## Archive for March, 2012

### Grin and Parrot!

There aren’t too many math humorists in the world, and, for a very brief period of time, I was in high demand. (Relatively speaking, of course.) A talent agent once saw me give a presentation called *Puns and Puzzles*, which mixes number tricks with math jokes. After the show, he told me about a cruise line that caters to a very intellectual audience, and he asked if I’d be willing to perform on a ship. A month later, I was sailing the seven seas and, occasionally, making people laugh.

Aboard the *Princeps Mathematicorum*, audiences were generally polite. But one night, a rude mathematician sat in the front row with an even ruder parrot on his shoulder. Halfway through my act, I presented the following puzzle:

- Look at the 16 digits from one of your credit cards. Create two eight‑digit numbers, one consisting of all the digits in odd positions (that is, the first, third, fifth, and so on), the other consisting of all the digits in even positions (second, fourth, sixth, and so on).
- Add the digits of the first eight‑digit number, and double the result. Write it down.
- Add the digits of the second eight‑digit number. Write it down.
- How many of the digits in the first eight‑digit number are 5 or greater? Write it down.
- Now add the results of steps 2, 3, and 4.

I then magically revealed the last digit of each audience member’s result.

The mathematician rolled his eyes at the trick, and the parrot squawked, “This is just the Luhn system — brrrraaaaaak! — for credit cards! Of course the result will be 0!”

His outburst surprised me. I don’t like when my secrets are revealed. I was frustrated, but there was little I could do. It’s not like I could have an argument with a parrot. The rest of the show went poorly, and I left the stage distraught.

The next night, I had another show. But the mathematician and his bird were back, once again seated in the front row.

Wanting to avoid a repeat, I used a different puzzle:

- Take a four‑digit number.
- Scramble the digits to form a different four‑digit number.
- Subtract the smaller number from the larger number.
- Now, take the digits of the result, and add them together.

I then showed the following chart and magically predicted the symbol that would be associated with each audience member’s result:

The parrot shrieked, “Of course it will be omega! Any two numbers created from the same digits — brrrraaaaaak! — will be congruent modulo 9! The sum of the digits will always be a multiple of 9.”

Once again, he ruined my trick. I was angry, and I went to bed that night completely dismayed. But my dejection was short-lived — during the night, we hit an iceberg, and the boat sank.

I was submerged into freezing cold water. A piece of the deck was floating on the surface, and I pulled myself onto it. To my dismay, the parrot was perched on the other end. I shot him an angry look. He said nothing. I said nothing. We floated for several hours in silence, shivering, just staring at one another.

Finally, he spoke.

“Okay,” he said, “I give up. How the hell did you make the ship disappear?”

### A Life of Pi

I fell asleep on the couch last night while watching *Modern Family*. At 3:14 a.m., I woke up, left the couch, and stumbled to bed.

Several hours later, my son Eli came into our room and woke me. That was at 6:28 a.m. My wife agreed to take the morning shift, so I fell back asleep.

When I woke again, it was 9:42 a.m.

Then, at 12:56 p.m., I received an email from my friend Pat Flynn, and I was cheered by the silliness of the subject line: “My new favorite quadratic formula song.” I smiled thinking about the possibility that anyone would have a favorites list containing more than one song about the quadratic formula.

This was a rather uneventful sequence… except that the times were π, 2π, 3π, and 4π. Sort of. To two decimal places, 4π = 12.57, not 12.56. So my theory that my life is ruled by π was discredited.

All was not lost, however. The link in Pat Flynn’s email made me smile. It featured two teachers singing a song about the quadratic formula to the tune of Adele’s *Rolling in the Deep*. The lyrics are decent, and the teachers are pretty good vocalists. Here, give it a listen yourself…

And here are a few quotes about π you might enjoy.

If equations are trains threading the landscape of numbers, then no train stops at π. – Richard Preston

The primary purpose of the DATA statement is to give names to constants; instead of referring to π as 3.141592653589793 at every appearance, the variable PI can be given that value with a DATA statement and used instead of the longer form of the constant. This also simplifies modifying the program, should the value of π change. – FORTRAN manual for Xerox Computers

So here we have π

^{2}, which an engineer would call 10. – Frank King

### Pigs and Fish

My sons were reading *Wild and Woolly Animal Jokes* by David McLaughlan, and they got a fair chuckle out of this one:

What do you call a pig with three eyes?

A piiig.

I had previously encountered a different version of the joke:

What do you call a fish with no eyes?

A fsh.

Clearly, the punch lines were not crafted by mathy folks, who I think would answer these questions as follows:

What do you call a pig with three eyes?

–pigWhat do you call a fish with no eyes?

Real.

### It’s Back to Prime Time

On Saturday, I turned 41 years old. I’ve been looking forward to this for a while. It’s a prime year, and its twin prime is two years away. In between, I’ll be a number of years that is “the answer to life, the universe, and everything.”

Forty-one is also cool because *f*(*x*) = *x*^{2} + *x* + 41 is a prime-generating function. That is, *f*(1) = 43, *f*(2) = 47, *f*(3) = 53, and so on.

What is the first value of

xfor whichx^{2}+x+ 41 isnotprime?

The following image might help you answer that question. The number 41 appears in the center, and consecutive positive integers then proceed in a spiral. Notice that all of the numbers highlighted in yellow are prime. A pattern of primes continues along the diagonal — at least for a little while.

It also turns out that 41 is the smallest number whose cube is the sum of three cube numbers in two different ways:

41

^{3}= 2^{3}+ 17^{3}+ 40^{3}= 6^{3 }+ 32^{3}+ 33^{3}= 68,921

And 41 is the sum of the first six prime numbers:

2 + 3 + 5 + 7 + 11 + 13 = 41

At 41, I still feel young. But you know you’re an old mathematician when…

- You report your age in hexadecimal. (I’m only 29!)
- You’re not dead, but you’ve lost most of your functions.
- The distance you walked to school as a kid is directly proportional to your age.
- Your age can be described as “countably infinite.”
- You regularly go off on tangents.
- The phrase “pulling an all nighter” means not getting up to pee.
- When asked your age, you reply, “I’m in the 99th percentile.”
- You use the term
*surd*, and you know how to calculate its value on a slide rule.

### Fruit Flies Like a Banana… or Maybe a Bacardi

The best Marx Brothers line ever?

Time flies like an arrow.

Fruit flies like a banana.

It turns out that male fruit flies are a lot like men — when their advances are rejected by females, they’ll seek comfort by getting drunk. So say researchers from UC-San Francisco who study reward pathways in the brain and addiction. The results don’t surprise me; I’m just trying to imagine what a fruit fly’s pick-up line sounds like…

What’s a nice gal like you doing in a rotten banana like this?

You have the most beautiful red eyes.

Is that fermented apple cider you’re wearing?

I used to have a shirt where a fly is sitting on a pile of feces at a bar. Next to her is another pile of feces, and a male fly asks her, “Excuse me… is this stool taken?” It was from a lab company that analyzed fecal samples.

And of course…

What do you get when you cross a tsetse fly and a mountain climber?

Nothing. You can’t cross a vector and a scalar.

### The Ides Have It

When my sons woke up today, I told them, “Beware the Ides of March.”

To which Alex responded, “What are ides?”

I explained that the ides are roughly the middle day of the month. But then Alex asked why the ides was the 15th of March instead of the 16th, since March has 31 days.

“I don’t know.”

Nor do I know why the word *ides* is used to refer to this date. It comes from the Latin word *idus*, which can be translated to — yep, you guessed it — the English word *ides*. Nor do I know why *ides* is singular.

I also don’t know why the ides of March, May, July and October occur on the 15th day, but the ides of every month occur on the 13th day. But it does lead to a fun math problem for a four-year-old to figure out:

What is the maximum number of days between the ides in consecutive months?

The following calendar may help you figure this out.

Here are some math jokes related to things in the middle:

A circle is a round straight line with a hole in the middle.

What was Zeno of Elea’s middle name?

Of.

And all this talk of ides made me think of a really stupid joke from a really stupid joke book that I read when I was in elementary school. (That was a long time ago, hence the dated references, but maybe some of my older readers will appreciate it.)

If a woman named Ida married Dan Rather, got divorced, then took Bill Knott as her second husband… she’d be Ida Rather Knott.

### What Dates are Mathier than Pi Day?

While I am grateful that Pi Day gives some much-needed publicity to math, it’s a contrivance like textbook problems about two trains approaching from opposite directions. (Honestly, rather than spend your time determining how long until two trains on the same track collide, why not use that time to inform someone about the imminent collision?) Other than containing the same digits that appear in 3.14, there’s nothing terribly special about 3/14. And it propagates the widely held belief that π is only known to two decimal places.

That said, the cultural significance of Pi Day cannot be overstated. (Or maybe it just was?) Consequently, there are **six cool Pi Day cards at Illuminations** for you to share with friends via Facebook, Twitter, and Pinterest, or download them and include them in an email, on your website, or in a blog post. This one is my favorite:

Recently, there has been a movement to replace π with τ = 2π. (See The Tau Manifesto.) That would suit me just fine, and then we could celebrate Tau Day, which occurs on the more mathematical date 6/28. In addition to 6.28 representing the value of 2π (to two decimal places, anyway), it is also the case that both 6 and 28 are perfect numbers (the sum of their proper factors is equal to the number itself), and this year the value of the month, date and year of 6/28/12 are all even.

Please understand, my disdain for 3/14/12 is not personal. It’s just that other dates this year are, well, *mathier*.

Christmas Eve is one of those mathier dates…

- When written as 12/24/12, all of
*mm*,*dd*and*yy*are even. *mm*+*yy*=*dd*- Each of the digits within the date (1, 2, and 4) are powers of 2.
- The sum of the digits is 1 + 2 + 2 + 4 + 1 + 2 = 12, and 122412 ÷ 12 = 10,201 = 101
^{2}.

…as is the ninth of June…

- The numbers 6, 9, 12 form an arithmetic sequence.
- All three numbers are multiples of 3.
- The month (6) is a perfect number, the date (9) is a square number, and the year (12) is the smallest abundant number.

What do you think is the mathiest date of 2012? And what criteria do you use to determine if a date is mathy?

### Random Number Generation

While playing Nurikabe, my sons completed the following puzzle:

The puzzle itself isn’t very interesting, but did you notice the Puzzle ID? Exactly 1,000,000. The boys thought this was pretty cool, and I did, too. Yeah, yeah, I know, the occurrence of 1,000,000 shouldn’t impress me more than the appearance of, say, 8,398,176 or 3,763,985. But there are just under 10,000,000 unique 5 × 5 puzzles on the site, and only nine of them contain six 0’s. How lucky were we to get that random number?

Generating random numbers can be a difficult proposition, especially for a computer. This article from WIRED magazine — which describes a pattern that inadvertently appeared on lottery tickets, making it possible to predict winning tickets before they were scratched — shows how difficult it can be to generate numbers that appear to be random. (The article really is worth a read, especially for math geeks. Truth be known, WIRED is the only magazine that I read cover-to-cover every month.)

Robert Coveyou, a mathematician who worked on the Manhattan project, was an expert in pseudo-random number generators. He is most famously remembered for the following quote:

The generation of random numbers is too important to be left to chance.

Of course, Randall Munroe at xkcd has a foolproof method for generating a random number:

I would hate for you to need a random number and then have difficulty generating one. I’m here to help, so I present the…

**MJ4MF Random Number Generator (PDF)**

Creating the MJ4MF RNG is quite simple. Just follow these steps:

- Download and print the PDF from the link above.
- Cut out all six squares, one for each number 1-6.
- For each square, make two folds: first, fold the paper to the center vertically; then, fold the paper to the center horizontally. The result of these two folds is shown, below left.
- When all six pieces are folded, interlace them to form a cube. This is shown, below middle. The assembled cube is shown, below right.

Finally, a joke about random numbers.

A student is asked for the probability that a random number chosen between 0 and 1 will be greater than 2/3. The student answers 1/3. The teacher says, “Great! Can you explain to the class how you arrived at your answer?” The student says, “There are three possibilities: the number is either less than, equal to, or greater than 2/3, so the probability is 1/3!”

### Be Careful!

The following are perhaps the most uttered, least helpful words that a parent says:

Be careful!

The uselessness of the statement derives from its timing. It is typically said *after* a young child has bumped his head, slammed his finger in a garbage can lid, or fallen down and skinned his knee. What good does it do then? Yet it must be an autonomic response; I cannot help but utter the phrase after one of my sons has an accident.

Apparently, I’ve said it enough times that it has been internalized by my sons. While looking at the cover of the *Into The Wild* soundtrack, my four-year-old son Alex said, “Daddy, he’s not very careful.”

“Who’s not?” I asked.

“This guy,” he said, pointing to the photo of Emile Hirsch sitting atop an abandoned bus. “He’s sitting on top of a bus. That doesn’t seem very safe.”

I suppose it’s not surprising that a mathy father would instill carefulness in his son. We mathy people are known to be a careful lot.

Student: How do you perform a Fourier transform?

Teacher: Very carefully.

The following professionals are also very careful when they do what they do.

How does a mathematician find the area under a curve? She carefully calculates the definite integral.

How does a chemist find the area under a curve? He carefully plots many points on a piece of paper, draws a line of best fit, cuts out the area under the curve, weighs the cut-out, and divides the mass by the density.

### My 5 Favorite Math Games

I’ve been thinking about games a lot recently. I’ve been reading *The Multiplayer Classroom*, in which Lee Sheldon describes his experience turning a college class into a multiplayer game. Students earn experience points (XP) and progress through levels; the grade a student receives at the end of the course reflects the level reached in the game. Awesome concept!

My sons have recently been playing a math game that my friend Barb Dougherty calls “Sums and Products.” I prefer the name given to it by Constance Kamii: *Salute*. The latter name captures the action performed by players, raising a card to their foreheads.

Here. Watch for yourself.

I like the game well enough, because my kids get to practice addition and multiplication facts in a non-routine way. But I much prefer games that teach new concepts rather than just reinforce things they already know.

So, a question for you:

**What’s your favorite math game?**

Here’s my top five.

**1.** **Sprouts.** Yeah, I know… how cliché, right? But I can’t help myself. It’s just a great game.

**2.** **Pig.** Players roll a pair of dice. On each turn, you can roll the dice as many times as you want, and you can stop whenever you want. Your score for that turn is the sum of all the rolls. However, if you roll a 1 on either die, you get 0 points for that turn; and if you roll double 1’s, your total score for the entire game returns to 0. First player to reach 100 points wins. How daring are you?

**3. Ker-Splash**. This is one of the games at Calculation Nation^{®}, a suite of math strategy games from NCTM. My co-worker Julia Zurkovsky designed the game, and I still think it’s the best game on the site.

**4.** **Theseus and the Minotaur**. I don’t know, perhaps this isn’t really a math game. But it’s too damned addictive not to include on this list. How addictive is it, you ask? In the middle of writing this post, I jumped to another tab to find the link to the game. It was 55 minutes before I returned to finish this post. Damn you, Toby Nelson!

**5. Deep Sea Duel.** This game goes by many names. If you can figure out its most common name, you’ll have no trouble winning. The answer can be found in the article “What’s the Name of this Game?” by John Mahoney, which originally appeared in the October 2005 issue of *Mathematics Teaching in the Middle School*.