A Muppet You Can Count On

A big MJ4MF thanks to Lindsey Witcosky, who directed me to a wonderful BBC article about one of my favorite Sesame Street characters, the Count! The link to the article is provided below, but first a quiz based on some trivia in the article.

1. What was the Count’s full name?

2. What was the Count’s favorite number?

3. Who was the voice of the Count from 1970 until 2011?

Answers are below, but you can also find them in the BBC article:
http://www.bbc.co.uk/news/magazine-19409960

The Count’s favorite number is equal to 187², and BBC Radio asked listeners of the show More or Less to speculate why. One listener noted that 187 = 94² ‑ 93² and, of course, 187 = 94 + 93. The BBC article referred to this coincidence as, “An embarrassment of riches!” But I prefer to think of it as, “An embarassment of algebra!”

Algebra can be used to show why this is true. The nth square number is equal to the sum of the first n positive odd integers. That is,

n² = 1 + 3 + 5 + 7 + … + (2n ‑ 1)

From this it follows that

94² = 1 + 3 + 5 + 7 + … + (2 × 94 – 1)

and

93² = 1 + 3 + 5 + 7 + … + (2 × 93 – 1)

so of course

94² – 93² = 2 × 94 – 1 = 187

Moreover, the difference of two squares is equal to the product of the sum and difference of the two numbers. That is,

a² ‑ b² = (a + b)(a ‑ b)

Consequently,

187 = 94² ‑ 93² = (94 + 93)(94 ‑ 93) = (94 + 93)(1)

So, saying that 187 = 94² ‑ 93² = 94 + 93 is kind of like saying the same thing twice, just in different ways.

Answers

1. Count von Count

2. 34,969

3. Jerry Nelson, who passed away on August 23. R. I. P.

Pringles: The Edible Hyperbolic Paraboloid

My wife forwarded an email with a link to a CNN article and subject line, “Your husband will love this.” Uh-oh. Even my closest friends cannot correctly predict what I will and will not love, so how would a colleague of my wife — who only knows me from an introduction at a professional reception — be able to make such a prediction?

But the article did not disappoint. The author wrote about the mathematically satisfying shape of Pringles^®, and she quoted her husband thus:

They [Lays Stax] set themselves up as a Pringles competitor, but it’s an entirely different curvature!

I have never met the author, but her last name was familiar. As luck would have it, her math professor husband and I taught together at a gifted camp for several summers. Small world, eh?

My favorite line of the article was from the last paragraph.

Flavor is subjective. Math is irrefutable.

Fact.

What I enjoyed most about this occurrence was the intersection of several math topics. The article discusses parabolic cylinders and hyperbolic paraboloids, which are topics in multivariable calculus; a colleague of my wife forwarded a link about an article written by the wife of a former colleague, which demonstrates social network theory; and, a colleague of my wife is not equivalent to the wife of my colleague, which shows non‑commutativity.

My two cents? Pringles^® rule.

22 Jokes From a Colleague

This past Sunday, I received 22 jokes from Keith Raskin, who has, among other things, too much time on his hands. Though some of them are clearly based on old chestnuts, Keith claims that they are all his original creations.

When I suggested to Keith that he had too much time on his hands, he responded, “Not much. Enthusiasm! Maybe brief obsession.” He proved his point by sending me another four jokes.

It wouldn’t be fair for me to keep these jokes to myself. I see no reason that you shouldn’t have to suffer, too.

Of the jokes he sent, there were six that I wasn’t sure I understood or just didn’t think were funny. The jokes that immediately follow are the 16 jokes that I understood and for which I saw possible inherent humorous value. The other six appear at the bottom of the post, along with Keith’s explanations. (Warning! When a joke has to be explained, it is no longer funny!)

A math student is told by his mother to set the table.
“To what?” he replies.

Which polygon is also a card trick?
Decagon.

I went to see Plane Meets Plane, but there was a long line. Not much point in seeing Plane Meets Line again.

Two barcodes go to a shady optometrist. They sit and stare at a light for half an hour. One of them says, “I think this is a scan.”

Two lines walk into a barcode. They hashed it out.

What does a vegan mathematician eat?
Roots, whole numbers, natural logs, tree diagrams, and stem-and-leaf plots.

Student: What’s infinity?
Math Teacher: Think of a number.
Student: Okay, I’ve got one.
Teacher: Good. That’s not it.

Student: What’s zero?
Teacher: The number of times something happens that doesn’t.
Student: What are the chances of that?
Teacher: Exactly.

How many mathematicians does it take to change a light bulb?
On average, or do you want the whole distribution?

In life, trees grow roots.
In math, roots split logs.

A guy goes into a math store exactly eleven times.

What did the 8 say to infinity?
Rise and shine, buddy!

What did the Venn diagram say to infinity?
Eat something, dude!

How do you solve any equation?
Multiply both sides by zero.

What did the trig teacher say to the triangles?
You’re all right.

Security Guard: I need some ID.
Math Teacher: Additive or multiplicative?
Security Guard: Yours.
Math Teacher: Ah, reflexive!

In the interest of full disclosure, here are the jokes that I didn’t understand or didn’t think were funny. But Keith admits he’s not going for funny. He’s a high school teacher and is “desperate for tension breakers and minor amusements or moments of actual  engagement.”

Yes, I know that I am not the official arbiter of what’s mathematically funny. But then again — if not me, who?

What is the binomial distribution?
A free lunch program.

(“Binomial” sounds like “Buy no meal.”)

Two circles walk into a club. They made a tree.

(The suit clubs in a deck of cards is made from three circles and looks like a tree. Add two more circles, it looks even more like a tree.)

How did every student get a score over 100 on the test?
They were percentages!

(Percentages are numbers over 100, literally: for example, a% = a/100.)

What are inequalities?
Read a newspaper.

(Social commentary.)

Student: What’s abstract reasoning?
Teacher: …

A guy goes into a beleaguered math store.
Guy: What happened?
Clerk: Well, we have wall-to-wall problems, our answers are still in boxes, and our solutions are leaking out.

Along with explanations for some of the jokes above, Keith sent along four additional jokes. Here you go…

What’s the ultimate epsilon delta argument?
Public education.

Did Pythagoras do the first PPT presentation?

What’s a pyramid scheme?
Death by triangulation.

How are filmmakers topologists?
There are open and closed sets, sequences that wind up in or out of the film, full attention to surface details, whether things are connected and continuity, and lots of coffee and doughnuts.

In Your Prime

I’ve got a prime number trick for you today.

1. Choose any prime number p > 3.
2. Square it.
3. Add 5.
4. Divide by 8.

Having no idea which prime number you chose, I can tell you this:

The remainder of your result is 6.

Pretty cool, huh?

I will now fill a bunch of space with quotes and jokes about prime numbers to prevent you from seeing the spoiler explanation below. But you can skip straight to the bottom if you’re not interested in the other stuff or if you just can’t control yourself.

Mark Haddon, author of The Curious Incident of the Dog in the Night-time, wrote the following:

Prime numbers are what is left when you have taken all the patterns away. I think prime numbers are like life. They are very logical, but you could never work out the rules, even if you spent all your time thinking about them.

(Incidentally, if you haven’t read that book, you should. Amazon reviewer Grant Cairns said it better than I could: “The integration of the mathematics into the fiction is better than any other work that I know of. The overall effect is a beautiful story that any maths fans will find hard to read without the tissue box close at hand.”)

Israeli mathematician Noga Alon said that he was interviewed on Israeli radio, and he mentioned that Euclid proved over 2,000 years ago that there are infinitely many primes. As the story goes, the host immediately interupted him and asked:

Are there still infinitely many primes?

And of course there’s this moldy oldie:

Several professionals were asked how many odd integers greater than 2 are prime. The responses were as follows:
Mathematician: 3 is prime, 5 is prime, 7 is prime, and by induction, every odd integer greater than 2 is prime.
Physicist: 3 is prime, 5 is prime, 7 is prime, 9 is experimental error, 11 is prime, …
Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, …
Programmer: 3 is prime, 5 is prime, 7 is prime, 7 is prime, 7 is prime, …
Marketer: 3 is prime, 5 is prime, 7 is prime, 9 is a feature, …
Software Salesperson: 3 is prime, 5 is prime, 7 is prime, 9 will be prime in the next release, …
Biologist: 3 is prime, 5 is prime, 7 is prime, the results for 9 have not yet arrived…
Advertiser: 3 is prime, 5 is prime, 7 is prime, 11 is prime, …
Lawyer: 3 is prime, 5 is prime, 7 is prime, there is not enough evidence to prove that 9 is not prime, …
Accountant: 3 is prime, 5 is prime, 7 is prime, 9 is prime if you deduct 2/3 in taxes, …
Statistician: Try several randomly chosen odd numbers: 17 is prime, 23 is prime, 11 is prime, …
Professor: 3 is prime, 5 is prime, 7 is prime, and the rest are left as exercises for the student.
Psychologist: 3 is prime, 5 is prime, 7 is prime, 9 is prime but tries to suppress it, …
Card Counter: 3, 5, and 7 are all prime, but I prefer 21.

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Explanation of the Prime Number Trick

We are trying to show that (p² + 5) mod 8 = 6. This is equivalent to showing that (p² ‑ 1) mod 8 = 0, or that (p + 1)(p ‑ 1) is divisible by 8.

Because p > 3 and is prime, then either p = 1 mod 4 or p = 3 mod 4. Consequently, it must be the case that (a) p + 1 = 2 mod 4 and p ‑ 1 = 0 mod 4 or (b) p ‑ 1 = 2 mod 4 and p + 1 = 0 mod 4. That is, both numbers will be even, and at least one of them will be a multiple of 4. For either (a) or (b), the product (p + 1)(p ‑ 1) will be a multiple of 8. Q.E.D.

When I Die…

Did you hear about Tommy Lasorda’s wish? The retired manager of the Los Angeles Dodgers is recovering from a heart attack he suffered in June, and he recently told the L. A. Times:

I’ve already told my wife that when I do go, I want our home schedule attached to my tombstone. I want people who are in the cemetery visiting their loved ones to say, “Let’s go to Lasorda’s grave and see if the Dodgers are playing home or away.”

That could get expensive, since it would have to be updated annually. I suggest one of these instead:

Old baseball players never die; they just go batty.

Old baseball players never die; they just do one more lap around the bases.

Old baseball players never die; they just get traded to the Blue Jays. (Sorry, Toronto!)

This got me thinking about what I’d want on my tombstone when I leave.

Don’t meet my end with gasps and shrieks;
I left you with a book (and blog) for geeks.

Given the likelihood of my eternal destination, I take comfort in the following advice:

Go to Heaven for the climate,
to Hell for the company.

Contemplating what should appear on my tombstone puts me in good company. Many mathematicians have pondered the same question.

At age 18, Carl Friedrich Gauss showed that it was possible to draw a regular 17‑gon with compass and straightedge. Proud of his accomplishment, he later requested that a 17‑gon be inscribed on his tombstone. Although his wish was not granted, a memorial to him in his hometown of Braunschweig, Germany, now bears a small 17‑gon just below his right foot.

Imagine a sphere inscribed in a cylinder whose height is equal to its diameter. Archimedes discovered that the ratio of the surface area of the cylinder to the surface area of the sphere is 3:2 and also that the ratio of the volume of the cylinder to the volume of the sphere is 3:2. Despite his many accomplishments in mathematics, this is the one for which he wished to be remembered. He asked that a cylinder with inscribed sphere be displayed on his tombstone, with the ratio 3:2 inside.

Jacob Bernoulli was so enamored with the logarithmic spiral that he wanted one inscribed on his tombstone. Unfortunately, the engraver mistakenly carved an Archimedean spiral (shown below).

An apocryphal story is that the following poem appears on the tombstone of Diophantus. In fact, this problem appeared in a fifth-century Greek anthology of puzzles.

Here lies Diophantus, the wonder behold.
Through art algebraic, the stone tells how old:
“God gave him his boyhood one-sixth of his life,
One-twelfth more as youth while whiskers grew rife;
And then yet one-seventh ere marriage begun;
In five years there came a bouncing new son.
Alas, the dear child of master and sage
After attaining half the measure of his father’s life, chill fate took him.
After consoling his fate by the science of numbers for four years, he ended his life.”

Good luck solving the puzzle before you meet your end!

What’s the Difference?

Today is 8/20/12, which makes it a difference day, because the difference between the date and the month is equal to the year: 20 – 8 = 12. In general, a day in the form mm/dd/yy is a difference day if dd – mm = yy.

Dates of this type are relatively rare. There will be exactly 12 per year through 2019, but from 2031 through 2099, there won’t be any. So here’s a question:

How many difference days will there be during the 21st century?

The answer is below.

Speaking of differences, here are some math jokes about differences.

What’s the difference between a narcoleptic and a math professor?
The narcoleptic is a slumber nut.

What’s the difference between a math Ph.D. and a large pizza?
A large pizza can feed a family of four.

What’s the difference between a lemma and a proposition?
You’ll never receive a lemma at a bar.

What’s the difference between a mathematician and a chocolate muffin?
One is a mathematician, and the other is a chocolate muffin.

Though I believe mathematicians are useful, I would much prefer a machine for turning theorems into coffee.

(Admittedly, that last one is in the wrong format. But it seems weird to ask, “What’s the difference between a mathematician and a machine for turning theorems into coffee?” The answer would be, “Nothing.”)

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There will be 281 difference days during the 21st century. There are 12 per year for 2000–19, but then the number per year starts to decrease. You might expect there to be 11 difference days in 2020, 10 difference days in 2021, and so on, with the number decreasing by 1 each year. But February, April, June, September and November cause problems because they have fewer than 31 days. So the total number of difference days during the 21st century is:

20(12) + 10 + 10 + 8 + 8 + 7 + 5 + 5 + 3 + 3 + 1 + 1 = 281

Class Absences

Alas, the end of summer is nigh, and students are returning to campus. To get you in the mood for school again, the following is a quiz about college course abbreviations.

Many courses are known by four‑letter abbreviations, such as PHIL for philosophy and ASTR for astronomy. Complete each of the words below by inserting the abbreviation for a common course. For example,

ALP __ __ __ __ OW

could be completed by inserting ENGL, the abbreviation for English courses, to make ALPENGLOW.

1. COELAC __ __ __ __
2. IN __ __ __ __ UE
3. W __ __ __ __ LER
4. IRR __ __ __ __ CILABLE
5. __ __ __ __ ICIAN
6. R __ __ __ __ TION
7. IMP __ __ __ __
8. __ __ __ __ INTON
9. SE __ __ __ __ ED
10. DE __ __ __ __ IFY
11. REIN __ __ __ __ E
12. WING __ __ __ __

(Answers below.)

What’s that you say? You’re not quite ready for a test yet. That’s fine. Then maybe just enjoy the following quotes, which could yield some insight about how to proceed during the upcoming semester.

Mathematician Richard Askey remembers the following advice about how many courses to take each semester:

When I was in graduate school at Princeton, I was told to take three courses. One of them to work on really hard, another to work on moderately hard, and the third one just to absorb. In my case, I never showed up to the latter class, taught by Robert Gunning on several complex variables. Several complex variables (Cⁿ) was starting to get very fashionable then, but I decided to specialize in n = ½.

The experience of comedian B.J. Novak (The Office) might offer some insight about choosing a major.

I learned nothing in college. It was really kind of my own fault. I had a double major: psychology and reverse psychology.

And you might want to pass on the following advice, courtesy of mathematician Ron Graham, to some of your math professors.

Someone has remarked that, “An ideal math talk should have one proof and one joke, and they should not be the same.”

Students often find that many topics in upper-level math courses are review from previous courses, thus proving that the mathematics curriculum is compact. That is, any material covered by an infinite number of math courses can be covered by some finite subset of those courses.

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Answers

1. ANTH (Anthropology)
2. TRIG (Trigonometry)
3. HIST (History)
4. ECON (Economics)
5. PHYS (Physics)
6. EDUC (Education)
7. ARTS (Arts)
8. BADM (Business Administration)
9. ARCH (Archaeology)
10. CALC (Calculus)
11. STAT (Statistics)
12. SPAN (Spanish)

Southpaw Summations

August 13 is Left Handers Day.

It’s interesting to me that left-handers chose the 13th as a day to honor themselves, since the number 13 is often associated with a lack of luck. After all, the word sinister, which implies that something evil or harmful is about to happen, derives from the Latin word sinistra, which means lefthanded. It comes from the Latin word sinus, referring to the pocket on a toga that always appeared on the left side.

My brother-in-law has a shirt that says:

Everyone is born right-handed.
Only the strongest can overcome it.

“Interesting theory,” I said.

“It’s only one of two possible theories,” he informed me. “The other is that everyone is born left-handed, but only the strongest can maintain it.”

Most sources say that about 10% of the population is left-handed, and most statistics show that there are more left-handed females than left-handed males.

Interestingly, some studies have found that left-handedness is higher in math teachers than in the general population. In particular, a statistically significant difference was found for male math teachers. It’s also the case that left-handed students score higher in math on the SAT.

It has been suggested that those without a language bias in the left hemisphere of the brain, who are left-handed at a higher rate than the general population, would have an advantage in mathematical ability. For this reason, one researcher said that there is not a higher than normal occurrence of left-handers among the mathematically gifted, but rather that there is a lower than normal occurrence of right-handers.

This reminds me of a famous syllogism:

• Ten percent of all car thieves are left-handed.
• All polar bears are left-handed.
• If your car is stolen, there’s a 10 percent chance it was taken by a polar bear.

Given this information about mathematical ability, it stands to reason that polar bears may have a penchant for both grand larceny and integral calculus.

Bad Joke Tolerance Test

This is a tolerance test. (If, upon hearing that, you thought, “Mine is about ±3%,” then you will probably do fairly well.) How many of the following bad jokes can you endure? If you…

• Have to close your browser after just one? You need some training. Read Math Jokes 4 Mathy Folks.
• Run from the room, shrieking, “Make it stop! Make it stop!” after just five? You, too, can be a bad joke survivor. Come back to this page every day for a week, and try to read just four jokes each day. Together, we’ll get through this.
• Make it halfway? Good effort. Many great men turned back sooner.
• Get through this entire list without groaning once? You, my friend, are a rock. The Army could use someone with your ability to tolerate pain.

Did you hear about the beautiful, cross-eyed math teacher who lost her job?
She was easy on the eyes, but she couldn’t control her pupils.

When chemists die, they barium.

I’m reading a book about anti-gravity. I can’t put it down.

A dyslexic man walks into a bra.

I don’t enjoy computer jokes. Not one bit.

Accountants watch their figures.

A math professor in an unheated room is cold and calculating.

The probability of someone watching you is proportional to the stupidity of
your action.

Mathematicians die when their number is up.

A gram cracker is a metric cookie.

Ten math puns appeared above, in the hopes that one would make you laugh.
Sadly, no pun in ten did.

Is ln(i) an imaginary lumber?

The volume of a robot character in Star Wars is V = r²d².

There is a fine line between numerator and denominator.

Pentagon. Hexagon. Oregon.

Sorry, it’s true: i > u.

General Calculus is able to differentiate between his friends and enemies.

You can miss one math class, you can miss another… but after a while, it’ll start to add up.

i², just keepin’ it real.

The international student was unfamiliar with algebra, so when asked
what 2n + 2n was, he replied, “It’s 4n to me.”

Two feuding math families were at odds over evens.