After winning their trip to Paris, the guests became elated and celebrated with the consumption of some adult beverages. Ever responsible, the host confiscated the keys to all cars to ensure that no one drove home drunk. Later on, when things started to calm down, party-goers started to request the return of their keys claiming to be sober enough for the drive home.

Having once been out-done by the guests, our host took another whack. He distributed all of the keys, but did so randomly. He then presented a challenge he felt sure they’d only be able to satisfy if they were indeed sober enough to drive. They were allowed to exchange keys, but only in rounds. During each round, each party-goer could either do nothing or pair up with another party-goer and exchange the sets of keys each was holding. (Each party-goer could be part of at most one pairing per round.) No one would be allowed to drive home unless everyone recovered their own keys.

The host wished to allow only a fixed number of rounds. To be fair, he wanted to be sure that it would indeed be possible to make the change. However, he also wanted to make it as difficult as possible for the party-goers. What is the minimum number of rounds must allow them to ensure that an exchange would be possible?

For clarity, all key recipients can discuss, share information such as who has the keys of whom, and agree upon a strategy. Also, careful readers will realize that there were 20 guests at the party originally. Sadly, it was a rather disorderly party and some guests did leave early, but many more appeared. Everyone present at the key ceremony had a key confiscated, and everyone with a key confiscated received a key for this challenge, but neither you nor the host knows just how many there are.

So, I’m teaching a new course, Math 2033, Mathematical Thought, and it’s going great! I’d like to take a moment to write about it!

(This is one reason the MF has been kinda slow lately; another is that I’m chair) When it’s fully up and running, we’ll have about 150 students in one large section each semester (we’re starting with about 100). In a nutshell, it’s the Math Factor, as a course.

So, the list of topics is pretty familiar; from the podcast you are pretty well acquainted with the kinds of things I like to share: game theory, encryption, a little number theory, theory of computation & godel’s theorem, cardinality/infinity; plus more visual hands on things like topology, graph theory, symmetry, four-dimensional geometry, and so forth, and some baby programming in a playground IDE (scratch.mit.edu)

The real thing though is that my co-teacher and I have taken a kind of radical approach to the structure of the course—and our crazy ideas are working out great! In a nutshell, the students are guaranteed a C just for showing up and doing what they’re told (more on that in a sec), but to get an A, they have to become active collaborators in the building of the course, adding to the long-term infrastructure. I didn’t expect to give out very many A’s at all, but a surprising number of students seem up to the challenge. Part of the point is that this reward structure aligns the interests of the course directly with the interest of the students. Another nice thing is that it is much more like the reward structure of Real Life, far more so than most academic experiences: you can coast and do ok, but to really succeed, initiative and imagination are required. Interestingly, 20% of the class can’t even rise to the minimal standard of showing up, and will fail. 

Another nice thing is that students can bring to bear any of their own interests and abilities; we need such a wide range of things done—photography, writing, editing, leaders on our discussion board, organizers, all kinds of stuff. It is in fact possible to get an A by dragooning other students into harvesting, trimming and delivering a huge load of bamboo for some math sculptures the class will be making soon. The fact is, I have large ambitions for this, and no way to do more than a fraction of the work; students that help bring this off will be the ones that get an A.

SO, how does it actually work? The basic daily rhythm is that we give a lecture, usually with some sort of hands-on fun and games component. We then post a prompt or two on the class discussion board (hidden to the outside world). The students have 24 hours to post, and then another 24 hours to comment on each others ideas. This is the real heart of the course and the activity has steadily grown, reaching 1700 posts a couple of weeks ago. Wow!  (I shouldn’t exaggerate this though: some are really into it, many are trying to get by with as little as possible. I am aiming for a culture where slacking is gently disgraced, and we’re on track to get there)

As you can guess, this has completely lifted out of my ability to monitor; we have a number of ways this is digested and managed. For example, about half a dozen of the more thoughtful students are responsible for reading all of posts and trying to raise the level of discourse, and for creating useful summaries of the best ideas.

BUT that’s just the “internal” part of the course. Externally, open to the world, is a wiki, math2033.uark.edu which is going pretty well. I view this as a multi-year project, so this is a pretty good start. Most of what you see there is the product of about twenty students, and a few really do A LOT of work, including having developed the basic organizational framework. (So they get A’s for sure)

I’ve been spending a lot of time developing solid materials for use in the course, such as this sample handout, on the Halting Problem.

The students seem pretty pumped. It’s working!!! 

(We passed an important milestone last week; several students told me they ended up fooling around way too much with one of the optional assignments, messing them up in other classes! Perfect!)

You have eight batteries and know that four are good and four are dead, but don’t know which are which.  Your only method of testing them is to insert two into a device that will work if you’ve put in two good batteries and not otherwise.  How many such “tests” are required in order to be sure that you’ve located two good batteries?

As of this posting, the answer to this question is not yet on the POTW website, but if you come to this later, the spoiler may be there, so be careful to avoid spoilers if you want to work this through.

Janine is one of twenty guests at a Christmas party.  Each guest is given a book as a present.  Janines’s book is called ‘Living with Crazy Buttocks’.  She isn’t sure what to make of that.

The guests are invited to play a game.  Each book is put into an identical cardboard box.  The boxes can be opened and closed without leaving a mark.  The twenty boxes are piled up around the Christmas Tree.

The guests are told that they will each have the opportunity to open half of the boxes.  Their objective is to find their own book.  If they all succeed the group wins and they will win a trip to Paris.  If any one of them fails then the group fails but they will each get a Twinkie to keep for life.

The guests are taken to another room and then taken to the tree one at a time.  They cannot see what any other guest does at the tree.  They are not able to communicate once  the game starts.  The boxes are put back after each guest, as though they had never been there.

You would think that the chance of the group succeeding was 1/2^20 but they can do much better than that.

The group must come up with a strategy before the game starts.  What is the best strategy to get the group to Paris, and let Janine keep her ’Crazy Buttocks’?

These books are all real.  They will be helpful if you have ever had any of the following thoughts:

We all know the Nazis killed millions of innocent people but what were they like on ecological issues?

I would like to speak Italian but can’t be bothered to learn any Italian words, can you help?

Aubergines are very flushed, just how angry are they?

I think I’m dead, how can I tell for certain?

I am rich but dead.  How should I pimp my coffin?

I am worried about running into large, slow moving objects; can you suggest any strategies to avoid this?

Just how boring was 1587?

I live thousands of miles from Versailles.  Will I get a good view?

I am English, am I human?

My buttocks are insane.  

Combine the four number 1,3,4,and 6 with operators of addition, subtraction, multiplication and division (and parenthesis to indicate order of operation) to yield an expression equal to 24.

I assure you that you can take this in the most straight-forward manner possible.  You aren’t mean to smoosh them together to get “13″ out of 1 and 3.  You aren’t meant to use “1″ as a problem number or something of that sort.  An answer will look something like this:

(4-1)*3/6

except that is equal to 1.5 .  Your expression must equal 24.

I’m interested to hear if this is as difficult for others as it was for me.

   Three Points are taken at random on an infinite Plane.  Find the chance of their being the vertices of an obtuse-angled Triangle.

Note: An obtuse-angled triangle is one that has an angle measuring more than 90 degrees.

Oh by the way, would you like a cool Math Factor Poster? Click on this to download: