31 December 2008

Best-seller encore impatiemment attendu

From Robert Sedgewick's web site: slides for a talk entitled Impatiemment Attendu, given at a conference in honor of Phillipe Flajolet's 60th birthday. The gist of this talk appears to have been something like "The book will be out soon, after about thirty years of gestation." (That's Analytic Combinatorics, in case you're wondering.)

I mention it because apparently, at the beginning of this month in Paris, this post I made in September was projected on a big screen in front of a bunch of important people. I am of course amused. (The title "impatiemment attendu" is not mine, though; I took it from a paper by Nicolas Pouyanne. I suppose the English translation "impatiently awaited" is mine, but this was not a translation that required some huge inspiration.)

At the time, Amazon said that the book would be out on December 31. It's December 31. It's not out yet, as far as I know. I'll be at ANALCO '09 on Saturday. A slide in the presentation says that the book will be available at SODA (ANALCO takes place the day before); maybe the book will be there?

E-mail address change

I have a new e-mail address.

To figure it out, concatenate the first three letters of my first name, my entire last name, and "@gmail.com".

The intrepid reader can figure out my "academic" e-mail address. Once I get things set up those should redirect to the same place anyway. (I'm tired of checking multiple addresses.)

And I apologize for obfuscating the address like this, but it's a new address, and I'd like to keep the spammers at bay for at least a little while.

Happy New Year! (Do I have any readers in Japan, Korea, Australia, or anywhere else where it's 2009 already? And Kate, if you're reading this, I urge you to remember that you're on vacation and you should get off the Internet.)

30 December 2008

Tetrahedra with arbitrary numbers of faces

While reading a paper (citation omitted to protect the "guilty"), I came across a reference to an "n-dimensional tetrahedron", meaning the subset of Rn given by

x1, ..., xn ≥ 0 and x1 w1 + ... xn wn ≤ τ

for positive constants w1,..., wn and τ.

Of course this is an n-simplex. But calling it a "tetrahedron" is etymologically incorrect -- that means "four faces", while an n-simplex has n+1 faces. This probably occurs because most of us tend to visualize in three dimensions, not in arbitrary high-dimensional spaces.

I'm not saying that "tetrahedron" shouldn't be used here -- I'm just pointing out an interesting linguistic phenomenon.

29 December 2008

A combinatorial problem from Crick

I recently read What Mad Pursuit: A Personal View of Scientific Discovery
, which is Francis Crick's account of the "classical period" of molecular biology, from the discovery of the double helix structure of DNA to the eventual figuring out of the genetic code. It differs from the more well-known book by James Watson, The Double Helix: A Personal Account of the Discovery of the Structure of DNA, which focuses more on the characters involved and less on the science.

Crick was trained as a physicist, and learned some mathematics as well, and every so often this pokes through. For example, back when the nature of the genetic code wasn't known, combinatorial problems arose to prove that a genetic code of a certain type was or was not possible. One idea, due to Gamow and Ycas was that since there are twenty combinations of four bases taken three at a time where order doesn't matter, perhaps each one of those corresponded to a different amino acid. This turned out to be false. Another, more interesting problem comes from asking how the cell knows where to begin reading the code. What is the largest size of a collection of triplets of four bases such that if UVW and XYZ are both in the collection, then neither VWX nor WXY is? The reason for this constraint is so that the "phase" of a genetic sequence is unambiguous; if we see the sequence UVWXYZ, we know to start reading at the U, not the V or the W. Thus the collection can't contain any triplet in which all three elements are the same, and it can contain at most one of {XYZ, YZX, ZXY} for any bases X, Y, Z, not necessarily distinct. There are sixty triplets where not all three elements are the same, thus at most twenty amino acids can be encoded in such a code. There are solutions that acheive twenty; see the paper of Crick, Griffith, and Orgel.

Note that the "20" in the two types of code here arises in different ways. If we assume a triplet code with n bases, then the first type of code can encode as many as n(n+1)(n+2)/6 amino acids, the second (n3-n)/3.

Crick says that the more general problem of enumerating the number of codes which imply their own "reading frame" was considered by Golomb and Welch, and separately Freudenthal. Based on the title and the date, I think the first of these is the paper I point to below -- but our library doesn't have that journal in electronic form, and the physical library is closed this week!

F. H. C. Crick, J. S. Griffith, L. E. Orgel. Codes Without Commas. Proceedings of the National Academy of Sciences of the United States of America, Vol. 43, No. 5 (May 15, 1957), pp. 416-421.

George Gamow, Martynas Ycas. Statistical Correlation of Protein and Ribonucleic Acid Composition Statistical Correlation of Protein and Ribonucleic Acid Composition. Vol. 41, No. 12 (Dec. 15, 1955), pp. 1011-1019.

Golomb, S.W., Gordon, B., and Welch, L.R., "Comma-Free Codes", The Canadian Journal of Mathematics, Vol. 10, 1958. (Citation from this list of Golomb's publications; I haven't read it.)

28 December 2008

Tao on calibration of exponents

Terence Tao: Use basic examples to calibrate exponents. This article, for the eventual Tricki, gives many examples of the following basic procedure. In many problems there is a "size" parameter N, and the problem has an "answer" that we believe for some reason behaves like Nk for some constant k. A quick way to find N is to look at "basic examples" (say, random graphs in a graph-theoretic problem).

The interesting thing about this article -- and about the Tricki as a whole, once it finally launches -- is that its organizational principles are not the same as most mathematical exposition. A typical lecture or section of a textbook gives problems with similar statements but not necessarily with similar proofs; the Tricki will group together problems with similar proofs but not necessarily with similar statements.

27 December 2008

Comfort with meaninglessness?

Comfort with meaninglessness the key to good programmers, from Boingboing.

Is this true for mathematics as well as computer programming?

23 December 2008

Housing prices drop 13 percent -- what does this mean?

The New York Times reports on bad housing news:
The median price of a home plunged 13 percent from October to November, to $181,300 from $208,000 a year ago. That was the lowest price since February 2004.
They mean that house prices have gone down 13 percent in a year, i. e. from November 2007 to November 2008. That's what the National Association of Realtors press release says.

But one sees this pretty often -- the confusion between monthly declines and annual declines. And sometimes a 1% decline in a month might be reported as a "12% per year" decline -- but then the "per year" gets dropped, the statement "prices of X dropped 12% this month" is made, and those who aren't familiar with how people who care about the price of X report their numbers get confused.

Don't get me wrong -- a drop of 13% in a year is still a big deal. But a drop of 13% in a month would be a much bigger deal.

22 December 2008

Is wind chill misleading?

From Daniel Engber at Slate: Wind Chill Blows. Back when nobody read this blog, I wrote about how the heat index doesn't make sense to me, because I know what 95 degrees with "typical" humidity for my location feels like, and telling me it "feels like" 102 is misleading. (That's Fahrenheit, not Celsius; we're not literally boiling here in the summer.)

Something similar is true for wind chill. Both of these measures only take into effect two of the many variables that effect comfort -- temperature and either humidity or wind speed. They assume that these are the only two variables which actually vary -- clothing, amount of sunlight, weight, etc. are held constant. In reality, comfort is a function of many variables, and it's misleading to create an index that assumes it's just a function of two variables. People know that they should take more than the temperature into account, but I've seen quantitiatively unsophisicated people think of the wind chill as some perfect index of the weather.

But let's face it, a wind chill of zero sounds scarier than a temperature of sixteen. (Those are approximately the numbers I heard reported this morning in Philadelphia.) That means more people watch the news.

Logarithmic jihadism

From the Wall Street Journal:
U.S. counterterrorism officials have been on guard for homegrown recruitment by radical groups. Intelligence analysts from the New York Police Department, in a study of radicalization in Western Muslim communities, warned that "jihadist ideology" is "proliferating in Western democracies at a logarithmic rate."
Maybe it's just me, but logarithmic proliferation doesn't seem all that scary.

19 December 2008

Uncyclopedia's list of proof methods

Uncyclopedia has a list of methods by proof. My favorite:


Of course, you have to be careful which letters you use as variable names in stating your result -- you can only use one of a, α, and A, for example.
And some of them are methods of proof that are actually used:
Proof by Diagram

Reducing problems to diagrams with lots of arrows. Particularly common in category theory.

And here's an interesting point about mathematical writing:
Proof by TeX

The proof is typeset using TeX or LaTeX, preferably using one of the AMS or ACM stylesheets. When laid out so professionally, it can't possibly have any flaws.

Erdos papers available online, and the Erdos-Turan law for the order of elements in the symmetric group

The collected papers of Paul Erdös are available online.

I'm glad I found this. There's a theorem of Erdös and Turan that I've been curious about for a while. Namely, let Xn be the order of a permutation in Sn selected uniformly at random. Then

\lim_{n \to \infty} {\rm Prob} \left( {\log X_n - {1 \over 2} \log^2  n \over \sqrt{{1 \over 3} \log^3 n} } < y \right) = \Phi(y)

where Φ is the cumulative distribution function of a standard normal random variable. Informally, log Xn is normally distributed with mean (log2 n)/2 and variance (log3 n)/3. Unfortunately the proof, in On some problems of a statistical group-theory, III, doesn't seem to explain this fact in any "probabilistic" way, so I'm not quite as excited to read the paper as I once was. But I had believed the proof was in the first paper in the (seven-paper) series, which is in storage at our library, and it's nearly the holidays, so I probably would have had to wait quite a while to get a copy just to see that it wasn't the one I wanted.

In fact, it was worrying that I had the wrong paper that led me to find this resource in the first place -- seeing the "I" in the citation I had got me curious, so I went to Google. What I expected to see in the Erdos-Turan paper, and what I actually wanted to see, was a "probabilistic" proof somehow based on the central limit theorem. This exists; it's in the paper of Bovey cited below. Also, Erdös seems to have not been good at titling papers; titles "On some problems in X", "Problems and results in Y", "Remarks on Z", "A note on W", etc. are typical. I guess he was too busy proving things to come up with good titles.

Bovey, J. D. (1980) An approximate probability distribution for the order of elements of the symmetric group. Bull. London Math. Soc. 12 41-46.
Erdos, P. and Turan, P. (1967) On some problems of a statistical group theory. 111. Acta Math. Acad. Sci. Hungar. 18 309-320.

Guessing the answer without knowing the question

Ian Ayres asks a question at Freakonomics. Which of the following is the correct answer? 4π, 8π, 16, 16π or 32π square inches?

No, I didn't forget the question. But it's possible to make a reasonable guess by trying to reverse-engineer the question.

(Don't read the comments. They're full of people who didn't get it.)

17 December 2008

Jean Chretien is secretly a mathematician?

"I don't know. A proof is a proof. What kind of a proof? It's a proof. A proof is a proof, and when you have a good proof, it's because it's proven."
-- Jean Chretien, former prime minister of Canada. The context appears to be something having to do with Canada's involvement in the Iraq war, but I'm having trouble finding details. It seems that this was a Big Thing in Canada when it happened, so perhaps I have Canadian readers who can explain?

NYT profiles of Jessica Fridrich

Specializing in Problems That Only Seem Impossible to Solve , by Bina Venkataraman, in yesterday's New York Times.

This is an article about Jessica Fridrich, a professor at Binghamton University, who at one point held the world record for the fastest solving of the Rubik's Cube. She currently specializes in the research of information hiding in digital imagery.

15 December 2008

Pairing up the states

A Ballot Buddy System, by Randall Lane, an op-ed in today's New York Times.

As you may remember, there was a presidential election six weeks ago in the United States. But Barack Obama isn't officially elected president until today; today is the day that the electors cast their votes. This is the first time since 1892 that a state will have electors voting for more than one candidate. Maine and Nebraska both have laws in which two electors go to the winner of the popular vote in the state and one goes to the winner of each congressional district. Nebraska went for McCain, but the 2nd congressional district (Omaha and some of its inner suburbs) went for Obama.

It's been suggested that all states should apportion their electoral votes in this way, on the assumption that less people live in "safe districts" than "safe states". (I'm not sure if this is the case, especially with the way some districts are gerrymandered these days.) But the problem with this is that the majority of people (and legislators) in any state would see their party hurt by the passage of such a law in their state.

Lane's suggestion is that Republican-leaning states and Democratic-leaning states with approximately the same number of electoral votes (say, Texas and New York) could agree to pass these laws together. The problem is that in each pairing, it seems that you'd want two states that are roughly of equal size and are equally far from the political center; it seems that it might not be possible to construct such a pairing. The obvious problem is what to do with California? It's easy to state a few plausible pairs, as Lane does, but I'm not sure that all the states could be paired off in this way. Furthermore, things probably get weird, in terms of how much "power" each state holds in presidential elections, if some substantial number of states have enacted such laws.

13 December 2008

π really does equal 3

Okay, not really. But here's a fake proof that π = 3, which I hadn't seen before.

12 December 2008

The Xie brothers' advice collection

Tao Xie and Yuan Xie (brothers, in case you're wondering) maintain an advice collection consisting of links to things other people have written about how to succeed in scientific careers -- on getting a PhD, writing papers, and so on. Those links that seem to be aimed towards people in certain subjects are mostly aimed at computer scientists, but at least some of what I'm finding in their links seems reasonable.

Of course, one piece of advice they probably should give is "don't spend lots of time reading this sort of advice".

Since I'm talking about brothers, I feel obliged to mention the following paper:
Michalis Faloutsos, Petros Faloutsos and Christos Faloutsos, On Power-Law Relationships of the Internet Topology, SIGCOMM 1999. It has nothing to do with the Xie brothers' advice collection, but I wanted to mention it anyway because I saw a citation to it and I was amused.

Where are the mathematicians?

Why are people in Iowa interested in combinatorics? Combinatorics is more popular in Iowa than in any state but Massachusetts.

Google now has a feature called "Google Insights"; you can type in a search term and see where people are searching for it, how frequency of searches varies with time, etc. In states where there's a lot of volume it's possible to zoom in; in Massachusetts it's possible, for example, and most of the interest is in Cambridge. Given that there is a Big Important University and a liberal arts school that has a well-known mathematics department in Cambridge, that's not surprising. But I can't zoom in on Iowa.

(It's possible to get results by country, too, but these results seem ridiculously skewed; I suspect that Google may be normalizing by the number of Internet users in a given area, and the user pool is different in different places.)

Another interesting one: "probability" is popular in Maryland, and among cities in that state it's most popular in College Park and Laurel. College Park is where the University of Maryland is. Laurel is where the NSA is. You can see similar things in other states; for example, in New York, "probability" is most common in Stony Brook, Troy (RPI), and Ithaca (Cornell). In Pennsylvania, it's University Park (Penn State), Bethlehem (Lehigh), and State College (Penn State again). The general pattern seems to be first a few college towns, then the big cities -- the places with the fourth and fifth highest numbers for "probability" in Pennsylvania are Pittsburgh and Philadelphia.

Most mathematical search terms I could think of are highly seasonal -- they're less common in the (Northern Hemisphere) summer, when schools aren't in session. That seems to imply that lots of the people doing the searching are students. I couldn't find a mathematics-related search term that didn't show this seasonality; I don't know if it can be done, because only search terms that receive a reasonably large amount of traffic are reported on the site at all, and things which are important enough to get lots of traffic are probably studied in schools.

11 December 2008

How do you pronounce ≤ and ≥?

I'm taking a break from proofreading a paper. I'm reading it out loud, because I find this is the best way to catch mistakes; it forces me to look at every word.

There are inequalities in this paper, so the signs ≤ and ≥ come up a lot. How do you pronounce these? When I was in college I pronounced them "less than or equal to" and "greater than or equal to". But sometime around the first year of graduate school I seem to have shifted to "at most" and "at least", which have the obvious advantage of being shorter.

Edit (11:15 pm): It appears I've mentioned this before.

McMullen on the geometry of 3-manifolds

The Geometry of 3-Manifolds, a lecture by Curt McMullen. This is one of the Science Center Research Lectures; in which Harvard professors talk about their research to the interested public; the series doesn't appear to have a web page, but here's a list of videos available online in that series; these include Benedict Gross and William Stein on solving cubic equations. There are non-mathematical things too, but this is at least nominally a math blog, so I won't go there.

McCullen apparently also gave this lecture at the 2008 AAAS meeting in Boston, and has a couple other video lectures available online.

And now I want a do(ugh)hnut-shaped globe with just North and South America on it. This is a fanciful example of what an "armchair Magellan" might suspect the world looked like if humans had reached the North and South poles starting from somewhere in the Americas but had never crossed the Atlantic or Pacific; they might suspect that the cold area to the north and the cold area to the south are actually the same. McMullen uses to illustrate that tori and spheres are not the same, since loops on the sphere are contractible but loops on the torus are not. The lecture, which leads up to telling the story of the Poincaré conjecture, begins by using this as an example of how topology can distinguish between surfaces.

Finally, here's an interesting story, which may be well-known to some people but wasn't to me: Wolfgang Haken, one of the provers of the Four-Color Theorem, may have intended that (famous, computer-assisted) proof to be a "trial balloon" for a brute-force proof of the Poincare conjecture.

09 December 2008

Danica McKellar, my mother, and Cuban food

In honor of my age now being a square, my parents took me out to dinner. (Okay, so it had nothing to do with my age being a square.)

My mother: "I read in the US Air in-flight magazine that... um... that girl from Boy Meets World is writing books about math now."

Me: "You mean the girl from The Wonder Years??"

Yes, this actually happened. The two shows are not all that different, and the male leads in them are brothers in real life and look similar, so it's a natural mistake to make.

Danica McKellar majored in math at UCLA (where Terence Tao was one of her teachers, and has written two books aimed at middle school girls to tell them that math doesn't suck, namely Math Doesn't Suck: How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail and Kiss My Math: Showing Pre-Algebra Who's Boss

By the way, if you're in Philadelphia and want "modern Cuban cuisine", Alma de Cuba has it. I especially recommend that you go to this restaurant if someone else is paying.

On birthdays

Yesterday, my age was a factorial.

Today, my age is a square.

This raises an interesting problem.

On translation of games

At Language Log recent discussion has gone on about how you can translate from one language to another, but you can't translate from one game to another. For example, you can't take a game of chess and translate it into poker.

I'm reminded of the Subjunc-TV in Douglas Hofstadter's Godel, Escher, Bach: An Eternal Golden Braid, which has characters tuning into a baseball game that has been made to look like a football game. Of course this doesn't work perfectly, which is intended to illustrate Hofstadter's points about the imperfection of analogies.

At Language Log, I learned that there are certain "logical games" for which a notion of translation is possible. These are apparently of interest to logicians; you can read more at the Stanford Encyclopedia of Philosophy.

But in combinatorial game theory, we can associate each position in certain games with a "number"; is it meaningful to say that positions in different games which have the same number are the "same position"? In this case, translations between games would become possible, except that those numbers are apparently difficult to calculate.

08 December 2008

Ken Jennings is not a mathematician

Ken Jennings, the 74-time Jeopardy! champion, has a blog.

Today he wrote about polyominoes, inspired by the question of how many pieces there would be in Tetris if it were played with pieces made of five or six squares instead of the canonical four. He's not a mathematician, and he finds it surprising that there's no nice formula for the number of polyominoes with n cells. I suppose it is kind of surprising to someone with no mathematical training; by this point in my life I've gotten used to the fact that things just don't work that way.


Via X=Why?, I found an article on how a criminal investigation lab in California is inviting students to come in and showing them that math is useful for solving crimes. (In the CSI way -- figuring out how blood splatters, say -- not in the Numb3rs way.) This is certainly a good thing to do.

But can you make sense of this?
Craig Ogino, the department's crime lab director, started the event by offering a prize of $10 to the student who could use trigonometry to determine the number in gallons of a mixture used to make methamphetamine, based on his sketch.
I'm assuming that trigonometry was actually used for something else -- like, say, the aforementioned blood splattering analysis, seen later in the article -- and that the reporter made a mistake. But I'm not totally sure. Any thoughts?

07 December 2008

Kids these days...

Shitload of math due Monday, from The Onion:
Making matters worse, students said, was their math textbook, which reportedly doesn't even have any of the freaking answers in the back.
How would these kids feel if they learned that eventually the questions aren't even in the book, but you have to come up with them yourself?

2008 Putnam problems

This year's Putnam exam problems, via 360.

I haven't thought about these, but I might as a break from writing things up over the next few days.

04 December 2008

How could you guess the formula for the sum of the first n fifth powers?

The following three formulae are reasonably well known:

1 + 2 + 3 + ... + n = n(n+1)/2

12 + 22 + 32 + ... + n 2 = n(n+1)(2n+1)/6

13 + 23 + 33 + ... + n 3 = (n(n+1)/2)2

(The sums of first and second powers arise pretty often naturally; the sum of cubes is rare, but it's easy to remember because the sum of the first n cubes is the square of the sum of the first n natural numbers.)

The first member of this series I can't remember is the following:

14 + 24 + 34 + ... + n 4 = n(n+1)(2n+1)(3n2+3n-1)/30

and generally, the sum of the first n kth powers is a polynomial of degree k+1.

I ran into these formulas, which I'd seen plenty of times before while perusing the book Gamma: Exploring Euler's Constant by Julian Havil, which I had heard of a few years ago, forgotten about, and then found while browsing the library shelves. (Also in German, under the title GAMMA: Eulers Konstante, Primzahlstrände und die Riemannsche Vermutung.) This is a most interesting book, at least if you're someone like me who pretends to know number theory but really doesn't.

Anyway, back to the main story. Say I wanted to know the sum of the first n fifth powers. Well, there's a general method for finding the formula of the first k powers; it involves the Bernoulli numbers. But let's say I didn't know that. Let's say somebody hands me the sequence

1, 33, 276, 1300, 4425, 12201, 29008, 61776, 120825, 220825

in which the nth term, sn, is the sum 15 + 25 + ... + n5 -- but doesn't tell me that's where the sequence comes from -- and challenges me to guess a formula for it in "closed form". (Smart-asses who will say that there are infinitely many formulas are hereby asked to leave.) How would I guess it?

Well, it can't hurt to find factorizations for these numbers. And if you do that you get

1, 31 111, 22 31 231, 22 52 131, 31 52 591, 31 72 831, 24 72 371, 24 33 111 131, 33 52 1791, 52 112 731

and this seems interesting; these numbers seem to have lots of small factors. Furthermore, a fair number of them seem to have one largeish prime factor, which I've bolded. (Yes, I realize, 11 times 13 isn't prime, but I actually did think of it as a large factor.) What are the large factors that I observe in these numbers? They are

?, 11, 23, ?, 59, 83, ?, 143, 179, ?

and the nth of these is easily seen to be 2n2 + 2n - 1. (Some terms don't show up from inspection of the factorizations because they get "lost in the noise", as it were.)

From there the rest is pretty easy. We see that often (but not always), sn is divisible by 2n2 + 2n - 1. You can check that for n = 1, 2, ..., 10, the term sn is always divisible by (2n2 + 2n - 1)/3. So now consider the sequence tn = sn / ((2n2 + 2n - 1)/3). The numbers t1 through t10 are

1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025

and I recognized that all of these are squares; in particular tn = (n(n+1)/2)2.

Putting everything together, I get the conjecture that the sum of the first n fifth powers

sn = n2(n+1)2(2n2+2n-1)/12

which could be proven by induction, but actually writing out the proof is best left to undergrads.

The method here is reminiscent of Enumeration of Matchings: Problems and Progress by James Propp. In that article, Propp lists various unsolved problems in the enumeration of tilings, and conjectures that some of them might have answers which are given by simple product formulas, because actually counting the tilings in question gave numbers with nice prime factorizations.

Edit, 9:13 pm: of course this is not the only method, or even the best method; it's just the method I played around with this morning. See the comments for other methods.

How to break into a keyless-entry car

Weak security in our daily lives (in English): basically, you can use a de Bruijn sequence to break into a car with keyless entry in what might be a non-ridiculous amount of time. I'm referring to the sort which have five buttons marked 1/2, 3/4, 5/6, 7/8, and 9/0, and a five-digit PIN that has to be entered. This trick takes advantage of the fact that the circuitry only remembers the last five buttons pressed, so if you press, say, 157393, then the car will open if the correct code is either 15739 or 57393. It is in fact possible to arrange things so that each key you press, starting with the fifth, completes a five-digit sequence that hasn't been seen before.

Of course, you shouldn't do this.

Via microsiervos (in Spanish).

03 December 2008

Logic as machine language

Gil Kalai mentions a metaphor I hadn't heard of before about the foundations of mathematics:
To borrow notions from computers, mathematical logic can be regarded as the “machine language” for mathematicians who usually use much higher languages and who do not worry about “compilation.”
Of course there would be analogues to the fact that certain computer languages are higher-level than others as well. To take an example dear to me, the theory of generating functions might be at a higher level than the various ad hoc combinatorial arguments it's often introduced to students as a replacement of. I don't want to press this metaphor too hard because it'll break -- I don't think there are analogues to particular computer languages. But feel free to disagree!

02 December 2008

You can't say you're a liar

From Family Guy:
"Chris, everything I say is a lie. Except that. And that. And that. And that. And that. And that. And that. And that."