God Plays Dice

MSRI (the Mathematical Sciences Research Institute) is located at 17 Gauss Way, Berkeley, California. Here's a picture.

Of course, Gauss constructed the 17-gon with ruler and compass and was very proud of this. This article says it's not a coincidence, and so does this official MSRI document.

And rather surprisingly, that's not the only thing on Gauss Way. The Space Sciences Laboratory is at 7 Gauss Way. I'm not sure what significance 7 has, if any.

I looked at my watch at 12:05. I wasn't sure, for a moment, whether it was 12:05 or 1:00; I had to carefully look to determine which of the two hands was the longer one.

A question for you: how many times in a given twelve-hour period could I have this problem? More rigorously, suppose I have an ordinary twelve-hour analog clock, with an hour hand and a minute hand but no second hand. Furthermore suppose I can measure the position of the hands absolutely precisely, and they're "sweep" hands (i. e. they move at a constant angular rate, without "ticks"). At how many times between (say) noon and midnight could I interchange the hands of the clock and still have the hands in a position that corresponds to some time -- but not the time that it actually is? Noon, for example, is not such a time; if I interchange the minute and hour hands at noon I get a valid position of the hands, but that's the position the corresponds to noon. (I won't give an example of a valid time because giving one would be a big hint.)

Bonus: what are these times?

Another bonus: Add a second hand; are there still times which give rise to ambiguous hand configurations? (I don't know the answer to this one.)

(No fair looking up a solution; this is actually a pretty well-known brainteaser. It's well-known enough that I probably knew it existed, somewhere in the back of my mind, before I reinvented it today.)

edit (1:14 pm): Boris points out that he wrote a very similar question as question 23 of this test (PDF).

Here's a quiz full of Fermi problems (like "how many people are airborne over the US at any moment?") and an article by Natalie Angier in today's New York Times, in which she suggests that some basic quantitative reasoning skills wouldn't kill people. The article was inspired by a recent book of such problems, Guesstimation: Solving the World's Problems on the Back of a Cocktail Napkin, by John Adam and Lawrence Weinstein. Adam also has a forthcoming book entitled A Mathematical Nature Walk which may be interesting.

John Cook quotes a definition of "classical", due to Ward Cheney and Will Light in the introduction to their book on approximation theory. Basically, something is "classical" if it was known when you were a student.

The problem with this definition is that it depends on the speaker, which is really not a good property for a definition!

I just wondered -- what is the typical age of a PhD recipient? A bit of Googling turned up this table from Inside Higher Ed, which conveniently sorts by discipline; it reports on an NSF brief. Mathematics and physics are tied for second lowest median age at 30.3; chemistry is the only discipline that's lower, at 29.6.

The table I linked to also gives the median time from getting the bachelor's degree to getting the PhD; by subtraction one can get some number that is a "typical" age of bachelor's degree receipt for students who eventually get a PhD. The median time from bachelor's degree to PhD in mathematics is 7.9 years. Subtraction, 30.3 - 7.9, gives 22.4 as a "typical" age (the difference of medians, which isn't really meaningful) for students getting a bachelor's degree who eventually go on to get a PhD in math. (The highest typical age at bachelor's degree is 25.3, for people getting PhD's in education.) This is the minimum among all eighteen disciplines covered here. It's hard to imagine a median much lower than that given the age at which students typically enter formal education and the number of years it takes.

I interpret this as saying that students who get PhD's in mathematics are less likely to take time away from formal education between high school and college or to take longer than the traditional four years to graduate from college. I'd be interested to see if this is because students who spend time away from formal education "lose" whatever mathematics they knew and have trouble picking it back up again; it's a popular conception that mathematics is more "hierarchical" and so this is more of a problem there than in other fields. (Not having much experience with other fields, I can't say.)

Also, chemistry has a median registered time to degree (time from entering a doctoral program to receiving the PhD) of 6.0 years; the next lowest is mathematics at 6.8. Why is chemistry such an outlier?

Possibly of interest: free mathematics books online. Many seem to be either quite recent (since, say, around 2000) or more than a few decades old, although I haven't systematically checked this statement; this isn't surprising, as the very old books tend to be in the public domain and the very new books tend to have been produced in a period when computers were more universal than they were in the past.

There are a couple hundred books listed here, which is not anywhere near the number of free mathematics books available (legally) online. Various other lists exist, with varying degrees of overlap. Sometimes I flirt with the idea of attempting a more complete list but I realize it would become out of date quite quickly.

Yes, I'm still alive. I got out of the blogging groove somehow.

Today's xkcd makes an interesting point about the difference between "billion" and "million".

And although this isn't about math, Carl Sagan's Cosmos can be watched online at hulu.com. (Thanks to Blake Stacey for the pointer.)