).

Seriously, folks, it’s high time we made the transition. People opposed to metrication are just being *de‑feet‑ist*.

### Drink It Green!

Today is St. Patrick’s Day, of course, but at 9:33am, my mother’s baby boy celebrated the 40th anniversary of his birth. (Yes, I’m a cliche. I was born on St. Patrick’s Day, and my parents named me Patrick. But it was for the best, really — before my exact birthdate was known, the leading name candidate was George William, Jr. *Yecch.*)

Here’s an appropriate math joke for a beer-filled holiday:

An infinite number of mathy folks walk into a bar. The first one goes up to the bar and orders a pint. The second one goes up and orders half a pint. The third orders 1/4 pint. The fourth orders 1/8 pint, and so forth. A little overwhelmed, the bartneder asks them, “Hey, are you guys sure you want to do this? Isn’t that a bit much?”

The mathy folks reply, “Don’t worry. We know our limits.”

And some other jokes for a day dedicated to drinking…

Charles Dickens walks into a bar and orders a martini. The bartender asks, “Olive or twist?”

René Descartes is sitting in a bar. The bartender asks if he’d like another. “I think not,” says Descartes, and he promptly disappears.

An absent-minded math professor walks up to an attractive woman at the bar. “So, tell me,” he says, “do I come here often?”

A guy walks into a bar with a lizard on his shoulder. “What do you call that?” asks the bartender. “Tiny,” says the guy, “because he’s my newt.”

Okay, enough of that nonsense. Happy St. Patrick’s Day! Have a pint or two for me.

### Prime Curio Redux

As I mentioned in yesterday’s review of *Prime Curios*, the book contains a lot of interesting facts about prime numbers. In fact, it contains so many intereting tidbits that I was still reading three hours after posting my review. On page 202, I discovered a rather interesting curio:

968,666,869

The smallest palindromic prime with embedded beast number whose digits contain circles, i.e., using only the digits 0, 6, 8, 9.

What struck me about this curio was the embarassingly small size of the set under consideration. If you consider the four subcategories contained within the description — prime, palindrome, embedded beast number, and using only the digits 0, 6, 8, and 9 — the intersection of those groups is miniscule.

Consider each piece in turn. To limit the discussion, let’s only worry about integers less than one billion, since the prime number described above falls below that threshold.

**Prime**

Rumor has it that there are 50,847,478 prime numbers less than 1,000,000,000. (That value seems reasonable, given that the Prime Number Theorem suggests that there should be about 10^{9}/ln(10^{9}) ≈ 48,254,952.) In other words, about 5.084% of the positive integers up to 1,000,000,000 are prime.

**Palindrome**

In general, there are 9 × 10^{[(n + 1)/2]} palidromes with *n* digits. That means that are there 9 + 9 + 90 + 90 + 900 + 900 + 9,000 + 9,000 + 90,000 + 90,000 = 199,998 palindromes less than 1,000,000,000. In other words, less than 0.020% of those numbers are palindromes.

**Embedded Beast Number**

Analyzing this part was pretty cool. How many numbers less than 1,000,000,000 have an embedded beast number, that is, how many numbers have a string of three consecutive 6’s? It took about an hour of playing to find a general formula. For positive integers with *n* digits, the number of positive integers with an embedded beast number is:

10^{n – 3} + 8 × 10^{n – 4} + (*n* – 4)(9^{2} × 10^{n – 5})

That formula revealed that there are 6,400,000 positive integers with an embedded beast number less than 1,000,000,000, or only about 0.640%.

[**update – 3/16/2011**]

As noted by Joshua Zucker in the comments, this formula is incorrect. It fails to remove numbers that have more than one string of three consecutive 6’s. As Josh noted, there are only 42,291 seven‑digit beast numbers (the formula gives 42,300), and there are only 503,748 eight‑digit beast numbers (the formula gives 504,000). I will try to correct this within the next several days.

**Circle Digits**

If the digits in a number are limited to just 0, 6, 8, and 9, then there are 262,143 positive integers with only circle digits less than 1,000,000,000, or a mere 0.026%.

So, what does all this nonsense get us? It says that the probability of a positive integer less than one billion being a palindromic prime with embedded beast number whose digits contain circles is approximately

*P* = 0.05084 × 0.00020 × 0.00640 × 0.00026 = 0.000 000 000 019 523,

or, in layman’s terms, really frickin’ small.

With the odds at about 1 in 50,000,000,000, it’s no suprise that the first occurrence of this type of number is just shy of a billion at 968,666,869.

### Book Review: *Prime Curios*

At the Virginia Council of Teachers of Mathematics (VCTM) conference in Richmond last Friday, I had an incredibly fun time giving an hour-long presentation titled (what else?) *Math Jokes 4 Mathy Folks*. It was a mixture of stand-up comedy, number tricks, and a brief explanation of why a laughing student is a learning student.

The joke that received the best response? I posted the following image…

…and then I said, “Holy shift! Look at the asymptote on that mother function!”

After my presentation, a gentleman approached me, turned to a page in a book, and read a joke to me:

What do you call someone who hunts for Mersenne primes only for the prize money?

A Mersennary.

He then handed me a copy of a fascinating little book, *Prime Curios: The Dictionary of Prime Number Trivia*.

Turns out, the gentleman was G. L. Honaker. He and Chris Caldwell, a professor of mathematics at University of Tennessee-Martin, are the authors of *Prime Curios*. Chris maintains The Prime Pages, a web site with 20,000 pages of prime number trivia.

The book contains quite a few nuggets worth mentioning:

- If you concatenate the positive odd integers from 1 to 97, the result is a prime number.
- The chance that no pair of 53 people in a room have the same birthday is approximately 1/53.
- The prime number 369,119 divides the sum of all prime numbers less than 369,119.

The book contains more curios like this… in fact, there are 2,148 more of them, according to the description on the back of the book. It’s a fun book, especially if you’re a big number dork like I am. Mathy folks might enjoy it, but for sure you should check out The Prime Pages.