## Archive for March, 2011

### 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.

### A Cool Quick Trick for Pi Day

To some extent, I’m anti‑Pi Day. I think it has to do with the predictability of celebrations — everyone serves pie, does circle problems, and says things like, “I’m like π: irrational, but well-rounded!”

So, I was thinking that I would boycott Pi Day this year by not posting anything about the holiday on the MJ4MF blog. Then I discovered a cool trick. It was attributed to Martin Gardner on a web site, but I can’t verify the source. I think I’ve read every book by MG, and I’ve never seen it before.

Anyway, here’s the trick.

Write all 26 letters of the alphabet, but start with the letter J:

JKLMNOPQRSTUVWXYZABCDEFGHI

Then, remove all the letters that have vertical symmetry:

JKL N PQRS Z BCDEFG

Now, count the letters that remain in each subset: 3 1 4 1 6.

When I did this trick at a K‑12 math teachers’ conference recently, I wrote the numbers under each group. But I wasn’t sure that everyone would recognize the digits. So I drew an exaggerated decimal point between the 3 and 1, and I stated, “If you don’t know why this is relevant with Pi Day just around the corner, you’ve really missed the *point*.”

### Happy Anniversary!

Do you know what happened one year ago today? Well, lots of things, actually…

- A major storm near Hot Springs, AR, dropped baseball-sized hail while tornadoes raged nearby. (Yikes!)
- The first legal gay marriages in Washington, DC, were performed.
- Teen idol Corey Haim, best known for his role as Sam Emerson in
*The Lost Boys*, died of an accidental overdose.

But perhaps most importantly…

- The first post appeared on MJ4MF.

That’s right — I started sharing math jokes, random thoughts, and senseless drivel on this blog exactly 365 days ago. To those of you who read the MJ4MF blog regularly, comment occasionally, and forward links to your friends and colleagues, I want to say one thing:

Gee, you sure have a penchant for wasting time.

But seriously, I’d like to say **Thank You**. When I started, I wasn’t sure I had enough material to last a year. But given the number of subscribers and all the positive comments I’ve received, I plan to keep doing this for as long as I can. I’ve really enjoyed the past year, and I hope that I can keep you entertained.

Some trivia:

- The most popular post during the past year was Smart Quarterbacks, the Super Bowl, and SAT Scores.
- The blog received 14,432 views during its first year, and February 6 was its busiest day.
- The growth in traffic can be approximated by the funtion
*y*= 33*x*^{2}– 143*x*+ 312, where*y*is the number of pageviews during the*x*^{th}month since inception. - If you drank one low-calorie beer every night since the first post on MJ4MF, that would be
**one lite year**.

And finally…

I said to my friend, “Did you know that only 57% of people are able to understand the mathematical content contained on the MJ4MF blog?” His response:

I belong to the other 33%.

Oish. Happy anniversary!

### Rock, Paper, Scissors

If you haven’t seen it yet, you’ve got to play the interactive Rock, Paper, Scissors (RPS) game that appears in the Science section of the *New York Times*. Go ahead, indulge yourself in a few games before you continue… when you’re ready to come back, we’ll still be here.

Ah, good. Welcome back.

In the *New York Times* simulation, the computer is armed with over 200,000 pieces of data regarding the choices that humans have made when playing RPS. What the computer does is rather simple — it looks at your last four throws; it considers its last four throws; it searches its database to determine what other folks have chosen following the same string of throws; and then it makes its choice based on that information. It’s rather simple AI, but it’s very effective.

Said another way: You won’t win. Your ability to mingle at cocktail parties might imply that you’re as socially awkward as a computer, but it doesn’t mean that you’re as smart as one.

To have a chance to win, you have to stop thinking like a human. I was able to beat the computer using a simple method: I used the sequence of triangular numbers, and I divided each term by 3.

- If the remainder was 0, I chose Rock.
- If the remainder was 1, I chose Paper.
- If the remainder was 2, I chose Scissors.

Using that strategy, I prevailed, but barely: 8 wins, 6 losses, 6 ties.

You might notice, though, that it was a come-from-behind victory. I was down 5‑4 before winning 4 of the last 7. Using a random strategy only ensured that I wouldn’t play like a typical human. But doing so only gave me even odds, given that RPS is a zero‑sum game. In no way did it give me a major advantage, and my victory was merely a result of good luck.

All this talk of RPS makes me think about Demetri Martin’s take on the game:

I like rock, paper, scissors — 2/3. Rock breaks scissors: these scissors are bent, they’re destroyed, I can’t cut stuff — I lose. Scissor cuts paper: this is strips, this is not even paper, it will take me forever to put this back together — you got me. Paper covers rock: rock is fine, no structural damage to rock. Rock can break through paper at any point, just say the word. Paper sucks. It should be “Rock, Dynamite with a Cuttable Wick, Scissors.”

As it turns out, I’m a card-carrying member of the World RPS Society. My title is *Senior Investigator, Theoretical Throws Bureau*. I got in on the ground floor. When I joined in 1996, my lifetime membership cost me a mere $7. It’s now $45 + S&H. Of course, you now get a DVD, an RPS strategy book, and a t‑shirt. Back when I joined all I got was a laminated card with my name, title, and membership number. But I joined primarily for one reason, the note on the back of the card:

By authority of the World RPS Society (through the authority granted by the World RPS Steering Committee), the bearer of this card is hereby entitled* to determine the number of rounds in a match of RPS before a decision is considered to be binding (i.e., best of 3, best of 5, best of 57, etc.).

Card must be shown prior to play.

* Not valid at officially sanctioned World RPS tournaments or in Myers County, Alaska.

I have no idea what that bit about Myers County is all about, but I love this kind of power. At a party, when my buddy and I both reached for the last Dogfish Head 90 Minute India Pale Ale, he said, “Let’s RPS for it.” I agreed, but first I reached for my wallet, showed him my World RPS membership card, and said, “Okay, but we have to play a ‘Best of 193’ match.” He shot me a look, asked if I was joking, and when I told him I wasn’t, he said, “You’re a _____. Take the beer.” Admittedly, pulling rank wasn’t as gratifying as if I had actually beaten him 97‑96 in a Best of 193 series — but the reward was drinking that last Dogfish Head instead of having to settle for a Pabst or Miller Lite.

RPS is full of inside jokes. A *gambit* is a series of three throws, and each one has a name. My favorites are:

- Paper, Paper, Paper –
*The Bureaucrat*** - Rock, Rock, Rock –
*The Avalanche* - Paper, Scissors, Scissors –
*Paper Dolls*

What a great game…

### Seek and Ye Shall Find… or Not

Some say that love is where you find it. These folks are The Whispers.

Others say that love is where you look for it. These folks troll **match**.com or, in some icky cases, sugardaddy.com.

Yet others can’t find a damn thing — love, their keys, a paper they’ve been working on for weeks. These folks would be described by my mother as people “who couldn’t find their head if it weren’t attached.” For simplicity, though, I just call them mathematicians.

A mathematician is looking for his car keys under a lamp post. A neighbor sees him and offers to help. They search for over an hour but don’t find the keys. The neighbor finally asks, “Are you sure you lost your keys under this lamp post?”

“Well, no,” says the mathematician. “I lost them in the alley.”

“Then why are you looking for them over here?” the neighbor asks.

“Because it’s too dark to find anything in the alley!”

For many mathy folks, though, it doesn’t matter where you look…

Why does a mathematician have three pairs of glasses?

One for nearsightedness, one for farsightedness, and one to wear when he’s searching for the other two.

And just for fun, here’s one about a different type of glasses.

A programmer occasionally wakes up and feels thirsty. He thinks for a while and determines a solution. He leaves two glasses next to his bed: one full, in case he wakes up and is thirsty; and one empty, in case he wakes up and is not thirsty.