The “God Plays Dice” blog has a nice little baseball-themed post built on the coincidence that a number of the teams in the postseason this year are from the same or at least neighboring markets — two from Los Angeles, a pair from San Francisco and Oakland, and another pair from Washington and Baltimore. It can’t be likely that this should happen much, but, how unlikely is it? Michael Lugo works it out in what’s probably the easiest way to do it.

The Major League Baseball postseason is starting just as I write this.

From the National League, we have Washington, St. Louis, Pittsburgh, Los Angeles, and San Francisco.
From the American League, we have Baltimore, Kansas City, Detroit, Los Angeles (Anaheim), and Oakland.

These match up pretty well geographically, and this hasn’t gone unnoticed: see for example the New York Times blog post “the 2014 MLB playoffs have a neighborly feel” (apologies for not providing a link; I’m out of NYT views for the month, and I saw this back when I wasn’t); a couple mathematically inclined Facebook friends of mine have mentioned it as well.

In particular there are three pairs of “same-market” teams in here: Washington/Baltimore, Los Angeles/Los Angeles, San Francisco/Oakland. How likely is that?

(People have pointed out St. Louis/Kansas City as being both in Missouri, but that’s a bit more of a judgment call, and St. Louis…

I honestly don’t intend this blog to become nothing but talk about the comic strips, but then something like this Sunday happens where Comic Strip Master Command decided to send out math joke priority orders and what am I to do? And here I had a wonderful bit about the natural logarithm of 2 that I meant to start writing sometime soon. Anyway, for whatever reason, there’s a lot of punning going on this time around; I don’t pretend to explain that.

Jason Poland’s Robbie and Bobby (September 25) puns off of a “meth lab explosion” in a joke that I’ve seen passed around Twitter and the like but not in a comic strip, possibly because I don’t tend to read web comics until they get absorbed into the Gocomics.com collective.

Bill Watterson’s Calvin and Hobbes (September 27, rerun) wrapped up the latest round of Calvin not learning arithmetic with a gag about needing to know the difference between the numbers of things and the values of things. It also surely helps the confusion that the (United States) dime is a tiny coin, much smaller in size than the penny or nickel that it far out-values. I’m glad I don’t have to teach coin values to kids.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 27) mentions Lagrange points. These are mathematically (and physically) very interesting because they come about from what might be the first interesting physics problem. If you have two objects in the universe, attracting one another gravitationally, then you can describe their behavior perfectly and using just freshman or even high school calculus. For that matter, describing their behavior is practically what Isaac Newton invented his calculus to do.

Add in a third body, though, and you’ve suddenly created a problem that just can’t be done by freshman calculus, or really, done perfectly by anything but really exotic methods. You’re left with approximations, analytic or numerical. (Karl Fritiof Sundman proved in 1912 that one could create an infinite series solution, but it’s not a usable solution. To get a desired accuracy requires so many terms and so much calculation that you’re better off not using it. This almost sounds like the classical joke about mathematicians, coming up with solutions that are perfect but unusable. It is the most extreme case of a possible-but-not-practical solution I’m aware of, if stories I’ve heard about its convergence rate are accurate. I haven’t tried to follow the technique myself.)

But just because you can’t solve every problem of a type doesn’t mean you can’t solve some of them, and the ones you do solve might be useful anyway. Joseph-Louis Lagrange did that, studying the problem of one large body — like a sun, or a planet — and one middle-sized body — a planet, or a moon — and one tiny body — like an asteroid, or a satellite. If the middle-sized body is orbiting the large body in a nice circular orbit, then, there are five special points, dubbed the Lagrange points. A satellite that’s at one of those points (with the right speed) will keep on orbiting at the same rotational speed that the middle body takes around the large body; that is, the system will turn as if the large, middle, and tiny bodies were fixed in place, relative to each other.

Two of these spots, dubbed numbers 4 and 5, are stable: if your tiny body is not quite in the right location that’s all right, because it’ll stay nearby, much in the same way that if you roll a ball into a pit it’ll stay in the pit. But three of these spots, numbers 1, 2, and 3, are unstable: if your tiny body is not quite on those spots, it’ll fall away, in much the same way if you set a ball on the peak of the roof it’ll roll off one way or another.

When Lagrange noticed these points there wasn’t any particular reason to think of them as anything but a neat mathematical construct. But the points do exist, and they can be stable even if the medium body doesn’t have a perfectly circular orbit, or even if there are other planets in the universe, which throws off the nice simple calculations yet. Something like 1700 asteroids are known to exist in the number 4 and 5 Lagrange points for the Sun and Jupiter, and there are a handful known for Saturn and Neptune, and apparently at least five known for Mars. For Earth apparently there’s just the one known to exist, catchily named 2010 TK_{7}, discovered in October 2010, although I’d be surprised if that were the only one. They’re just small.

Elliot Caplin and John Cullen Murphy’s Big Ben Bolt (September 28, originally run August 23, 1953) has been on the Sunday strips now running a tale about a mathematics professor, Peter Peddle, who’s threatening to revolutionize Big Ben Bolt’s boxing world by reducing it to mathematical abstraction; past Sunday strips have even shown the rather stereotypically meek-looking professor overwhelming much larger boxers. The mathematics described here is nonsense, of course, but it’d be asking a bit of the comic strip writers to have a plausible mathematical description of the perfect boxer, after all.

But it’s hard for me anyway to not notice that the professor’s approach is really hard to gainsay. The past generation of baseball, particularly, has been revolutionized by a very mathematical, very rigorous bit of study, looking at questions like how many pitches can a pitcher actually throw before he loses control, and where a batter is likely to hit based on past performance (of this batter and of batters in general), and how likely is this player to have a better or a worse season if he’s signed on for another year, and how likely is it he’ll have a better enough season than some cheaper or more promising player? Baseball is extremely well structured to ask these kinds of questions, with football almost as good for it — else there wouldn’t be fantasy football leagues — and while I am ignorant of modern boxing, I would be surprised if a lot of modern boxing strategy weren’t being studied in Professor Peddle’s spirit.

Bill Amend’s FoxTrot (September 28) (and not a rerun; the strip is new runs on Sundays) jumps on the Internet Instructional Video bandwagon that I’m sure exists somewhere, with child prodigy Jason Fox having the idea that he could make mathematics instruction popular enough to earn millions of dollars. His instincts are probably right, too: instructional videos that feature someone who looks cheerful and to be having fun and maybe a little crazy — well, let’s say eccentric — are probably the ones that will be most watched, at least. It’s fun to see people who are enjoying themselves, and the odder they act the better up to a point. I kind of hate to point out, though, that Jason Fox in the comic strip is supposed to be ten years old, implying that (this year, anyway) he was born nine years after Bob Ross died. I know that nothing ever really goes away anymore, but, would this be a pop culture reference that makes sense to Jason?

Jef Mallet’s Frazz (September 28) wonders about why trains show up so often in story problems. I’m not sure that they do, actually — haven’t planes and cars taken their place here, too? — although the reasons aren’t that obscure. Questions about the distance between things changing over time let you test a good bit of arithmetic and algebra while being naturally about stuff it’s reasonable to imagine wanting to know. What more does the homework-assigner want?

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 28) pops back up again with the prospect of blowing one’s mind, and it is legitimately one of those amazing things, that . It is a remarkable relationship between a string of numbers each of which are mind-blowing in their ways — negative 1, and pi, and the base of the natural logarithms e, and dear old i (which, multiplied by itself, is equal to negative 1) — and here they are all bundled together in one, quite true, relationship. I do have to wonder, though, whether anyone who would in a social situation like this understand being told “e raised to the i times pi power equals negative one”, without the framing of “we’re talking now about exponentials raised to imaginary powers”, wouldn’t have already encountered this and had some of the mind-blowing potential worn off.

I had thought the folks at Comic Strip Master Command got most of their mathematics-themed comics cleaned out ahead of the end of the school year (United States time zones) by last week, and then over the course of the weekend they went and published about a hundred million of them, so let me try catching up on that before the long dry spell of summer sets in. (And yet none of them mentioned monkeys writing Shakespeare; go figure.) I’m kind of expecting an all-mathematics-strips series tomorrow morning.

Jason Chatfield’s Ginger Meggs (June 12) puns a bit on negative numbers as also meaning downbeat or pessimistic ones. Negative numbers tend to make people uneasy, when they’re first encountered. It took western mathematics several centuries to be quite fully comfortable with them and that even with the good example of debts serving as a mental model of what negative numbers might mean. Descartes, for example, apparently used four separate quadrants, giving points their positions to the right and up, to the left and up, to the left and down, or to the right and down, from the origin point, rather than deal with negative numbers; and the Fahrenheit temperature scale was pretty much designed around the constraint that Daniel Fahrenheit shouldn’t have to deal with negative numbers in measuring the temperature in his hometown of the Netherlands. I have seen references to Immanuel Kant writing about the theoretical foundation of negative numbers, but not a clear explanation of just what he did, alas. And skepticism of exotic number constructs would last; they’re not called imaginary numbers because people appreciated the imaginative leaps that working with the square roots of negative numbers inspired.

Steve Breen and Mike Thompson’s Grand Avenue (June 12) served notice that, just like last summer, Grandma is going to make sure the kids experience mathematics as a series of chores they have to endure through an otherwise pleasant summer break.

Mike Twohy’s That’s Life (June 12) might be a marginal inclusion here, but it does refer to a lab mouse that’s gone from merely counting food pellets to cost-averaging them. The mathematics abilities of animals are pretty amazing things, certainly, and I’d also be impressed by an animal that was so skilled in abstract mathematics that it was aware “how much does a thing cost?” is a pretty tricky question when you look hard at it.

Jim Scancarelli’s Gasoline Alley (June 13) features a punch line that’s familiar to me — it’s what you get by putting a parrot and the subject of geometry together — although the setup seems clumsy to me. I think that’s because the kid has to bring up geometry out of nowhere in the first panel. Usually the setup as I see it is more along the lines of “what geometric figure is drawn by a parrot that then leaves the room”, which I suppose also brings geometry up out of nowhere to start off, really. I guess the setup feels clumsy to me because I’m trying to imagine the dialogue as following right after the previous day’s, so the flow of the conversation feels odd.

Eric the Circle (June 14), this one signed “andel”, riffs on the popular bit of mathematics trivia that in a randomly selected group of 22 people there’s about a fifty percent chance that some pair of them will share a birthday; that there’s a coincidental use for 22 in estimating π is, believe it or not, something I hadn’t noticed before.

Pab Sungenis’s New Adventures of Queen Victoria (June 14) plays with infinities, and whether the phrase “forever and a day” could actually mean anything, or at least anything more than “forever” does. This requires having a clear idea what you mean by “forever” and, for that matter, by “more”. Normally we compare infinitely large sets by working out whether it’s possible to form pairs which match one element of the first set to one element of the second, and seeing whether elements from either set have to be left out. That sort of work lets us realize that there are just as many prime numbers as there are counting numbers, and just as many counting numbers as there are rational numbers (positive and negative), but that there are more irrational numbers than there are rational numbers. And, yes, “forever and a day” would be the same length of time as “forever”, but I suppose the Innamorati (I tried to find his character’s name, but I can’t, so, Pab Sungenis can come in and correct me) wouldn’t do very well if he promised love for the “power set of forever”, which would be a bigger infinity than “forever”.

Mark Anderson’s Andertoons (June 15) is actually roughly the same joke as the Ginger Meggs from the 12th, students mourning their grades with what’s really a correct and appropriate use of mathematics-mentioning terminology.

Keith Knight’s The Knight Life (June 16) introduces a “personal statistician”, which is probably inspired by the measuring of just everything possible that modern sports has gotten around to doing. But the notion of keeping track of just what one is doing, and how effectively, is old and, at least in principle, sensible. It’s implicit in budgeting (time, money, or other resources) that you are going to study what you do, and what you want to do, and what’s required by what you want to do, and what you can do. And careful tracking of what one’s doing leads to what’s got to be a version of the paradox of Achilles and the tortoise, in which the time (and money) spent on recording the fact of one’s recordings starts to spin out of control. I’m looking forward to that. Don’t read the comments.

Max Garcia’s Sunny Street (June 16) shows what happens when anthropomorphized numerals don’t appear in Scott Hilburn’s The Argyle Sweater for too long a time.

Nick Emptage, writing for puckprediction.com, has the sort of post which I can’t resist: it’s built on the application of statistics to sports. In this case it’s National Hockey League playoffs, and itself builds on an earlier post about the conditional probabilities of the home-team-advantaged winning a best-of-seven series, to look at the most unlikely playoff wins of the last several years. Since I’m from New Jersey I feel a little irrational pride at the New Jersey Devils being two of the most improbable winners, not least because I remember the Devils in the 1980s when the could lose as many as 200 games per eighty-game season, so seeing them in the playoffs at all is a wondrous thing.

It’s the start of a fresh month, so let me carry on my blog statistics reporting. In February 2014, apparently, there were a mere 423 pages viewed around here, with 209 unique visitors. That’s increased a bit, to 453 views from 257 visitors, my second-highest number of views since last June and second-highest number of visitors since last April. I can make that depressing, though: it means views per visitor dropped from 2.02 to 1.76, but then, they were at 1.76 in January anyway. And I reached my 14,000th page view, which is fun, but I’d need an extraordinary bit of luck to get to 15,000 this month.

March’s most popular articles were a mix of the evergreens — trapezoids and comics — with a bit of talk about March Madness serving as obviously successful clickbait:

What Are The Chances Of An Upset, which introduces some of the interesting quirks of the bracket and seed system of playoffs, such as the apparent advantage an eleventh seed has over an eighth seed.

There’s a familiar set of countries sending me the most readers: as ever the United States up top (277), with Denmark in second (26) and Canada in third (17). That’s almost a tie, though, as the United Kingdom (16), Austria (15), and the Philippines (13) could have taken third easily. I don’t want to explicitly encourage international rivalries to drive up my page count here, I’m just pointing it out. Singapore is in range too. The single-visitor countries this past month were the Bahamas, Belgium, Brazil, Colombia, Hungary, Mexico, Peru, Rwanda, Saudi Arabia, Spain, Sri Lanka, Sweden, Syria, and Taiwan. Hungary, Peru, and Saudi Arabia are the only repeat visitors from February, and nobody’s got a three-month streak going.

There wasn’t any good search-term poetry this month; mostly it was questions about trapezoids, but there were a couple interesting ones:

“first [basketball] team to 100 wins what percent of the time” (it’s near 90 percent)

“john venn interests that don’t have to do with math or science” (according to his biography at the School of Mathematics and Statistics, University of Saint Andrews, Scotland, he had interests in history and wrote a Biographical History of Gonville and Caius College as well as a treaties on the life of John Caius, one of the founders of his college; he also had a knack for building machines, including an automated cricket bowler that apparently clean bowled one of the Australian cricket team’s top stars four times, which I take to be impressive to people who speak cricket)

“what’s the gag in the name of the counselor in the wife of pi cartoon” as well as “scot hiburn wife of pi meaning of counselors name” (the joke is that he’s named “Hugh Jripov”, when he could have been named “Obelus” and let people know what the name of the division symbol there is)

I’d wondered idly the other day if a number-16 seed had ever lost to a number-one seed in the NCAA Men’s Basketball tournament. This finally made me go and actually try looking it up; a page on statistics.about.com has what it claims are the first-round results from 1985 (when the current 64-team format was adopted) to 2012. This lets us work out roughly the probability of, for example, the number-three seed beating the number-14, at least by what’s termed the “frequentist” interpretation of probability. In that interpretation, the probability of something happening is roughly how many times the thing you’re interested in happens for the number of times it could happen. From 1985 to 2012 each of the various first-round possibilites was played 112 times (28 tournaments with four divisions each); if we make some plausible assumptions about games being independent events (how one seed did last year doesn’t affect how it does this year), we should have a decent rough idea of the probability of each seed winning.

According to its statistics, and remarkable to me, is that apparently the number-one seed has never been beaten by the number-16. I’m surprised; I’d have guessed the bottom team had at least a one percent chance of victory. I’m also surprised that the Internet seems to have only the one page that’s gathered explicitly how often the first rounds go to the various seeds, although perhaps I’m just not searching for the right terms.

From http://bracketodds.cs.illinois.edu I learn that Dr Sheldon Jacobson and Dr Douglas M King of the University of Illinois (Urbana) published an interesting paper “Seeding In The NCAA Men’s Basketball Tournament: When is A Higher Seed Better?” which runs a variety of statistical tests on the outcomes of March Madness tournaments and finds that the seeding does seem to correspond to the stronger team in the first few rounds, but that after the Elite Eight round there’s not the evidence that a higher seed is more likely to win than the lower; effectively, after the first few rounds you might as well make a random pick.

Jacobson and King, along with Dr Alexander Nikolaev at SUNY/Buffalo and Dr Adrian J Lee, Central Illinois Technology and Education Research Institute, also wrote “Seed Distributions for the NCAA Men’s Basketball Tournament” which tries to model the tournament’s outcomes as random variables, and compares how these random-variable projections compare to what actually happened between 1985 and 2010. This includes some interesting projections about how often we might expect the various seeds to appear in the Sweet Sixteen, Elite Eight, or Final Four. It brings out some surprises — which make sense when you look back at the brackets — such as that the number-eight or number-nine seed has a worse chance of getting to the Sweet Sixteen than the eleventh- or twelfth-seed does.

(The eighth or ninth seed, if they win, have to play whoever wins the sixteen-versus-one contest, which will be the number-one seed. The eleventh seed has to beat first the number-six seed, and then either the number-three or the number-14 seed, either one of which is more likely.)

Meanwhile, it turns out that in my brackets I had picked Connecticut to beat Villanova, which has me doing well in my group — we get bonus points for calling upsets — apart from the accusations of witchcraft.

I did join a little group of people competing to try calling the various NCAA basketball tournament brackets. It’s a silly pastime and way to commiserate with other people about how badly we’re doing forecasting the outcome of the 63 games in the match. We’re competing just for points and the glory of doing a little better than our friends, but there’s some actual betting pools out there, and some contests that offer, for perfect brackets, a billion dollars (Warren Buffet, if I have that right), or maybe even a new car (WLNS-TV, channel 6, Lansing).

Working out what the odds are of getting all 63 games right is more interesting than it might seem at first. The natural (it seems to me) first guess at working out the odds is to say, well, there are 63 games, and whatever team you pick has a 50 percent chance of winning that game, so the chance of getting all 63 games right is , or one chance in 9,223,372,036,854,775,808.

But it’s not quite so, and the reason is buried in the assumption that every team has a 50 percent chance of winning any given game. And that’s just not so: it’s plausible (as of this writing) to think that the final game will be Michigan State playing the University of Michigan. It’s just ridiculous to think that the final game will be SUNY/Albany (16th seeded) playing Wofford (15th).

The thing is that not all the matches are equally likely to be won by either team. The contest starts out with the number one seed playing the number 16, the number two seed playing the number 15, and so on. The seeding order roughly approximates the order of how good the teams are. It doesn’t take any great stretch to imagine the number ten seed beating the number nine seed; but, has a number 16 seed ever beaten the number one?

To really work out the probability of getting all the brackets right turns into a fairly involved problem. We can probably assume that the chance of, say, number-one seed Virginia beating number-16 seed Coastal Carolina is close to how frequently number-one seeds have beaten number-16 seeds in the past, and similarly that number-four seed Michigan State’s chances over number-13 Delaware is close to that historical average. But there are some 9,223,372,036,854,775,808 possible ways that the tournament could, in principle, go, and they’ve all got different probabilities of happening.

So there isn’t a unique answer to what is the chance that you’ve picked a perfect bracket set. It’s higher if you’ve picked a lot of higher-ranking seeds, certainly, at least assuming that this year’s tournament is much like previous years’, and that seeds do somewhat well reflect how likely teams are to win. At some point it starts to be easier to accept “one chance in 9,223,372,036,854,775,808” as close enough. Me, I’ll be gloating for the whole tournament thanks to my guess that Ohio State would lose to Dayton.

[Edit: first paragraph originally read “games in the match”, which doesn’t quite parse.]

Rather than wait to read today’s comics I’m just going to put in a fresh entry going over mathematical points raised in the funny pages. This one turned out to include a massive diversion into the wonders of the ancient Roman calendar, which is a mathematical topic, really, although there’s no calculations involved in it just here.

Bill Hinds’s Cleats (March 7, rerun) calls on one of the common cultural references to percentages, the idea of athletes giving 100 percent efforts. (Edith is feeling more like an 80 percent effort, or less than that.) The idea of giving 100 percent in a sport is one that invites the question, 100 percent of what; granting that there is some standard expectable effort made, then, even the sports reporting cliche of giving 110 percent is meaningful. Cleats continued on the theme the next day, as Edith was thinking more of giving about 79 percent of 80 percent, and it’s not actually that hard to work out in your head what percent that is, if you know anything about doing arithmetic in your head.

Jef Mallet’s Frazz (March 14) was not actually the only comic strip among the roster I normally read to make a Pi Day reference, but I think it suffices as the example for the whole breed. I admit that I feel a bit curmudgeonly that I don’t actually care about Pi Day. I suppose that as a chance for people to promote the idea of learning mathematics, and maybe attach it to some of the many interesting things to be said about mathematics using Pi as the introductory note the idea is fine, but just naming a thing isn’t by itself a joke. I’m told that Facebook (I’m not on it) was thick with people posting photographs of pies, which is probably more fun when you think of it than when you notice everybody else thought of it too. Anyway, organized Pi Day events are still pretty new as Internet Pop Holidays go. Perhaps next year’s comics will be sharper.

Jenny Campbell’s Flo and Friends (March 15) comes back to useful mental arithmetic work, in this case in working out a reasonable tip. A twenty-percent tip is, mercifully, pretty easy to remember just as what’s-her-name specifies. (I can’t think of the kid’s name and there’s no meet-our-cast page on the web site. None of the commenters mention her name either, although they do make room to insult health care reform and letting students use calculators to do arithmetic, so, I’m sorry I read that far down too.) But as ever you need to make sure the process is explained clearly and understood, and Tina needed to run a sanity check on the result. Sanity checks, as suggested, won’t show that your answer is right, but they will rule out some of the wrong ones. (A fifteen percent tip is a bit annoying to calculate exactly, but dividing the original amount by six will give you a sixteen-and-two-thirds percent tip, which is surely close enough, especially if you round off to a quarter-dollar.)

Steve Breen and Mike Thompson’s Grand Avenue (March 15) has the kids wonder what are the ides of March; besides that they’re the 15th of the month and they’re used for some memorable writing about Julius Caesar it’s a fair thing not to know. They derive from calendar-keeping, one of the oldest useful applications of mathematics and astronomy. The ancient Roman scheme set three special dates in the month: the kalends, which seem to have started as the day of the new moon as observed in Rome; the nones, when the moon was at its first quarter; and the ides, when the moon was full.

But by the time of Numa Pompilius, the second (traditional) King of Rome, who reformed the calendar around 713 BC, the lunar link was snapped, partly so that the calendar year could more nearly fit the length of the time it takes to go from one spring to another. (Among other things the pre-Numa calendar had only ten months, with the days between December and March not belonging to any month; since Romans were rather agricultural at the time and there wasn’t much happening in winter, this wasn’t really absurd, even if I find it hard to imagine living by this sort of standard. After Numa there were only about eleven days of the year unaccounted for, with the time made up, when it needed to be, by inserting an extra month, Mercedonius, in the middle of February.) Months then had, February excepted, either 29 or 31 days, with the ides being on the fifteenth day of the 31-day months (March, May, July, and October) and the thirteenth day of the 29-day months.

For reasons that surely made sense if you were an ancient Roman the day was specified as the number of days until the next kalend, none, or ide; so, for example, while the 13th of March would be the 2nd day before the ides of March, II Id Mar, the 19th of March would be recorded as the the the 14th day before the kalend of April, or, XIV Kal Apr. I admit I could probably warm up to counting down to the next month event, but the idea of having half the month of March written down on the calendar as a date with “April” in it leaves me deeply unsettled. And that’s before we even get into how an extra month might get slipped into the middle of February (between the 23rd and the 24th of the month, the trace of which can still be observed in the dominical letters of February in leap years, on Roman Catholic and Anglican calendars, and in the obscure term “bissextile year” for leap year). But now that you see that, you know why (a) the ancient Romans had so much trouble getting their database software to do dates correctly and (b) you get to be all smugly superior to anyone who tries making a crack about the United States Federal Income Tax deadline being on the Ides of April, since they never are.

(Warning: absolutely no one ever will be impressed by your knowledge of the Ides of April and their inapplicability to discussions of the United States Federal Income Tax. However, you might use this as a way to appear like you’re making friendly small talk while actually encouraging people to leave you alone.)

Tom Horacek’s Foolish Mortals (March 17), an erratically-published panel strip, calls on the legend of how mathematicians “usually” peak in their twenties. It’s certainly said of mathematicians that they do their most important work while young — note that the Fields Medal is explicitly given to mathematicians for work done when they were under forty years old — although I’m not aware of anyone who’s actually studied this, and the number of great mathematicians who insist on doing brilliant work into their old age is pretty impressive.

Certainly, for example, Newton began work on calculus (and optics and gravitation) when he was about 23, but he didn’t publish until he was about fifty. (Leibniz, meanwhile, started publishing calculus his way at about age 38.) It’s probably impossible to say what Leonhard Euler’s most important work was, but (for example) his equations describing inviscid fluids — which would be the masterpiece for anybody not Euler — he published when he was fifty. Carl Friedrich Gauss didn’t start serious work in electromagnetism until he was about 55 years old, too. The law of electric flux which Gauss worked out for that — which, again, would have been the career achievement if Gauss weren’t overflowing with them — he published when he was 58.

I guess that I’m saying is that great minds, at least, don’t necessarily peak in their twenties, or at least they have some impressive peaks afterwards too.

The God Plays Dice blog has a nice piece attempting to model a baseball question. Baseball is wonderful for all kinds of mathematics questions, partly because the game has since its creation kept data about the plays made, partly because the game breaks its action neatly into discrete units with well-defined outcomes.

Here, Dr Michael Lugo ponders whether games are more likely to end at any particular spot in the batting order. Lugo points out that certainly we could just count where games actually end, since baseball records are enough to make an estimate from that route possible. But that’s tedious, and it’s easier to work out a simple model and see what that suggests. Lupo also uses the number of perfect games as a test of whether the model is remotely plausible, and a test like this — a simple check to whether the scheme could possibly tell us something meaningful — is worth doing whenever one builds a model of something interesting.

Tom Tango, while writing about lineup construction in baseball, pointed out that batters batting closer to the top of the batting order have a greater chance of setting records that are based on counting something – for example, Chris Davis’ chase for 62 home runs. (It’s interesting that enough people see Roger Maris’ 61 as the “real” record that 62 is a big deal.) He observes that over a 162-game season, each slot further down in the batting order (of 9) means 18 fewer plate appearances.

Implicitly this means that every slot in the batting order is equally likely to end the game — that is, that the number of plate appearances for a team in a game, mod 9, is uniformly distributed over {0, 1, …, 8}.

Can we check this? There are two ways to check it:

1. find the number of plate appearances in every game…

I don’t intend to transform my writings here into a low-key sports mathematics blog. I just happen to have run across a couple of interesting problems and, after all, sports do offer a lot of neat questions about probability and statistics.

benperreira here makes mention of “Lawler’s Law”, something I had not previously noticed. The “Law” is the observation that the first basketball team to make it to 100 points wins the game just about 90 percent of the time. It was apparently first observed by Los Angeles Clippers announcer Ralph Lawler and has been supported by a review of the statistics of NBA teams over the decades.

benperreira is unimpressed with the law, regarding it as just a restatement of the principle that a team that scores more than the league average number of points per game will tend to have a winning record in an unduly wise-sounding phrasing. I’m inclined to agree the Law doesn’t seem to be particularly much, though I was caught by the implication that the team which lets the other get to 100 points first still pulls out a victory one time out of ten.

To underscore his point benperreira includes a diagram purporting to show the likelihood of victory to points scored, although it’s pretty obviously meant to be a quick joke extrapolating from the data that both teams start with a 50 percent chance of victory and zero points, and apparently 100 points gives a nearly 90 percent chance of victory. I am curious about a more precise chart — showing how often the first team to make 10, or 25, or 50, or so points goes on to victory, but I certainly haven’t got time to compile that data.

Well, perhaps I do, but my reading in baseball history and brushes up against people with SABR connections makes it very clear I have every possible risk factor for getting lost in the world of sports statistics so I want to stay far from the meat of actual games.

Still, there are good probability questions to be asked about things like how big a lead is effectively unbeatable, and I’ll leave this post and reblog as a way to nag myself in the future to maybe thinking about it later.

Lawler’s Law states that the NBA team that reaches 100 points first will win the game. It is based on Lawler’s observations and confirmed by looking back at NBA statistics that show it is true over 90% of the time.

Its brilliance lies in its uselessness. Like NyQuil helps us sleep but does little to help our immune systems make us well, Lawler’s Law soothes us by making us think it means something more than it does.

Why is it so useless, one may venture to ask?

This is a graphical representation of Lawler’s Law. Point A represents the beginning of a game. This team (which ultimately wins this game) has roughly a 50% chance of winning at that point. As the game goes on, and more points are scored, the team depicted here increases its chance of victory based on the number of points it has scored. Point B…

I’ve been reading a book about the innovations of baseball so that’s probably why it’s on my mind. And this isn’t important and I don’t expect it to go anywhere, but it did cross my mind, so, why not give it 200 words where they won’t do any harm?

Imagine one half-inning in a baseball game; imagine that there’s no substitutions or injuries or anything requiring the replacement of a batter. Also suppose there are none of those freak events like when a batter hits out of order and the other team doesn’t notice (or pretends not to notice), the sort of things which launch one into the wonderful and strange world of stuff baseball does because they did it that way in 1835 when everyone playing was striving to be a Gentleman.

What’s the maximum number of runs that could be scored while still having at least one player not get a run?