## Archive for December, 2010

### On To The Next…

Another year almost over, the next one about to begin. Which makes me think of sequences…

Math jokes make all my Cauchy sequences converge.

And here’s a Fox Trot cartoon with sequences:

You undoubtedly can identify the first sequence: 1, 1, 2, 3, 5, 8, 13, …

Of course, it’s the Fibonacci sequence.

But do you recognize the sequence from the last panel? It begins 3, 0, 2, 3, 2, 5, …

If not, here’s your first question:

What’s the next term in that sequence?

And your second question:

What is the general formula for the terms in that sequence? (A recursive formula is completely acceptable. The explicit formula is quite a beast.)

Like the Fibonacci sequence, this sequence is defined by a recurrence relation. In particular,

P(0) = 3, P(1) = 0, P(2) = 2, and P(*n*) = P(*n* – 2) + P(*n* – 3)

This sequence has an amazing property: For any natural number* n*, if *n *is prime, then* n* | P(*n*). No, really. You can check for yourself. P(3), P(5), and P(7) are trivial, since P(3) = 3, P(5) = 5, and P(7) = 7. But…

P(11) = 22, and 11|22

P(13) = 39, and 13|39

P(17) = 119, and 17|119

P(19) = 209, and 19|209

P(23) = 644, and 23|644

Also like the Fibonacci sequence, the ratio of consecutive numbers in this sequence have a constant ratio. As we all know, the ratio of consecutive Fibonacci numbers is approximately 1.618034, better known as the golden ratio. For the Perrin sequence, the ratio of consecutive numbers is approximately 1.324718, known as the plastic constant.

Cool stuff.

Here are a couple other sequences for you to ponder as you prepare for the new year. Can you determine the next term?

O, T, T, F, F, S, S, E, …

3, 3, 5, 4, 4, 3, 5, 5, …

6, 14, 24, 36, 50, …

### Permutations

Tonight at dinner, my wife told Alex that he had to eat his carrots. “How many?” he asked. There were three on his plate. “Two,” she responded. (For what it’s worth, I deplore these dinnertime negotiations. I put three carrots on his plate — I expect him to eat three carrots. If I only wanted him to eat two, I would have given him only two. On the other hand, my kids are 3½ and they actually ask for broccoli, edamame, carrots, kale, and a host of other veggies, so I can’t really complain.)

Alex then turned to his brother and asked, “Which two should I eat, Eli?” He picked up Carrot A and Carrot B and asked, “These two?”; he then returned Carrot B to his plate and picked up Carrot C and asked, “These two?”; and, finally he picked up Carrot B and Carrot C and asked, “Or these two?”

I was pretty psyched about Alex’s “proof without words” that _{3}*C*_{2} = 3.

Of course, we all know how cool permutations are, since all of our inboxes have been filled by the text of a letter written by Graham Rawlinson to *New Scientist* in 1999. You know the one, which purported “that randomizing letters in the middle of words had little or no effect on the ability of skilled readers to understand the text.”

For iancstne, you utadnnersd this snenctee celalry eevn tughoh the ltetres of msot wodrs are out of oedrr.

It is inaccurate to say that there is “no effect,” however. A follow-up study showed that college students experienced a 12% decrease in overall reading speed when confronted with sentences containing transposed letters. Quite a few of the other statements in the email that we received — for instance, that there was a study done at Cambridge (there wasn’t) — were also inaccurate. Even the email itself is misleading, ostensibly written to enhance the desired effect and further prove its point; in truth, almost half of the words contain only two or three letters and are spelled correctly.

But don’t blame Graham Rawlinson for all of that. That’s just the way things work in cyberspace.

Okay, enough already. How ’bout some math?

How many permutations exist for the word PERMUTATIONS?

And last but not least, some permutation jokes…

Combinatorists do it in every possible permutation, but they do it discretely.

What do you get if you add Daytona Beach, Pismo Beach, Palm Beach, and South Beach in various permutations?

Sums of beaches.

### Square Deal

Recently, my twin sons Alex and Eli have taken a shine to crossword puzzles. They’re only 3½, but they love letters and words, so I started making up crosswords for them using a free online crossword puzzle maker. I construct clues based on things they know — for instance, REMY is the answer to the clue OUR DOG, and IDAHO is the answer to the clue STATE NAME WE CAN SPELL WITH OUR BATH TUB LETTERS. (They have a set of foam alphabet letters, but each letter A‑Z occurs only once, so it’s not possible to spell any words with repeated letters.) They don’t quite have the motor skills to write the letters, so they read the clues and spell the answers aloud as I fill in the grid.

Tonight, I asked if they’d like to help me make a crossword puzzle. I drew a 3 × 3 grid, and I asked, “To make a crossword puzzle, you have to fill in words both down and across. Can you give me a word with three letters?” Eli suggested TOW, which I used in the first row of the grid:

T | O | W |

Then I asked, “Okay, so the word in the first column starts with a T. Can you think of a three‑letter word that starts with a T?” Never one to overlook the obvious, Eli suggested TOW again. I filled it in, and we moved to the middle column. “Can you think of a three‑letter word that starts with an O?” Alex suggested ONE. Eli immediately realized that we could now add an E at the end of the middle row to make ONE. At that point, eight of the nine squares were filled. I pointed to the third column. “Can you think of a three-letter word that starts with a W and an E?” Eli shouted WET, and the grid was complete:

T | O | W |

O | N | E |

W | E | T |

Eli then pointed out that ELI contains three letters. “Let’s make another one!” he said. So we did:

W | E | T |

E | L | I |

T | I | E |

Alex then requested that we make a 4 × 4 grid that included his name, and I was very impressed with the grid that they concocted:

I | O | W | A |

O | V | A | L |

W | A | V | E |

A | L | E | X |

I told you that story mainly because I like talking about my sons, but also because it leads me to a cool puzzle that I think you’ll enjoy.

Use the point values for each letter as in the word game SCRABBLE:

- 1 point: A, E, I, O, L, N, R, S, T, U
- 2 points: D, G
- 3 points: B, C, M, P
- 4 points: F, H, V, W, Y
- 5 points: K
- 8 points: J, X
- 10 points: Q, Z

Create a 4 × 4 grid composed of common English words (you can use the list of official Scrabble four‑letter words** as a reference) such that the sum of the point values of the 16 letters is as high as possible.

The 4 × 4 grid that my sons created would be disqualified, because IOWA and ALEX are proper nouns. That aside, it contains four A’s; two O’s, W’s, V’s, L’s and E’s; one I; and, one X. If it were acceptable, it would be worth 35 points.

I was able to create a grid with a sum of 62 points. I’m sure that better grids are possible. What’s the best that you can do?

** Special thanks to Veky Edgar, who pointed out that the list of four-letter words appearing on my web site was incorrect. The list has been updated, and I’ve stolen it from a more credible source this time, so I believe it is now correct.

### 2 × 5 Hottest Mathematicians

The WordPress admin page lets me know what terms folks are searching for when they find the MJ4MF blog. This past week, one explorer was looking for “very attractive mathematicians.” It occurred to me that I had nothing to offer this person. Well, let’s rectify that immediately!

I considered posting a list of the 10 Hottest Mathematicians, but I decided not to intermingle boys and girls. There are (at least) two good reasons for keeping them separate. First, my selection method is completely arbitrary, and without objective criteria, I’m not sure how I’d determine whether a man is more attractive than a woman. I mean, I know my preference, but that doesn’t really seem fair. Second, there’s an issue with population sample — there have been far more male mathematicians in history, and a mixed-gender list of the 10 Hottest Mathematicians would undoubtedly have only one or two women.

So instead, here are two separate lists, one of the hottest female mathematicians, followed by a list of the hottest male mathematicians. The lists are, of course, entirely subjective, and I welcome an open debate on the subject.

**Hottest Female Mathematicians**

*Danica McKellar (1975 – )*

Yes, I know I’m gonna catch grief for including her on this list. But (a) she’s beautiful, (b) she’s the McKellar in the Chayes-McKellar-Winn theorem, which appeared in a paper she published as an undergrad, and (c) her Erdos number is 4. If you still don’t like her inclusion on this list, then protest by refusing to look at the picture below. (Yeah, right — I double dog dare you to look away!)

*Anneli Cahn Lax (1922 – 1999)*

She was a gifted mathematician and a first-rate student, but could her beauty have been another reason she was Richard Courant’s only female student?

*Sofia Vasilyevna Kovalevskaya (1850 – 1891)*

I’m sure the first female with a full professorship in mathematics in Northern Europe made her male students’ hearts flutter.

*Hypatia of Alexandria (370 – 415)*

A woman with truly classical beauty, Hypatia is believed to be the first woman to write about mathematics.

*Sophie Willock Bryant (1850 – 1922)*

In addition to an impressive mathematical career, this minister’s daughter published works on Irish history, religion, and philosophy. She was also a rugged outdoorswoman with a penchant for mountain climbing.

**Hottest Male Mathematicians**

*Renato Caccioppoli (1904 – 1959)*

With classical European good looks, Caccioppoli was “one of the most interesting and charming mathematical figures of the 20th century.” Perhaps he is all the more attractive because he was so intriguing. A movie about the events leading to his eventual suicide (*Morte di un Matematico Napoletano*) won seven awards and was nominated for an eighth.

*Donald C. Spencer (1912 – 2001)*

A former chair of the math department at Princeton once described Spencer as “the most attractive mathematician in America.” Admittedly, he didn’t have much competition for the title in the mid‑1950’s, but who I am to argue with a former chair of the Princeton math department?

*Evariste Galois (1811 – 1832)*

This striking young Frenchman had good looks a-plenty. But if that’s not enough for you, how about this? He died in a duel defending the honor of his love. Many women (including my wife) thinks that’s downright sexy.

*Andrew Wiles (1953 – )*

I’ve heard several women refer to him as “an adorable dork.” Bonus points for proving Fermat’s Last Theorem.

*Carl Friedrich Gauss (1777 – 1855)*

Admittedly, he didn’t age well — but he was quite a looker in his younger years, and his brilliance shone brightly till the end of his days.

### 5 New (Mathy) Words from the OED

The OED is the *Oxford English Dictionary*, the world’s largest dictionary of the English language (though not, however, the world’s largest dictionary — that distinction belongs to the Dutch dictionary *Woordenboek der Nederlandsche Taal*.) The OED attempts to “present in alphabetical series the words that have formed the English vocabulary from the time of the earliest records down to the present day.”

Forgive me if you already knew that. I just never assume that mathy folks know (or care) about the OED, just as I don’t assume that literary people are familiar with the Fundamental Theorem of Algebra.

The OED is revised four times a year. Over 2,400 entries were added during the most recent revision (December 1), and the following are new words that entered the dictionary during the past year:

**coordinate geometry**– system in which points, lines, shapes, and surfaces are represented by algebraic expressions. This term was added as a “subordinate entry,” meaning that it appears under the main entry “coordinate.” Still, it’s surprising that it took four millennia to get the word into the OED (Descartes introduced the coordinate plane in*Discourse on the Method of Reasoning Well and Seeking Truth in the Sciences*in 1637).**cyberslacking –**using internet access at work for personal reasons while maintaining the appearance of working. For example, updating a math joke blog instead of revising the budget per your director’s request.**ego-surfing**– searching the web for instances of your own name. Or checking Amazon daily to determine the sales ranking of your book.**Richard Snary**– as Dick is a shortened form of Richard, this is a pun for “dictionary.” (Go ahead, say, “Dick Snary” out loud and listen to what it sounds like.) This is a 17th‑century slang term that has somehow hung around for several hundred years, not unlike the math pun, “Pie aren’t square, pie are round!”**Rolle’s theorem**– a theorem which says that a differentiable function with equal values at two points must have a point somewhere in between where the first derivative is zero. It’s good to see that calculus is getting some props.**rope’s length**– in knot theory, the minimal length of an ideally flexible rope needed to tie a given knot. What’s unclear is whether this is the intention for the entry in the OED. Knot theorists use*ropelength*, not*rope’s length*, to describe this concept, but a Google search fails to reveal any common use of “rope’s length.”**techy**– an informal way of designating technological sophistication. This is heartening — with techy now officially recognized, “mathy” can’t be far behind.

The following are entries that *should* be added but probably never will be:

**decagon**– what a croupier says after being fired.**dilemma**– a lemma with two results.**paradox**– two wharves.**protractor**– in favor of farm machinery.**tangent**– a sun-burned gentleman.**Calvin Culus, Albert Jabra, and Paulina Hedron**– hey, if Richard Snary gets in for “dictionary,” then it’s only reasonable that we math folks get some stupid puns, too.

### Infinite Gifts

A mathematician came across a lamp. He rubbed it, and a genie appeared. “Can I have three wishes?” asked the mathematician.

The genie had decided long ago that granting three wishes for his release was passé, and he generally refused such requests. With the holidays just around the corner, however, he was feeling charitable. “Okay, fine,” he said. “But agreeing to grant you three wishes takes care of your first wish, so you have only two remaining.”

The mathematician thought for a moment. It sure would be helpful to have a little extra money with which to buy gifts for my friends and family, he thought. “Okay,” said the mathematician, “I’d like a coin bag that is always full of gold; no matter how many gold coins I remove, there will always be some left.”

“Done,” said the genie as he handed the mathematician a coin bag.

The mathematician tried it out and, sure enough, no matter how many coins he removed from the bag, it always remained full. He was absolutely delighted.

“So, then, what would you like for your third wish?” asked the genie.

“Ah, well, that coin bag is awesome,” said the mathematician. “In fact, it’s so good, I’d like another one just like it!”

### WISE in Doha, Qatar

I spent last week in Doha, Qatar, at the World Innovation Summit for Education (WISE). The architecture in Doha is amazing, and one of the highlights is the Museum of Islamic Art, designed by I. M. Pei. The photo below shows the west courtyard, with the skyline of Doha’s business district visible through the arches:

But I was there for a summit, not to play tourist, and I spent three action-packed days attending sessions. The quote of the week came from Mushtaq Chappra of The Citizens Foundation:

A problem unrecognized cannot be solved.

Mushtaq is an amazing man. He founded The Citizens Foundation, which has raised private funds and built 660 schools across Pakistan. He believes that enough money from private contributors can be raised so that all 180 million children who currently do not have access to primary education can be sent to school. His argument is simple: There are 10,000,000 millionaires in the world, and he only needs 330,000 of them to contribute to make it happen.

For his work, Mushtaq received a WISE Award. (You can read more about Mushtaq’s project as well as the five other award-winning projects here.) Mushtaq was only one of many education visionaries who attended the conference. Other attendees included:

**Dennis Littky**, who formed his own high school where students choose what they want to learn instead of being told what to learn. Two days a week consist of structured classes, but the other three days each week are open for students to use the library, local businesses, nearby colleges, and the community to learn more about their chosen topic. The best part? These students are motivated because they’re learning what they want to learn*and*their standardized test scores are higher than those of students at any nearby school.**Steen Jorgenson**, Sector Director of the MENA Region for the World Bank, who argued that a carbon-credit model works for business, so why not adopt a “human capital credit” for education?**Tim Rylands**, an amazing educator who showed us Visuwords (a visual thesaurus) and Tag Galaxy (which may be the coolest site I’ve ever seen) and lots of other awesome educational tools for the classroom.**Stephen Heppell**, a professor at the Center for Excellence in Media Practice, who argued that a school of education should not be allowed to train future teachers unless they run a model school with demonstrated excellence. (Hear, hear!)**Sara de Freitas**, of the Serious Games Institute, who stated that wikipedia is the second most used research tool in the United Kingdom. (Sadly, I failed to ask her which one was the most used.)

The presenters’ stories were awesome, and my passion for education was renewed.

But perhaps my favorite part of the summit was the announcement made by the organizers at the closing session. At WISE 2011, they will be awarding the WISE Prize for Education, which will reward an individual or teams of individuals who have made an outstanding contribution to any field or level of education. The winner, as selected by an international jury, will receive a gold medal and $500,000. Finally — an award to recognize the accompishments of educators in the same way that the Nobel Prize recognizes scientists.

Finally, let me tell you about a great personal moment. Riding the elevator, I began chatting with a British educator. When he saw National Council of Teachers of Mathematics listed as my affiliation, he said, “Oh, I used to teach maths, and I used to love to use some of the resources on an NCTM site. Now, what was it called?” I asked if he was thinking of Illuminations, and he said, “Yes, yes, that’s it!” I then told him that Illuminations was my project. “Well, jolly good,” he said, “What an honour to meet the man who runs Illuminations. My week is complete!”