## Archive for January, 2012

### Depressing Expressions

Did you hear that Monday, January 23, 2012, was the most depressing day of the year? That’s according to Cliff Arnall, a British life coach who, for a little while, was a tutor at Cardiff University. He used the following formula to make his prediction:

In that expression, *W* = weather, *D* = debt, *d* = days until next payday, *T* = time since Christmas, *Q* = time since a failed quit attempt (such as abandoning a New Year’s resolution), *M* = motivation level, and *N _{a}* = need to take action.

When I heard that a psychologist was creating mathematical expressions, I had just one thought:

Why did the psychologist send the expression to a doctor?

Because he wasn’t being rational.

When I read the formula, my first thought was, “Wow, that’s an incredible bunch of rubbish!” (Funny, I don’t normally use the word *rubbish*. Maybe it happened because I read about the expression in a British newspaper?) Only *W* and *T* are universally measurable variables. While *D*, *d*, and *Q* are also measurable, they vary from person to person and shouldn’t be used to predict a global most depressing day. And what’s this nonsense about time from Christmas? Is that really a factor for Jews, Sikhs, and other non-Christians? (Note: Many online sources incorrectly state that *d* = monthly salary. But that would cause the formula to make even less sense.)

This expression has been used for several years to predict the most depressing day, and a similar expression has been used to predict the happiest day of the year. The happiest day expression, which is similarly unintelligible, regularly predicts a date in June. Interestingly, the “research” was sponsored by Wall’s Ice Cream. (Hmm, now why would an ice cream company have an interest in people being happy during the summer?)

It’s a little too early in the year to say that January 23 will be the most depressing day of 2012. In fact, January 23 isn’t even the most depressing day *so far* in January — that distinction belongs to Sunday, January 8.

As for crazy expressions, the following equation contains my favorite:

The value of *k* doesn’t matter, but the equation doesn’t hold if the placeholder variable *k* is not included.

Incidentally, this equation is related to the following problem: Raise *n* + 1 consecutive integers to the power *n*. Subtract the first from the second, the second from the third, and so on, until you’re left with a set of *n* integers. Then subtract the first from the second, the second from the third, and so on, until you’re left with a set of *n* ‑ 1 integers. Continue this process until you’re left with just one integer. Its value may surprise you.

### Super Bowl Squares Contest

Laurence Tynes, the hero; Billy Cundiff, the goat. And so we head to Super Bowl XLVI with a rematch of the game four years ago. One can only hope that this game will be half as exciting as that one.

Your math/football trivia for the day? **Super Bowl XLVI is the second to require each of the first four Roman numerals (I, V, X, L)**; the first was Super Bowl XLIV two years ago. [Thanks to Eric Langen for pointing out my previous error.] Personally, I’m looking forward to Super Bowl LXVI, when the first four Roman numerals will occur in decreasing order. A real treat will occur in 3532, when Super Bowl MDLXVI will be played, wherein all six of the Roman numerals will appear in decreasing order. While I’m fairly certain I won’t be around to see that one, I hold out hope that I am reincarnated as a star football player who earns that game’s MVP honors; though it’s far more likely that I will return as a football to be used by adolescents in a backyard game.

Buoyed by the success of the online version of my favorite game, I’ve decided to run another online contest. This one relates to Super Bowl XLVI, and you’re asked to predict the units digit of each team’s score at the end of each quarter when the Patriots and Giants square off on Sunday, February 5.

Probably the most common type of office betting pool is a square football pool, which is often referred to as just The Squares. The pool is played on a 10 × 10 grid, and contestants can buy squares within the grid for a certain amount of money. **After all 100 squares have been purchased**, the numbers 0‑9 are randomly assigned to each row and column. The numbers for each row represent the units digit of the score for one team, and the numbers for each column represent the units digit of the score for the other team. The winners are the four people whose squares correspond to the units digit of the actual score of the game at the end of the 1st, 2nd, 3rd, and 4th quarters.

Feel free to use this Excel spreadsheet if you’d like to run your own version of this game. (Though be sure to check all applicable laws, to ensure that you’re not in violation of local or state gaming laws.)

The difference between the typical version of this game and the version I’m running here is that **you get to pick which pairs of numbers you want**. Consequently, winning isn’t solely a matter of random luck. But there’s a catch — you can pick the most likely number pairs, but chances are other folks will pick those numbers, too, and the winnings are divided among everyone who picked that pair. So, should you pick 0‑0 and divide the pot with a thousand others; or should you pick the highly unlikely 5‑2 and have the winnings all to yourself?

*Please note that the game I’m running is for entertainment only. No money is required to play, and there will be no pay-out to the winners.* If all goes well this year, perhaps next year there will be a real version that allows you to wager your hard-earned money in such a silly manner — assuming, of course, that I can find a way to skirt the myriad state gaming laws that would prevent me from running such a contest.

In case you’re wondering, “Why are you doing this?” remember that I’m the author of a math joke blog. Why do I do any of the things I do? For fun, mainly, and because I’m a certifed math geek. I like the math psychology of this game, and I’m just interested in the numbers that people will pick.

Here are the official rules:

- Imagine that you have $5, and each square costs $1, so you can buy up to five squares. It’s your money, spend it how you like — if you want to choose the same pair of numbers for all five bets, go ahead, knock yourself out. And what the hell do I care? Enter as often as you like; if you’ve got nothing better to do with your time than repeatedly submit entries for this contest, well, that’s your problem.
- All money bet will be divided equally among the four quarters, so the total amount will be equal to $5
*n*, where*n*is the number of contestants. (Should a contestant enter fewer than five choices, the last entered choice will be repeated multiple times to get the total to five.) - If you pick a winning square, you will share the winnings with everyone else who picked the same square. (For example, if 200 people play this game, there will $1,000 in the pot, so the winning amount for each quarter will be $250. If ten people choose 7-3 and it hits for one quarter, each person will receive $25.)
- Enter your five choices as two-digit numbers,
**where the tens digit represents the Patriots’ score and the units digit represents the Giants’ score**. (For instance, if you want Patriots 7, Giants 3, enter 73; but if you want Patriots 0, Giants 7, enter 07.)

That’s it. Access the form via the link below:

My friends Andy and Casey Frushour have been keeping data about which pairs of numbers occur most often. Before making your picks, you might want to check out their analysis of data from six years of NFL games as well as from all 45 Super Bowls.

Bets will be accepted until 11:59 p.m. ET on Saturday, February 4, and an image showing the number of times each square was chosen will be posted at:

**Super Bowl Squares Contest – Summary of All Bets**

The complete results for this contest will be posted on Monday, February 6, at the URL below. (But note that this link will return a “404 Error – File not Found” message prior to February 6.)

**Super Bowl Squares Contests – Results**

Good luck!

### Math, Unconventionally

Okay, boys and girls, here are some warm-up problems for today’s lesson…

- What is the product of all single-digit prime factors of 143?
- What is the value of 6 / 2 (1 + 2)?
- When is a DECADE not equal to 10 years?

The correct solution to each of these problems relies on a mathematical convention.

A convention is an action that is deemed acceptable by other members of a group. There are plenty of conventions — most social, but many mathematical — by which you probably already abide.

- If, while passing through a door in a public building, you notice a person less than
*k*feet behind you who will be passing through the same door momentarily, you should continue to hold the door open at least until that person reaches the door or, to be most proper, until she passes through it. (While we’re on the subject, I’d be happy to hear your opinions for the value of*k*. I was sure that*k*< 25, but I recently got a dirty look from a woman for whom I did not hold the door, and she was at least 40 feet away.) - You should use X’s in place of a stick figure’s eyes to indicate unconsciousness or death. (See image below.)

- You should drive on the right side of the road — unless you’re in Britain, Australia, Suriname or Guyana, where the right side is the wrong side. Incidentally, about 2/3 of the countries in the world follow the right-hand rule, accounting for nearly 3/4 of all traffic.
- You should assume that numbers are represented in base 10 (unless otherwise specified).
- You should use the plus sign (+) to indicate addition.

The order of operations is another accepted mathematical convention. Bon Crowder of Math Four says that it’s like driving on one side of the road or the other. “Doesn’t really matter which one, just as long as everyone else involved agrees to play by the same rules.”

But the following problem, which has been traversing the cyber-circuit recently, indicates that perhaps not all of us follow the same convention:

What is the value of 6 / 2 (1 + 2)?

There is an implied multiplication symbol just before the parentheses. In the order of operations, division and multiplication have equal precedence, so the value of the expression can be calculated as follows:

6 / 2 × (1 + 2)

6 / 2 × 3

3 × 3

9

But several references suggest that implied multiplication takes precedence over explicit multiplication. Perhaps you agree with this? If so, you’re not alone. Many have argued that the value should be found this way:

6 / 2 (1 + 2)

6 / 2 (3)

6 / 6

1

If you’d like to get involved in this argument, then feel free to join the discussion at Spiked Math.

Similarly, there is not universal consensus for the definitions of many math terms. One example is *whole number*. The *James and James Mathematical Dictionary* included three definitions for whole number:

- The non-negative integers 0, 1, 2, 3, …
- The positive integers 1, 2, 3, 4, …
- All positive and negative integers …, -3, -2, -1, 0, 1, 2, 3, …

The first definition is the one I learned in school, but apparently it’s not used in every school.

A similar thing occurs for the definition of *proper divisor*. A *proper subset* of a set is any subset that is not the original set itself. It would seem appropriate, then, that a proper factor would be any positive integer factor other than the number itself. But this definition is only used in some cases. For instance, when discussing perfect numbers, this definition is convenient: a number is perfect if the sum of its proper factors is equal to the number itself (e.g., the proper factors of 6 are 1, 2, and 3, and since 6 = 1 + 2 + 3, then 6 is a perfect number.)

Sometimes, however, the definition indicates that a proper factor of *n* should exclude both 1 and *n*. This definition is convenient when describing prime numbers; a positive integer is said to be prime if it has no proper divisors.

Which one is correct? Depends who you ask.

Finally, if you enjoy conventions, you might enjoy attending the annual convention of the Barbershop Harmony Society, which will occur July 1‑8, 2012, in Portland, Oregon. Mathy folks are sure to love it, as it is bound to be a *harmonic function*.

(No, in fact, this entire post wasn’t written as an elaborate set-up for that one joke. It just worked out that way.)

As to the problems at the start of this diatribe, here are the solutions. Sort of.

- Since 143 = 11 × 13, it has no single-digit prime factors. But mathematical convention dictates that the empty product is 1, so the product of all single-digit prime factors of 143 is 1. Crazy, huh? (Incidentally, another mathematical convention is at play here; namely, that 1 is not considered a prime number. If 1 were a prime number, then the Fundamental Theorem of Arithmetic would fail, because integers would not have unique prime factorizations.)
- See above. I agree with Spiked Math that the answer is 9, but some people still argue that the answer is 1.
- When it’s written in hexadecimal: DECADE
_{16}= 14600926. (And there’s*another*convention, although by no means is this one universal. Uppercase letters in normal font are used for the “numbers” in hexadecimal, whereas uppercase letters in italics are typically used to indicate sets or points in geometry. On the other hand, lowercase letters in italics are used to represent algebraic variables, such as*x*+*y*= 7, and they often are used to indicate a geometric length — for instance, many textbooks say that the side opposite angle*C*in a triangle has length*c*.)

### Which Calculus Joke is Funniest? Nun of the Above

My friend Pat Flynn, a teacher at Olathe East High School, recently told me about his childhood experience with math education.

Sister Mary Constance only used her ruler to measure pain, not distance.

That’s one of the funniest lines I’ve heard in a long time! Along similar lines…

What do you get if you cross a zero and a pigeon?

A flying none!

Pat is a calculus teacher, and I once heard some students discuss his humor.

When our calculus teacher would tell us a joke, my friend would laugh twice: once when he first heard it, then again when he got it.

Here are some jokes that Pat would surely like his calculus students to suffer through.

What did the calculus teacher ask the dazed and confused student?

“Young man, have you been taking derivatives?”What’s the difference between a mathematician and a physicist?

A physicist will take the average of the first three terms of a divergent series.

But it’s not just calculus… Pat enjoys making students groan at every level, so here are some all-purpose jokes.

Why did the variable break up with the constant?

The constant was incapable of change.Did you hear about the bodybuilding mathematician who was always positive?

He had niceabs().

### Dave is a Four-Letter Word

Q: What are these?

- EA + EA + EA + EA + EA + …
- EA + EA + EA + EA + EA + …
- EA + EA + EA + EA + EA + …
- EA + EA + EA + EA + EA + …

A: Four E A series.

I was reminded of this joke when I received a holiday card from my friend DAVE. His wife is ANNE, his daughter is LENA, and his son is AXEL. It struck me as interesting that all four names in their family (1) consist of four letters and (2) contain the letters E and A. That led me to create the following puzzle for my sons.

Can you find a name that fits each of the following patterns?

A __ __ E | A __ E __ | A E __ __ | |||

__ A __ E | __ A E __ | E A __ __ | |||

__ __ A E | __ E A __ | E __ A __ | |||

__ __ E A | __ E __ A | E __ __ A |

I was able to complete 75% of the puzzle on my own, and I was able to complete 100% of it with some help from Google. No doubt — the one with AE in the third and fourth positions was the toughest. Good luck!

Incidentally, comedian Penn Jillette (of Penn & Teller) believes the names Dave (my friend) and Alex (my son) were for losers.

### Math Jokes for National Sleep Day

If you like sleep, boy, have we got some holidays for you.

Today is National Sleep Day. eHow.com has a list of things to do today, and the first thing on their list — shocker! — is sleep. (Okay, technically they list “sleep in,” but doesn’t that seem obvious for this particular holiday?)

In the U.S., February 19 is National Sleep In Day; in Britain, it’s October 31. One blogger declared that May 11 should be National Sleep Naked Day.

March 3-9, 2012, is National Sleep Awareness Week, which occurs annually the week before the change to Daylight Savings Time.

And if you’re one of those folks who really likes to extend your holidays, you don’t need to limit your love of sleep to just one day or even a week. November is National Sleep Comfort Month, and May has been dubbed National Sleep Better Month.

Holy criminy! Is all of this really necessary? Luckily, mathy folks really like to sleep.

What do mathematicians sleep on?

Matrices.

Mathy folks also appreciate that others need sleep, too.

A math teacher is someone who talks in someone else’s sleep.

Married mathy folks have a keen awareness of how much sleep they need.

A single mathematician was asked, “If you go to bed eight hours before you have to wake up, and your girlfriend wants to have two hours of sex, how much sleep will you get?” He answered, “6 hours.”

When a married mathematician was asked the same question about having two hours of sex with his wife, he responded, “7 hours, 57 minutes. Why does it matter what she wants?”

Finally, a joke about the other meaning of the word *sleep*.

Wife:“If I died, would you get married again?”

Mathematician:“No.”

Wife:“Why not? Don’t you like being married?”

Mathematician:“Of course, I do.”

Wife:“Then why wouldn’t you remarry?”

Mathematician:“Fine, I’ll remarry.”

Wife:“You would?”

Mathematician:(groan)

Wife:“Would you live in our house, too?”

Mathematician:“Sure, it’s a great house.”

Wife:“Would you sleep with her in our bed?”

Mathematician:“Where else would we sleep?”

Wife:“Would you let her drive my car?”

Mathematician:“Probably. It’s brand new.”

Wife:“And would you let her use my golf clubs?”

Mathematician:“No, she’s left-handed…”