5 – 1 = 4, unless it equals 5

The Union Internationale de Pentathlon Moderne held their 2008 Congress in Guatemala in November, and the main headline from the Congress was the decision to combine two of the five Modern Pentathlon events into a single competition.

From its introduction at the 1912 Olympics until the end of 2008, the five events of the Modern Pentathlon have included running, fencing, swimming, show jumping, and shooting. (The phrase Modern Pentathlon is meant to distinguish this competition from the Pentathlon, an ancient Olympic event involving competitions in javelin, shot put, wrestling, discus, and running.)

The recent change has involved combining the running and shooting events into a combined event:

Athletes will begin … with a short run to the shooting range, here they take 5 shots before a 1km run. They repeat this 2 further times, in total 15 shots and 3km run. [link]

Unlike the winter biathlon, athletes will not carry their sidearms while running.

The Yahoo News article on this describes it as changing the Pentathlon from five to four events, which raises some obvious doubts as to nomenclature. (Quadrathlon, perhaps?) The UIPM insists, though, that the Pentathlon is not being reduced to four events, since “All five disciplines are still equally represented and competed”.

I anticipate a fantastic Final Jeopardy question: “The combined number of events in the Decathlon, the Heptathlon, and the Modern Pentathlon.”

Nominations for the 2008 Eddies

Here are 360’s nominations for the 2008 Edublog Awards:

Best New Blog: Division by Zero

Best Resource Sharing Blog: Teaching College Math Technology Blog

Best Teacher Blog: JD2718

Each of these three blogs is one we read regularly here at 360.  Check them out, and don’t forget to nominate your favorites by the 30th!

Perfection

In the episode “The Boneless Bride in the River” in Season 2 of the TV show Bones, the body of a young woman is found in a river, and it’s discovered that she was likely brought over to the US on a fiancée visa. About 14 ½ minutes into the episode, two of the main characters (Special Agent Seeley Booth and Forensic Anthropologist Dr. Temperance “Bones” Brennan) have this conversation:

Booth: Homeland Security says the fiancée visa was expedited by a lawyer on retainer into a smaller bride agency here in town called “The Perfect Wife”.
Brennan: Oh that sounds archaic.
Booth: No, you know, in therapy I learned that superlatives like perfect are meaningless.
Brennan: Not in science. A perfect number is a number whose divisors add up to itself, as in one plus two plus three equals six.
Booth: Well, in therapy I learned that definitive statements are by their very nature, wrong.
Brennan: Isn’t the statement “definitive statements are by their very nature wrong”, definitive, and thus wrong?

Speaking of wrong, Brennan was a little bit wrong in her definition: a perfect number is one whose proper divisors add up to itself. But still, neat math in a neat show is always worth a mention. And perfect numbers are pretty neat, because like so much in number theory they’re simple but there are still open problems about them.

A bit of history [where “a bit” apparently means “a lot because I don’t know how to edit today”]: perfect numbers were studied by Pythagoras, which makes the concept at least 2500 years old. Euclid also talked about perfect numbers a few hundred years later in Book IX of The Elements. In particular, Thomas Heath’s translation of Proposition 36 states:

If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect.

As a translation of this translation, this is saying that if 1+2+2²+…+2^k is prime, then that sum times the last number in the sum (2^k) must be perfect. For example, 1+2=3 is prime so 3·2=6 is perfect. Likewise, 1+2+4=7 is prime, and 7·4=28 is perfect. [As a note, the formula is is sometime written out algebraically with 2^k+1-1 used instead of 1+2+2²+…+2^k; in addition, when it’s written that way it’s sometimes reindexed so that k is used in place of k+1, and the statement becomes “If 2^k-1 is prime, then (2^k-1)·2^k-1 is perfect.”]

A few hundred years after Euclid, Nicomachus wrote some more information about perfect numbers. According to the St. Andrew’s web site, he made
five claims:

(1) The nth perfect number has n digits.
(2) All perfect numbers are even.
(3) All perfect numbers end in 6 and 8 alternately.
(4) Euclid’s algorithm [described above] to generate perfect numbers will give all perfect numbers
(5) There are infinitely many perfect numbers.

Nicomachus’s word was law as far as perfect numbers were concerned, and his claims, while unproven, were believed for decades centuries a really really long time. Of course, there were only four perfect numbers known at that time (6, 28, 496, and 8128), so they really didn’t have much to go on. In reality, the 5^th and 6^th perfect numbers (33,550,336 and 8,589,869,056 respectively) disprove claims (1) and the “alternately” portion of (3), but it took a while for someone to discover those.

(Which leads to a little tangent: who did discover those numbers? Ismail ibn Ibrahim ibn Fallus (1194-1239) made a list of ten numbers he thought were perfect. Three of them weren’t, but the other seven were. Sadly, a lot of other mathematicians had no idea about this list: mathematicians in Western Europe had to wait another 350 years for those numbers to enter their collective psyche, during which time a few other mathematicians found the 5^th and 6^th perfect numbers and were equally ignored.)

But even after it was known that Nicomachus’s claims weren’t themselves perfect (ba DUM!), mathematicians continued to study the numbers. In the 1600s Pierre Fermat tried to find patterns, and ended up discovering his Little Theorem as a consequence. Marin Mersenne also spent some time on it, and in fact his exploration of when 2^k-1 is prime, as a part of that theorem of Euclid’s mentioned above, led to the notion of Mersenne primes (primes of the form 2^k-1 where k itself is prime).

In the 1700s Leonhard Euler entered the fray. He couldn’t prove that Euclid’s formula generated all perfect numbers, but he did show that it generated all even perfect numbers. And a bunch of other mathematicians spent a lot of time trying to show that numbers were or were not perfect (which was related to showing that specific numbers of the form 2^k-1 were or were not prime), a challenging task in the pre-computer days.

Not that we’re doing much better now. As of this moment, we still only know of 46 perfect numbers, and they’re pretty big. We do know a few cool things about perfect numbers in general:

• Even perfect numbers end in 6 or 8.
• Even perfect numbers are triangular numbers (e.g. 6=1+2+3 and 28=1+2+3+4+5+6+7) where the ending digit is one less than a power of 2.
• The reciprocals of the divisors of perfect numbers all add up to 2:
  
  

But there’s a lot we still don’t know:

• We don’t know if all perfect numbers are even.
• We don’t know if there are a finite or infinite number of perfect numbers.

In other words, perfection continues to eludes us.

Nicolaus Copernicus: lost and found

He was born 535 years ago as Mikolaj Kopernik or Nicolaus Koppernigk, and he died 70 years later. In between, he proposed that the sun and not the earth is at the center of the universe, which was a bit of a shock at the time.

When he died in 1543, he was buried in Frombork Cathedral in Northern Poland but his exact grave was never marked. Then four years ago the Bishop (Jacek Jezierski) requested help from archaeologist Dr. Jerzy Gassowski in finding the grave. It took a few years, but a grave was indeed found in an appropriate spot. But was this Copernicus? The body was the right age, but that’s hardly conclusive. What is fairly conclusive is DNA evidence, except that there weren’t exactly databases set up at the time. What they needed was something like a piece of his hair.

And that, it appears, is exactly what they had. Some of Copernicus’s books (that he himself owned, not that he wrote) are still around and on display. In one of those books were four hairs. Dr. Marie Allen tested the DNA, and it turned out that two of those hairs belong to the body under the cathedral.

Proof? Maybe not. But as CSI meets De revolutionibus, it’s pretty cool.

For more information, see the Post-Gazette or CNN.

A Paltry Geometric Dilemma, Part 2

Via Internet Time-Wasters III at Making Light, here’s the World’s Hardest Easy Geometry Problem:

From the site:

Using only elementary geometry, determine angle x. Provide a step-by-step proof.

You may only use elementary geometry, such as the fact that the angles of a triangle add up to 180 degrees and the basic congruent triangle rules (side-angle-side, etc.). You may not use more advanced trigonomery, such as the law of sines, the law of cosines, etc.

Small hints are available on the site, but they’re very small. Think of this as something to keep you busy while you’re digesting all that turkey.

(The first Paltry Geometric Dilemma.)

Carnival of Mathematics #44

It’s two months before the 44^th president will be in the White House, but the 44^th Carnival of Mathematics is already here!  It’s got everything from controversy and fraud to games and a request to include some good mathematics in the list of 1,000,000 good things!

This carnival is being hosted by Maxwell’s Demon, the blog of Edmund Harriss, who does both mathematics and art.    On his blog he has a neat post on rep-tiles here (which leads to fractal dragons, on which is based the crocheting project Here Be Dragons at Woolly Thoughts), and on his homepage he shows some of his artwork here and here, including the Octagonal Gasket (like Sierpinski’s Triangle, but with octagons)

and these wooden fractal puzzle pieces that can be put together in different ways

(Both are licensed under Creative Commons)

Enjoy!

A complicated proposal

Matt Snyder and Tanya Hein are getting married this Saturday in Madison. No, I don’t know them, but I read about Matt’s proposal in this afternoon’s Wisconsin State Journal. They’d been dating a few years and were both ready to get engaged, but Matt decided to make it special. And nothing says special like a mathematical treasure hunt. The first key (to one of several locked drawers in a treasure chest) led to a coded message, which ended up instructing Tanya to drive to another location. There she found another key, which fit into another drawer, which had another puzzle in it. Each puzzle, when solved, instructed her to go to a different location where there was another key to another drawer that held — you guessed it — another puzzle.

Each location had a reason — the J.T. Whitney’s parking lot was the site of their first kiss. Some of the encrypted messages, involving complex letter codes, linear algebra and trigonometry, were highly difficult to solve.

Presumably there was no partial credit for getting close to a solution. But neither was there a time limit — this was no weekend jaunt. It took two months before Tanya finally got to the proposal. Her response was to give Matt an envelope with a puzzle inside to decipher.

I can only imagine what their seating chart will be.

Speaking of proposals, are there really a thousand yellow daisies in this scene (about a minute in)? It says there are, and estimation is not my own personal strength in mathematics, but it still seems to me that there’s actually quite a few more than that. Anyone know?

KenKen

Barbie might think math is hard, but perhaps she’d be willing to try this game.  I just heard about it yesterday around the water cooler copier (Thanks Lynn!).  It’s a little bit like Killer Sudoku, but with operations other than just addition.

The initial grid is a square, perhaps 4×4 or 6×6.  Like Sudoku, the digits 1-4 or 1-6 are put in cells so that each row contains exactly one of the digits and each column contains exactly one of the digits.

In addition, there are groups of cells (like the cages in Killer Sudoku).  In the corner is a number and an operation.  If the digits in a group are put together with that operation, they form the given number.  For example, if there are two cells with “8x” in the corner, the digits multiply to 8, and so must be 1&8 or 2&4 in some order.  Likewise, “2/” (using / for division) could be 1&2, 2&4, 3&6, or 4&8 where the pairs are in either order.  While you can’t repeat digits in any row or column, you can repeat digits in a group of cells.

Here’s one to try, courtesy of SGBailey.

(You can find the answers in the place that always gives you answers.)

If you search for KenKen there seem to be several sites where you can get your daily fix (including the New York Times), but most require registration.  And apparently in some of the harder versions the operation is left off so that you know the answer, but not whether it is a sum, product, quotient, or different.  I can totally imagine a variation that uses multiple operations in a single group of cells, but as far as I know that hasn’t been invented yet.

Math Careers: Computer Animation (Games and Movies)

Video gaming.  Creating cool movies.  If you want to do computer animation, you need math.

One of our recent graduates ended up working in just this field.  Actually, he wasn’t even a math major:  he was a theater major who picked up a math minor in his last two years.   He’s now doing graphics programming, and it requires a ton of math:  he mentioned that the most important course turned out to be Linear Algebra.   In all fairness, since he’s officially a software engineer I have to assume he had some computer science as well.  But we don’t even offer a computer science degree, and he started work in this right out of college, so I don’ t know how much formal computer science training he had.

The article Math in the Movies from 2007 gives similar information.  There’s a 90-second video there, which I can’t seem to reproduce here, but in part of it the announcer Cindy Demus(?) says:

Trigonometry helps rotate and move characters, while algebra creates the special effects that make images shine and sparkle.  Calculus helps light up a scene and new math techniques turn images like this [flat and blocky] into this [smooth and more realistic].

Then Tony DeRose, a computer scientist from Pixar Animation Studios, added

I remember as a mathematics student thinking, “Well, where am I ever going to use simultaneous equations?” And I find myself using them every day, all the time now.”

So with some sort of computer background and good math skills, that job could be yours as well.

Generating Pythagorean Triples

A recent post at jd2718 noted that for all Pythagorean triples , that is to say, for all positive integer solutions to , it turns out that 60 divides the product .

In the course of exploring proofs of that result, I suggested a proof using the representation , which generates all primitive Pythagorean triples (triples which have no common factors).  Proving that 60 divides their product proves the general result, as every Pythagorean triple has the form , where k is a positive integer and is primitive.

In the ensuing discussion, a question arose as to how we know that all possible Pythagorean triples can be found in this way.   Since it is easier for me to post LaTeX code to this blog rather than in the comments on the jd2718 blog, I’ll present the proof here.

Suppose that , with positive integers having no common factors.  This implies that exactly one of a and  b is even (since if both are even, then so is c, leading to a contradiction, while if both a and b are odd, then both and are congruent to 1 mod 4, while would be congruent to 0 mod 4, not 2 mod 4).

So without loss of generality, assume that a is even.  Thus is also even, and since and differ by an even number, both of those factors must be even.

Furthermore, exactly one of and is 2x(an odd number), since if both factors are divisible by 4, it would follow that both b and c would be even (since b and c are .)   Likewise, any odd common factor of and would also then be a common factor of both b and c, from which it follows that

• and have no odd common factors
• one of these expressions is of the form (for some odd u)
• and hence the other expression is of the form , where , where u and v are positive integers

It now follows that , and .  We can find a using , and thus .

Conclusion:  every primitive Pythagorean triple has the form , for positive integers u and v.

Justifying an addictive game: it’s GEOMETRY!

One of my students recently sent me an email with a link to the game Shinju. (Thanks Chris Z!) He included the note, “I thought you would enjoy this game I came across. It deals with Maxi geometry!”

The game is played on an 8×8 board that is partially covered with shells, one of which has a golden pearl inside. You click on a shell, and it either reveals the hidden pearl (You Win!) or it gives a number indicating how many steps away the pearl is, where each step can be horizontal, vertical, or diagonal. It’s fun to play the game, and I found it surprisingly addicting.

Math Confusion in the News: percent

Last week Governor Arnold Schwarzenegger proposed a temporary (3-year) sales tax increase in California to help close the budget deficit. Some newspapers, however, are mixing up the amount of the increase in an effort to get the news out.

Error #1: “Governor Proposes 1.5 Percent Sales Tax Hike” from MyFox Los Angeles

The proposed increase isn’t actually 1.5 percent (which wouldn’t be all that much). It’s 1.5 percentage points, which makes it about a 20% increase from the current 7.25% state sales tax. I suspect that most people understand what the headline intends, however, because using “percent” instead of “percentage point” is fairly common. (Kudos to the LA Times for being precise in their story!)

Error #2: “Schwarzenegger proposes 1.5-cent sales tax increase to close budget gap” from the San Jose Mercury News

This headline is just wrong. A 1.5¢ tax? And it’s not just in the title, but in the body of the story. Several other newspapers made the same mistake, either running the Mercury News story without correction or writing their own story about the 1.5-cent increase (I’m looking at you, Sacramento Bee). Indeed, these 1.5-cent increase stories were common enough that I actually double checked that it wasn’t some new terminology for “percentage points”.

Incidentally, the word “per cent” is only 440 years old, and “per centage” only 222 years old. Tidbits from the OED!

Some WWI tidbits for Veterans Day

I was just reading the article “Dr. Veblen Takes a Uniform:  Mathematics in the First World War” by David Alan Grier (from the American Mathematical Monthly, Dec 2001).  The full article is about the Captain Oswald Veblen and the math folk who worked with him at a military facility in Aberdeen, Maryland, but there are plenty of additional tidbits  For example:

Over 150 mathematicians served in the First World War. Many took conventional military roles but half found ways to employ their mathematics. They worked as surveyors, assisted cartographers, and taught trigonometry to officer candidates.  (p. 923)

A lot of the work involved making ballistic tables, which was not such a simple task:

These mathematicians did part of their computing at the actual test ranges, where they served as observers, data collectors, and range officers. On the water range, a large range that ended in the Chesapeake Bay, the mathematicians were stationed on towers along the shore. After they observed the splash of a shell hitting the water, they would compute the range of the shot and telephone their result to the firing station. At the firing station, a second mathematician would adjust the range for changes in temperature, humidity, and wind. Once the series of shots was completed, a mathematician at a central office would compute ballistics coefficients and create range tables.  (p. 927)

Which sounds about as exciting as alphabetizing names for the phone book.  By hand.  In the mud.  Nonetheless, it had to be done and so it was.  Another set of computations, not as cold as the above,  involved solving differential equations numerically to computer trajectories.  The main staff started doing this, but then they got bored and made the enlisted men take over.  In Washington DC, however, there weren’t enough men to do they job and so the Diff Equ was handed off to women:

By early summer, [Forest] Moulton hired eight women. All had graduated from prominent universities during the prior two years. All had been mathematics majors. They came from University of Chicago, Brown University, Cornell University, Northwestern University, Columbia University and the George Washington University. For these women, the war was an opportunity to play a role, perhaps only briefly, on the public stage.  (p. 929)

Rosie the Riveter, meet Connie the Computer!  Actually, the only women mentioned by name is Elizabeth Webb Wilson, who turned down nine other job offers before taking on this job.  And apparently in their spare time, at least some of the women involved also fought for suffrage.

All in all, it was a good read.  Happy Veterans Day!

Math Careers: Astronaut

Additional training required.

Witness Lieutenant Colonel Robert Shane Kimbrough (who goes by Shane). He’s heading up into the sky Friday on the Space Shuttle Endeavor with Colonel Eric Boe and flight engineer Sandra Magnus. Lt. Col Kimbrough didn’t get his undergraduate degree in math (it was in Aerospace Engineering), but he got his MS in Operations Research from Georgia Tech and taught math at West Point. He’s given talks to school kids about how important math and science were in his being able to become an astronaut. And most recently he recognized his high school Calculus teacher (Sandy Sturgeon) in an official NASA letter because she was “instrumental in helping him get where he is today” and made calculus both real and fun.

For more information, see his official NASA biography or this recent article on ajc.com.

This is the premier post in a new series inspired by the common question, “What can I do with a degree in Math?”

Zero isn’t nothing

When there are a lot of zeros after a number, it’s easy to add or drop some. Apparently.

Two examples: there’s an email that was going around in September about the proposed Federal Bailout Plan. The plan at the time was proposed to be around $700 billion, and the author of the email pointed out that if this was divided by the 200 million or so adults in the United States, it amounted to around $350,000 per person, and the author suggested that a better solution would be to mail a check for $350,000 to each person instead. Now before you go looking for ways to spend the money, you need to double check the math. It turns out that $700 billion divided by 200 million adults is only $3500/person. No small amount, to be sure, but not quite enough to pay off the mortgage. [Note: Variations on this suggestion appeared on several websites with different numbers, but they were all off by a factor of 10 or 100.]

The web sites that posted this quickly caught the mistake and made a correction. And all was well. Except that this pesky problem of the wandering zero just appeared again. This article in The Guardian (and repeated elsewhere) begins:

Nuclear power plants smaller than a garden shed and able to power 20,000 homes will be on sale within five years, say scientists at Los Alamos, the US government laboratory which developed the first atomic bomb….’Our goal is to generate electricity for 10 cents a watt anywhere in the world,’ said John Deal, chief executive of Hyperion. ‘They will cost approximately $25m [£13m] each. For a community with 10,000 households, that is a very affordable $250 per home.’

Feelings about neighborhood nuclear reactors aside, there’s a little problem with the math. $25 million divided by 10,000 households is $2,500 per home, not $250.* Perhaps still affordable compared to the cost of heating one’s home in the wintertime, but illustrative of the care that one needs to take when dealing with really large numbers. Especially when it comes to bailouts or nuclear reactors.

*an error originally caught by Classical Values