A New Year’s Puzzle

Happy New Year!

Now that you’ve caught up on your sleeping and filled yourself with too much food, you probably have a hankerin’ for a math problem.  Here is one we particularly like because it only uses Calc III sequence and series stuff:

Everybody knows that the harmonic series

diverges (since it’s partial sums grow without bound, it can’t converge to any fixed number).  One amusing consequence of this is the fact that you can build a bridge from New York to London by simply stacking playing cards on top of each other.

Now let’s consider a closely related series.  Pick your favorite digit between 2 and 9.  We’ll pick 2.  Make a new series by taking the harmonic series and deleting all terms which have your fixed digit in it one or more times.  So we would delete 1/2, 1/12, 1/20, 1/21, 1/22, …., 1/8591721, etc.   Of course, the majority of the terms of the harmonic series will remain undeleted.   This new series is the one we want to think about.

Question:  Does this new series converge?

Question:  Does whether or not it converge depend on which digit you delete?

Question:  If it converges, say to the number S, you know that it can be very difficult to compute what S is for most series.  Instead, can you at least give a number which you know is larger than S?

Solutions are welcome in the comments.  Hint:  The answer to these questions only require Calc III mathematics. We’ll post the solution in a few days or, if you’re impatient, you can google the solution without too much difficulty.

Below is an interesting related question:

Perelman, Part II

We failed to mention that all the press about Perelman might have overshadowed the tireless work of Dr. Stephen T. Colbert, DFA in this area of doughnut mathematics:

Check it out here.

Grigori Perelman and the Poincaré Conjecture

In 1904, the famous mathematician Poincare made a reasonable conjecture:  If a 3-dimensional geometric object had the same homology groups as a 3-dimensional sphere and, and is compact, without boundary, and if every loop in the object can be squeezed down to a point (just like the 3-dimensional sphere), then the object must be the 3-dimensional sphere.

It’s a bit technical to describe homology, but beyond that Poincare’s conjecture is pretty simple to understand:  It’s the mathematical version of “If it looks like a duck, then it’s a duck.”

But, just like Fermat’s Last Theorem and the Riemann Hypothesis, being easy to state often means it’s hard to solve.  So much so that it was one of the problems with a $1,000,000 bounty on its head as part of the Millennium Prize Problems!

Grigori Perelman

In 2002/2003 Grigori Perelman posted a sketch of a solution to Poincare’s Conjecture (and much more!) on the ArXiv using Ricci Flow (part of math having to do with differential equations).   Although Perelman was known to be a very good mathematician, his solution came out of the blue.  Although his work builds on the work of others, he made a number of important advances entirely on his own.

Terence Tao (who knows a thing or two about solving hard math problems) spoke about Perelman’s work and said:

They [the Millennium Prize Problems] are like these huge cliff walls, with no obvious hand holds. I have no idea how to get to the top. [Perelman's proof of the Poincaré Conjecture] is a fantastic achievement, the most deserving of all of us here in my opinion. Most of the time in mathematics you look at something that’s already been done, take a problem and focus on that. But here, the sheer number of breakthroughs…well it’s amazing.

–Terence Tao

It took several years for the best mathematicians in the world to go through Perelman’s proofs carefully and check all the details.  At the 2006 International Congress of Mathematicians in Madrid, Spain it was declared that Perelman’s proof was correct.  At that time he was offered the Fields Medal (which is arguably the most famous prize in math and we’ve discussed it here).  Amazingly, Perelman declined the award!  He has since withdrawn from mathematics.  If and when he is offered his share of the $1,000,000 Millennium Prize, it’s not clear that he’ll accept that either!

If you’d like to read more about Perelman and the Poincare Conjecture, or if you’re looking for a gift for someone who might be interested, then you should take a look at Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century by Masha Gessen, a new book about the human side of this story.  You can read a review of it here. Or, if you’d like to read more about the math, then check out The Poincare Conjecture: In Search of the Shape of the Universe by Donal O’Shea.

Or, you can just rap:

Martin Gardner in the News

Now it just may be a coincidence, but not long after we post about Martin Gardner and his book signing, this week’s OKC Gazette has an article about him as well!  You can see it here. Besides being a very interesting article, there are quotes by the OU Math Department’s own Marilyn Breen and Boris Apanasov.

If nothing else, you should impress your family over the break by folding a hexaflexagon and showing it them:

You can find instructions on how to fold you own hexaflexagon here.

Summer Research Opportunity

NIST Summer Undergraduate Research Fellowship (SURF)

Deadline: February 16, 2010

http://www.surf.nist.gov/app.htm

Synopsis

The SURF program is an opportunity for undergraduates to spend part of their summer working elbow to elbow with researchers at the National Institute of Standards and Technology (NIST), one the world’s leading research organizations and home to three Nobel Prize winners, Gain Valuable hands-on experience, work with cutting-edge technology, and sample the Washington, D.C. area. The SURF program is for students majoring in science, mathematics, and engineering. Note that applications for participation in the SURF program are only accepted from colleges or universities, and not from individual students. Students can participate in any one of the nine NIST laboratories:

·         Building and Fire Research

·         Center for Nanoscale Science and Technology

·         Chemical Science and Technology

·         Electronics and Electrical Engineering

·         Information Technology

·         Manufacturing Engineering

·         Materials Science & Engineering

·         NIST Center for Neutron Research

·         Physics

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To begin a proposal for this program, fill out the Infosheet at http://research.ou.edu/proposal/infosheet/infotype/InfoMain_2008.asp

Search for additional funding opportunities at http://fundingopps2.cos.com

Contact Research Information Services at 325-5868

Contact Proposal Services at 325-3901

Contact Research Administration at 325-4757

Math and Baseball, Part II

Yesterday John Paul Cook talked in the Math Club about sabermetrics.  For those who weren’t able to make it, we’d like to tell you two of the many things he talked about.

First, the father of sabermetrics, Bill James, discovered his “Pythagorean Winning Percentage Formula“:  If R is the number of runs scored in a season by a baseball team, and A is the number of runs they allowed the other team to score, then you should expect that the fraction of the games they won to be

Winning Fraction =

There is a mathematical basis for this formula, but Bill James found it by looking at many, many, many baseball teams’ records.   Actually, the more accurate formula is to replace all the 2’s by 1.82, but it’s much easier to square things.

For example, the Minnesota Twins scored 817 runs and allowed 765 runs in 2009. By the above formula, their theoretical winning rate is approximately .5328.  That is, they should have won 53.28% of the 163 games they played.  So their record should have been 86.8 wins and 76.2 losses.  What was their actual record?  87 wins and 76 losses!  (And the MN Twins were chosen at random from the 2009 season, by the way)

One interpretation is that the Twins won about as many games as they “should” have won.  That is, that their record fairly reflects how well they were playing.  On the other hand, if the Twins record had been 93 and 70, then you would know that they had won more games than they “should” have.  That is, that some of those games were due to luck.  There is a lot of close games in baseball and on average you expect a team to win about as many of them as is justified by their quality of play, but sometimes a team gets a bit lucky (or unlucky) and a few more games go their way.

Teams use this formula in the real world to help determine what to do in between seasons.  In the Twins case, they won a lot of games because they’re a good team, so the management will probably try to keep the same players and maybe try to improve a bit by getting one or two new players in key positions.  Another team might have the same win-loss record, but when they do the math, discover that they only have that record because they were “lucky”.  Which means that more than likely next year they will do worse and it will take more than 1 or 2 new players if they want to keep having a good record.

There's always next year.

John Paul also told us that the same formula applies in other sports.  R is the points the team earns over the season, A is the number of points the team lets its opponents score.  The only change is that the exponent changes from sport to sport. For example, in football the 2 should be replaced with a 2.37.  John Paul did the calculation for the Oklahoma football team.  He first tossed out the Tulsa and Idaho State games since those weren’t serious opponents and the data from those games would unfairly skew things.  Applying the formula, the Sooners “should” have had a record of 7 wins and 3 losses.  In reality, they had a record of 5 and 5.  So the math says that two of their losses can be blamed on bad luck.  Of course, if you’re paying Bob Stoops $4 million per year, that’s small consolation.

John Paul also told us a Bill James quote which seemed especially appropriate on a day when a politician is in Norman selling their book.  Bill James was talking about sportswriters (and their opinion on whether or not Jim Rice should be in the Hall of Fame), but he could have been talking about politicians:

Virtually all sportswriters, I suppose, believe that Jim Rice is an outstanding player.  If you ask them how they know this, they’ll tell you that they just know; I’ve seen him play.  That’s the difference in a nutshell between knowledge and BS; knowledge is something that can be objectively demonstrated to be true, and BS is something that you just “know.”  If someone can actually demonstrate that Jim Rice is a great ballplayer, I’d be most interested to see the evidence.

–Bill James

Addendum:  On the theme of politicians and math, in recent testimony to Congress about Afghanistan, Secretary of Defense Robert Gates said:

[July 2011] will be the beginning of a process, an inflection point, if you will, of transition for Afghan forces as they begin to assume greater responsibility for security.

Robert Gates

We’re pretty sure Mr. Gates is using “inflection point” in a different way then us math folks do.

POM Solutions

The results are in for September and October!

For September, the winner of Totally Radical was Thai Dinh.  Click here for the solution and a quick explanation.

For October, the winning submission of Rüdiger’s Rutabagas was joint work by T. Rhyker Bernandez and Nathan Thomas.   The solution was the Catalan Numbers, and we can hardly give a better explanation than that found here.  (The problem translates into counting rooted plane trees.)  Check it out; you’ll see that tons of interesting problems have Catalan numbers as the solution.  They should definitely be in your problem-solving toolkit.

We’ll announce the December winner soon, and return in February with a new problem!

Congratulations to all!

Card Shuffling Will Blow Your Mind

We’re continuing our discussion of things Dr. Su’s visit made us think about.

In our previous post, we learned that if you do the Riffle Shuffle, then you should do it 7+ times to get close to perfectly mixing the deck.  Remember, by perfectly mixed we said that all 52! configurations are equally likely.   We’d like to revisit that statement.  First, let us remind you that

Why is that the number of configurations?  Well, for each configuration there is 52 possibilities for the first card, and once that’s been chosen, there is 51 possible cards for the second card, and so on.  To get the total possible number of configurations you have to multiply all these possibilities together.

Okay, but how big of a number is 52!?  Pretty big, you say.   For this and all other calculations in this post we’ll use Wolfram Alpha.  That’s a search engine recently released by the makers of Mathematica which is especially good at working with numbers.  In particular, it can do most of the things Mathematica can do (like integrals! Try typing “integrate sin(x)” into Alpha.).

Using Alpha, we get that 52! is (after rounding down) around .  That is, 8 with 67 zeros after it.  Okay, that sounds pretty big.  To give you some idea, there is an estimated atoms in the universe.  So it’s up in that range.

Here is the amazing thing.   Let’s estimate (actually we’ll way overestimate) how many different configurations of the standard 52 card deck have occurred since shuffling was invented.   Here is the numbers we will use (hopefully you’ll agree that they are all reasonable or, more likely, overestimates):

• Let’s say playing cards have been around for 10,000 years (actually, it’s more like 1,500 years).
• Let’s say each of those years has had  100,000,000 seconds (actually 365*24*60*60 = 31,536,000, but there was leap days, etc., so we’ll round way up).
• Let’s say during each of those seconds there was 10,000,000,000 people living on the Earth (actually right now there is around 6,800,000,000, and that’s the most it’s ever been).
• Let’s say that for every second of every one of those 10,000 years all those people were doing one riffle shuffle a second.
• Let’s also assume that every single shuffle resulted in a brand new configuration that never occurred before.

Hopefully you agree that every assumption is a ridiculous overestimate.  Multiplying it all together we see that there has been a total of

That’s a huge number!  Or is it?  In scientific notation, that is different configurations which have happend.  Said a different way, as a fraction of the total possible only

of the total number of configurations have occurred.

Said a different way, if you have shuffled your deck a good 10+ times so that is well mixed, then the configuration your deck is in has a one in chance that it has occurred before.  In comparison, if you buy one Powerball lottery ticket, your odds of winning is one in 195,249,054.

That is, you are more likely to win the next 5 Powerball drawings then to have a configuration which has occurred before!

The odds are staggeringly high that your well mixed deck of cards are in a configuration which has never, ever occurred before in the history of cards!

If that doesn’t blow your mind, then you are probably better off playing cards with these guys:

Sabermetrics

A friend of ours once described baseball as a “statistics generating device“.  What with at bats, hits, runs, RBIs, wins, losses, saves, ERA, putouts, assists, errors, etc.  there is no shortage of data you can collect from your average baseball game.  Countless hours are spent arguing over “Was Willie Mays faster than Mickey Mantle?” or “Who was a better hitter: Barry Bonds or Babe Ruth?“

No Contest.

Invented by Bill James, Sabermetrics is the use of statistics to analyze players/teams in baseball (and other sports).  In the last 10 years it has become popular to use this mathematical analysis (instead of just experience and gut feelings) to decide things like which players to hire and fire.  Some teams swear by this approach, but it’s also controversial since sabermetricians argue that things like RBI aren’t very important in studying players.

This

Wednesday, December 2nd at 5pm in PHSC 1105

John Paul Cook, a grad student in the OU math department, will talk about some of the math and history behind sabermetrics.  He has promised that you don’t need to know about baseball or statistics!  He also claims that some of the same mathematics can be used to study other sports (maybe he’ll tell us why the OU football team has been stinking it up this year!).

As always, Free Pizza!

Also, the flyer is here.

Sabermetrics not Sabertoothmetrics

Found Math — Neolithic Edition

By now you surely know of the OU Math Club logo (to the left).  You may even know that the non-OU part of the design is Sierpinski’s Triangle (p.s. The entire Triangle is in the logo — you just have to look really, really close ).  Dr. Brady recently let us know that the ancient Irish may have had a similar design in mind over 5000 years ago!  Namely, one of the oldest buildings still standing in the world is the Newgrange Tomb which is located here in Ireland.

Besides being over 500 years older than the Great Pyramids, and being of somewhat mysterious origins (If you ask an Irishman, they may say it was built by Dagda), it has beautiful carvings in it’s stonework.  It is most famous for the Triple Spiral image:

However, there are many other images, including one on Kerb 67 which some argue is an early version of Sierpinski’s Triangle:

A closeup of Kerb 67

Here is another look at the same stone:

Another image of Kerb 67 with Sierpinski's Triangle

Now it may be that Dr. Brady has a pro-Irish, pro-Math Club logo, pro-Neolithic bias, but there definitely some similarity.  You’ll have to judge for yourself if an artist 5000+ years ago was the first to think about a pattern of nesting triangles.

However, we can tell you of our two favorite ways of making the Sierpinski Triangle:

1. Make a very large (20+ rows) Pascal’s Triangle, ideally on a sheet of graph paper with one number per square so everything stays symmetric.  Color in all the even numbers and see what pattern emerges.  Bonus:  Do the same, but use one color for those which are evenly divisible when you divide by 3, and another color for all the numbers which have remainder 1 when you divide by 3.  How can you generalize this?  See the papers of Dr. John Holte.
2. Start with an equilateral triangle.  Label the corners with 1, 2, 3.  Pick one corner as your starting position.  Randomly (e.g. roll a die and divide by 2, or use a random number generator) choose between 1,2,3.  Go half the distance between your current position and the corner you randomly selected.  Make a dot.  This is your new position.  Randomly choose between 1,2,3 and go half the distance between your current position and the corner you randomly selected.  Make a dot.  Repeat 100+ times.  What pattern emerges? Bonus:  Write a script for a computer to do this for you and save yourself a lot of work.  Then use it to see what you get for a square, pentagon, etc.