My Favorite Math Party Trick

    I have a minor personal triumph to relate today.  So, I try to have several cute, quick math puzzles/facts on hand to bust out at parties, because I am cool like that.  The most popular of these is a participation-based trick that goes as follows (feel free to play along at home):

    Take a pen and paper and draw a quadrilateral.  There are no restrictions (it can be concave or self-intersecting), but don’t make it too close to the sides of the paper.  Now, for each edge, draw the square containing that edge that is outside the quadrilateral.  Put a dot in the center of each of the four squares, and draw a line connecting opposite dots, ie, those that came from opposite edges.

The Punchline: The lines you just drew are the same length, and perpendicular.

(If you lack pen and paper, theres an applet here)

    It works pretty well because it isn’t very sensitive to sloppy geometry on the part of the artist; so you will pretty consistantly get ‘perpendicular-looking’ lines.  Also, I often bend the truth a bit and attribute it to Napolean Bonaparte, even though he proved something different but closely related.  The usual attribution of this result is to van Aubel, who to my knowledge conquered very little of Europe.

    I’ve been using this for a couple of years now, and periodically I attempt to find a nice geometric proof without passing to coordinates.  Such a proof eluded me up to last night, when I came up with a reasonably nice vector-based proof.  Its a little cheap since using vectors isn’t totally different from passing to coordinates, but in my mind a geometric proof is one which can be done only with pictures (though I will use words for laziness’ sake).  For the record, I was unaware of van Aubel’s proof until this morning, which is a more traditional geometric proof, but a little bit more indirect.

Chord Diagrams: Understanding the 4T Relation

   Last time, we used knot theory as a way of motivating these funny things called chord diagrams, which were circles with a collection of chords.  They came up as ways of writing singular isotopy classes of singular knots, but there are many other ways of thinking of them.  They can also be thought of as fixed-point free involutions on elements (modulo cyclic permutations) or a trivalent graph with a distinguished Hamiltonian cycle.  I list these other incarnations more as a way of appealing to a broader base, though the latter perspective will be relevant today.

    The topic of the day is the 4T relation, which I mentioned at the end of the previous post.  It, together with the 1T relation, determined which functions on the set of all chord diagrams of a given degree came from knot invariants.  The 1T relation came from a fairly straight-forward observation, while I didn’t even attempt to defend the 4T relation.

    One way of thinking of this relation is as a symptom of a larger structure.  We will generalize chord diagrams to a slightly broader context, and impose a relation there that is more natural.  From there it is obvious to see that these generalized chord diagrams can always be reduced in terms of the usual chord diagrams, and that the only evidence of this broader structure is the 4T relation.

Chord Diagrams

    Well, its been quite awhile since the last post.  The real problem is that this blog can’t be a higher priority for me than, you know, important stuff like learning, teaching and research.  So, when time and energy get tight, its one of the first things to go.  It also doesn’t help that the more I bend my mind towards research, the less time I spend thinking about things that would actually make good posts (I have started and abandoned several posts on uninteresting research-type things in the last couple months).

     I’ll try to get back into the swing of things by talking about some stuff I really enjoy, but is far from my research: knots.  I’ve always had a soft spot for knot theory, since its like a poor man’s number theory; a source of simple problems which require techniques from advanced math to solve.  A good example of this are Tait’s conjectures, three basic conjectures from the 19th century that resisted proof until the discovery of the Jones invariant using techniques from analysis and representation theory (and secretly physics).

    Today, though, I’d like to talk about ‘chord diagrams’, a type of object subtly related to that of knots, and whose study can yield some interesting new knot invariants.  They also come up in a number of different areas (solving Feynman diagrams, the representation theory of lie algebras) that I know very little about.  If anyone reading is more familiar with some of the places these come up, please send me a reference.

Hat Guessing Puzzles, The Revenge

    I guess since my previous hat color guessing problem was so popular, I might as well talk about the other one I know.  However, this one isn’t meant to attack the foundations of mathematics.  The problem is as follows:

Three people are sitting in a circle.  Black or white hats (50% chance of each) will all be placed on their heads, and they will be able to see everyone’s hat color but their own.  They will all simultaneously write down on a piece of paper either “Black”, “White”, or “Pass”, trying to guess their own hat color.  All the people collectively win (whatever that means) if at least someone guesses their hat correctly and no one guesses incorrectly.  They lose if anyone guesses incorrectly, or everyone passes.  If they can agree on a strategy beforehand, what is their best chance of winning?

    Again, there is the problem that no information can be conveyed to someone about their own hat color, so they would seem to be guessing blindly (talking and facial expressions are prohibited).  However, they can still win 75% of the time.  Figure it out!

    Once you solve the easy version of this puzzle, the harder version is with larger numbers of people.  As a partial spoiler, stick to , where the best win rate is out of .  How is this possible?  (Answer below the fold)

The Axiom of Choice is Wrong

    When discussing the validity of the Axiom of Choice, the most common argument for not taking it as gospel is the Banach-Tarski paradox.  Yet, this never particularly bothered me.  The argument against the Axiom of Choice which really hit a chord I first heard at the Olivetti Club, our graduate colloquium.  It’s an extension of a basic logic puzzle, so let’s review that one first.

100 prisoners are placed in a line, facing forward so they can see everyone in front of them in line.  The warden will place either a black or white hat on each prisoner’s head, and then starting from the back of the line, he will ask each prisoner what the color of his own hat is (ie, he first asks the person who can see all other prisoners).  Any prisoner who is correct may go free.  Every prisoner can hear everyone else’s guesses and whether or not they were right.  If all the prisoners can agree on a strategy beforehand, what is the best strategy?

The answer to this in a moment; but first, the relevant generalization.

A countable infinite number of prisoners are placed on the natural numbers, facing in the positive direction (ie, everyone can see an infinite number of prisoners).  Hats will be placed and each prisoner will be asked what his hat color is.  However, to complicate things, prisoners cannot hear previous guesses or whether they were correct.  In this new situation, what is the best strategy?

Intuitively, strategy is impossible since no information can be conveyed from anyone who knows your hat color to you, so it would seem that everyone guessing blindly.  However, all but a finite number of prisoners can go free!

Unfortunate Mathematical Names

  It has long been one of my pet peeves that, as a discipline, physicists seem to be way better than mathematicians at giving things cool and useful names.  This disparity appears to have grown in the last few decades, as physicists have started naming quarks (charm, strange, flavor, red/blue/green) and dark matter (WIMPs and MACHOs).  I hang my head in shame when I realize that the average mathematician would probably have named dark matter particles ‘pseudo-massive quasi-particles’ and called it a day.  Mathematicians, of course, can’t even stop giving things the same name – bundle, sheaf, stack, gerbe (french for bundle, sheaf or stack) – or just tacking on more prefixes to an existing name… I’m looking at you, deformed pre-projective algebras.

  What mathematical term has always bothered you, for the uselessness, obtuseness or unfortunateness of its name?  I’m hoping to see what has always rankled other people.