A recent article on Wired’s website discusses the mathematics of Lego – more specifically, it highlights an article on the complexity of Lego systems.  As any child will tell you, Lego sets can vary from very simple, small sets, to much larger and more complicated ones.  As a simple corollary, smaller sets will have fewer pieces, and larger sets will have more pieces.  But how does the number of types of pieces grow as the size of the set grows?  For example, if a 100 piece set consists of 10 different types of pieces, is it reasonable to guess that a 1000 piece set will consist of 100 different types of pieces?

In a word, no.  Though the number of different types of pieces will grow as the size of a set grows, it will grow slower than the size of the set (in other words, it will grow sub-linearly).  To put it another way, as the size of the Lego set grows, rather than building more and more new types of pieces, the same types of pieces that are present in smaller sets tend to be used in new ways.  The effect is that the proportion of distinct piece types decreases as the size of the set grows.  From a mathematical standpoint, if we let y denote the number of different types of pieces, and x be the number of pieces, then this power law is giving us the following equation:

for some constants A and b, with b between 0 and 1.  While y grows as x grows, it does not grow as quickly as x itself.

Taken in a broader context, though, this should not be surprising.  Examples of similar phenomena are prevalent throughout nature, as well as in made-made phenomena such as urban planning.  One example cited in the Wired article is Kleiber’s Law, which states that the ratio of an animal’s metabolic rate to its mass tends to decrease as mass increases (in other words, larger animals are capable of metabolizing more efficiently).  Here’s an article that discusses an analogue of this power law in the context of brain development, and relates this to the development of cities.

So the next time you give a Lego set to a child, feel free to explain this connection – I’m sure any child will welcome the math lesson (at least, any child worth giving a Lego set to in the first place).  It’s also worth noting that this phenomenon is most likely not unique to Lego sets – I am eagerly awaiting a similar report on the mathematics of Tinkertoys – though unfortunately, in this case the number of piece types seems not to have increased in nearly a century.

While combining the arts with the sciences is nothing new, it’s cool to see a program embrace the intersection of these disciplines with such gusto.  Of course, it can be difficult to squeeze educational content out of a song with a science focus, but if School House Rock has taught me anything, it is that education and fly jams need not be mutually exclusive.  If you feel, however, that NIMBioS’s song on sexual selection doesn’t quite make the cut on the education front, here are some other math and science songs to ease you through your hump day.

Here is “The First and Second Law” by Flanders and Swann (this one is highly recommended), a song about thermodynamics:

The University of Tennessee isn’t the only school to join songwriting and science.  Here’s an evolutionary jam courtesy of the University of Chicago:

For the mathematically inclined, there isn’t much that can beat “Finite Simple Group (of Order Two)”:

Though, if those jokes don’t make much sense, one can always listen to muppets singing about math instead. Who knew there was math in tube socks?

(Hat tip to Meg for the NPR link!)

I’ll be back next week with something more substantive.  In the meantime, enjoy your weekend, and if you’re in Los Angeles, Happy Carmageddon!

It was the first time in 18 years that such a quirky thing happened with a full schedule. On Aug. 10, 1993, all seven NL games featured one team scoring precisely two runs, STATS LLC said.

The last time it occurred with five or more runs was July 20, 1955, when all four AL games had at least one team score exactly six, STATS LLC said.

When I read this article, some questions immediately came to mind: exactly how rare is it for one team in a collection of 7 baseball games to have a common score of 5?  Also, if 7 teams in 7 games have the same score, which score are they most likely to share?  Are the 7 games with a common score 0f 2 more or less likely to occur than the 7 games with a common score of 5?

We can answer these questions with some (relatively) simple probability models, given some caveats.  I’d like to estimate these probabilities using only one parameter: the average number of runs a team scores during a game.  Of course, that average will vary from team to team, and also from year to year (in particular, runs per game have declined from the heyday of steroid-mania that gripped baseball at the turn of the millennium).  Due to different rules, there may also be variation between the American and National Leagues.  Let me ignore this, though, and consider only an average number of runs per game overall – what we lose in precision we will more than make up for in clarity.

Ahh, the late 90's, when it was easier to sock a few dingers.

The question remains: how many runs are scored on average in a baseball game?  I found some data online which is somewhat outdated, but I’ll stick to it for convenience (and, more importantly, out of laziness) – any alteration in this number is easy to propagate throughout the following discussion.  In this article from 2005, the author tabulated the average number of runs per game in MLB over a 5 year span from 2000-2004 (that’s over 12,000 games!).  He has a nice looking graph of the distribution of scores as well:

A savvy probability student might see the long tail of this probability distribution and liken it to the Poisson distribution, a distribution encountered in many probability courses, and which is frequently motivated by a desire to model “rare events.”  I put the term in quotations since what constitutes “rare” is frequently left undefined, and in any event, is not really pertinent to this discussion.

Let us suppose, then, that the number of runs scored per game by each team follows a Poisson distribution.  French aside, this means that the probability a team will score n runs is equal to

,

where A is the average number of runs scored per game – in this case, 4.82, and e is the unsung hero sometimes known as Euler’s number.  Don’t worry too much about this formula; if you prefer, the graph of the function looks like this (courtesy of Wolfram Alpha):

Note that the fit isn’t perfect – this graph starts much lower at 0 than the graph of the actual data pictured above, for example – but there is precedence for using the Poisson distrubtion to model runs in a baseball game (this article provides one such example, but a subscription is required to view it in its entirety).  More careful analysis is possible, and can be found in resources like this one, but again, I want to keep things relatively simple.

So, let us suppose that the probability that a team scores n runs is .  What then, is the probability than in a baseball game, one of the teams will score n runs?  Either team A can score n runs or team B can score n runs, but they can’t both score n runs since baseball games can’t end in a tie.  This means that the probability of A or B scoring n runs is simply the probability that A scores n runs plus the probability that B scores n runs, or

For the odds that this happens 7 times, we then multiply this number by itself 7 times (lurking under this is the assumption that runs scored in different games are independent, which seems like an entirely reasonable assumption to make).  To summarize, we estimate the probability that one team in each of 7 games scores n runs is

If n = 5 (as it did earlier this month), the probability is roughly .064%.  In other words, if 7 AL games were played every day, you would expect this outcome once every 1,560 days or so.  Having said that, with more careful analysis it’s possible to show that in fact, if 7 games will have teams scoring the same number of runs, 5 is the most likely number.  For comparison, when n = 2 the probability is only a paltry 0.00812%, making what happened on May 8th over 75 times more likely than what happened on August 10, 1993.  Of course, it’s not fair to compare these records to the 6 run record in 1955, since in that case only 4 games were played, rather than 7.  Nevertheless, it’s not difficult to adjust this model from 7 games to 4 games (or an arbitrary number of games).

So, rather than some murky intuition telling us this event should be unlikely, with a little more effort we can attempt to quantify exactly how unlikely this event should be.  More sophisticated models for runs could be used, but perhaps that is a topic I will save for another day.

In that spirit, I would be remiss if I did not take a moment to mention this article from Wired last month on the man who cracked the code for several scratch lottery ticket games.  Mohan Srivastiva, geological statistician by day and mathematical rogue by night, discovered a pattern in certain scratch lottery tickets back in 2003, but I’m sure (as this article suggests) he’s received a bit more publicity since the Wired article hit.

I highly recommend reading the whole article, but I’ll outline the gist of his discovery here.  In order to do so, I’ll need to specify a type of scratch game he cracked.  The article focuses primarily on a tic-tac-toe themed scratcher shown below.

The left side of the ticket is what gets scratched – below each X and each O lies a number.  Once all of ticket has been scratched, you can compare the uncovered numbers to the numbers on the eight 3×3 grids.  If, in any of those grids, you can find three numbers in a row, column, or diagonal that match the hidden list, you are a winner.  Note the craftiness here – much like the McDonald’s monopoly game, it’s much more likely to get two numbers in a row rather than three, so that a ticket can seem tantalizingly close to being a winner.

In each square of each grid sits a number between 1 and 39.  Also, within each grid, no number repeats more than once; however, since there are 72 squares total on the scratcher, some numbers must repeat themselves between grids (by the pigeonhole principle, if you like).  Some numbers may repeat several times (for example, 17 appears three times in the ticket on the left), while others will appear only once (such as 08).  The key to cracking the ticket, Srivastiva realized, is to take note of the numbers that appear only once on the ticket.  Such numbers will be called “singletons.”

There are several singletons on the ticket presented here, and a more thorough analysis is given in the Wired article.  Most importantly, though, one of the grids has a row of singletons: 24, 12, and 29 are all singletons, and this sequence makes an appearance in the second grid in the in the third row.

What Srivastiva observed was that if a ticket has a sequence of singletons in a winning row, column, or diagonal, then that ticket is likely to be a winner.  In particular, since you can determine all the singletons without scratching off the ticket, he realized that this game reveals information about the likelihood of winning!  In theory, one could (at least in 2003, before the game was pulled) make a career out of buying these tickets in bulk, scratching off the identified winners, and returning the remainder – Srivastiva even went so far as to ask if lottery tickets could be returned, and found that indeed they could be (in fact, it seems as though this is not such an uncommon occurrence). Ultimately, the only reason why he exposed this fault was that he decided the effort involved in sticking it to the man wasn’t worth it – he thought he could earn roughly $600 a day by going through lottery tickets, but he earned more money and had more fun at his day job.

With the right lotto strategy, this could be you!

Kudos to Mr. Srivastiva for his foray into mathematical badassery.  From a mathematical standpoint, there are a number of questions one can ask about this particular type of ticket.  Here’s one: what’s the probability that a number is a singleton?  Of course, these tickets can’t be completely random, as Srivastiva observed, since “the lottery corporation needs to control the number of winning tickets. The game can’t be truly random. Instead, it has to generate the illusion of randomness while actually being carefully determined.”  Nevertheless, for argument’s sake let’s suppose the numbers on the ticket are random.

In this case, a number is a singleton if it appears on one grid and doesn’t appear on the remaining 7 grids.  What is the probability that a given number appears on a 3 x 3 grid?  Since there are 9 numbers in the grid, this probability equals 9/39 = 3/13.  If we fix a number between 1 and 39, and let X denote the number of times that number appears in the grids, then X satisfies a binomial distribution with n = 8 and p = 3/13.  In particular, X = 1 means that the number is a singleton, and P(X = 1) = 8*(3/13)*(10/13)⁷, which is approximately 29.4%.  We also see that the expected value of X (i.e. the expected number of times any given number will occur) is np = 24/13, which is around 1.85.  One can also find the probabilities that X takes on some other value.

Of course, one can also ask what happens if we vary the number of grids, the sizes of the grids, or the size of the number pool from which we draw.  Other questions abound as well: what are the odds of getting two singletons in a row?  Three singletons in a row?  How do the odds of winning change if the number of hidden values change (either in absolute terms, or as a proportion of the total pool of values)?  These questions, gentle reader, I will leave for you.

Then again, maybe this guy was putting his own spin on the latest dance craze.

In and of itself, this might not seem like a particularly newsworthy achievement – as any Pi Day aficionado can tell you, there are people who have memorized more digits.  Perhaps what makes it newsworthy is the fact that the student is only twelve years old, or, more perversely, the fact that his accomplishment came in response to the challenge of his math teacher, who asked his students to memorize as many digits of pi as possible.  By far the most depressing part of the video is a brief clip that shows all the students in the classroom mindlessly rattling off successive digits of pi.  The lack of enthusiasm is almost palpable.

Now, I don’t want to come off as too much of a curmudgeon here.  I have no doubt that this student is stoked that he made it on to TV for an academic achievement, regardless of the actual merits of that achievement (at least the student is aware enough to remark that the information he’s memorized will probably never be put to use).  That’s fine – he has every right to be proud of himself for making it onto the local news.  What really gets my goat is the fact that this teacher thought it would be a good idea to make students memorize digits of pi.  I can think of few better ways to dampen a natural enthusiasm for mathematical learning than by asking students to memorize a series of digits that will have no practical value for any of them, ever.  It would be like having an English teacher ask students to memorize a random string of words which, taken collectively, didn’t teach the student anything about vocabulary or grammar.

Is there any benefit to this exercise?  According to the teacher, “The ability to memorize that much stuff has to help tremendously.”  Well, ok.  But aren’t there more important things to learn about in math class?  Is math class really the best venue to discover a talent like this?  I am fairly certain that students in Singapore aren’t spending class time and homework time memorizing digits of pi.  I’m sure this teacher has good intentions, but I fail to see much value in this apparently newsworthy event.  The mystique of the number pi, I suppose, never fails to attract attention.

If this exercise is what gets this sixth grader interested in math, then by all means he should memorize as many digits of pi as he can.  For the vast majority of students, however, such an exercise is probably beyond tedious.  I can only hope that this news story doesn’t inspire other teachers to compel other students to do the same thing.