From Ian I got to this Observer article. The story is that university education is becoming less attractive as the costs increase and the salaries for graduates don’t correspondingly increase. The net result is that universities might be in an increasingly competitive game as western universities progressively lose their edge in the world wide market and as students look for more cost efficient alternative.

(…) more and more A-level students ask about alternatives to university, said the author of the research, Peter Brown, director of Gabbitas Educational Consultants. (…) we are seeing Chinese universities [are also] more financially attractive.

Asian universities stand to win big. In the western world, the first university to offer high quality, but significantly cheaper university education, by essentially cutting down on the fat and keeping what really matters, is going to win big time.

We are at a pivotal point where it might be good timing for a radical rethinking of university education.

Did I mention that you can listen to Stanford lectures on your ipod? See http://itunes.stanford.edu/. Chances are good that Stanford will be among the winners.

I just thought it was a cool title for a paper (if anyone has read it, let me know if it is any good):

On moving a sofa around a corner

in Geometriae Dedicata, Volume 42, Number 3, June 1992

Joseph L. Gerver

A necessary condition is given for a region of the plane to have the greatest possible area of any region able to move around a right-angled corner in a hallway of unit width. A region is constructed, with area 2.2195… and bounded by 18 analytic pieces, which satisfies this condition. It is conjectured that this is the unique region of maximum area.

Ian recalls some of the basic problem solving heuristics:

• If you are having difficulty understanding a problem, try drawing a picture.
• If you can’t find a solution, try assuming that you have a solution and seeing what you can derive from that (“working backward”).
• If the problem is abstract, try examining a concrete example.
• Try solving a more general problem first. This is the “inventor’s paradox”: a more ambitious plan may actually have more chances of success.

While I never studied these heuristics, I think I use them all. I probably learned them by trial and error. Maybe we ought to teach those.

I would add a few which I feel are very potent:

• Try to sketch a solution hastily, then try to find faults in your solution.
• If you can’t solve a problem, try to solve a related, but simpler problem.
• If you can’t solve a problem, try dividing into smaller problems (divide-and-conquer).

It looks like it is quite old, but I found the Combinatorial Object Server for the first time this week and I thought I’d share it with my readers. I was looking for irreducible polynomials with binary coefficients (don’t ask why) and I found that this server can generate them on the fly for you! A beautiful application of the web.

Here are some things it can do:

• Permutations and their restrictions
• Subsets or Combinations
• Permutations or Combinations of a Multiset
• Set Partitions
• Numerical Partitions and relatives
• Binary, rooted, free and other trees
• Necklaces, Lyndon words, DeBruijn Sequences
• Irreducible and Primitive Polynomials over GF(2) to GF(5)

This reminds me a bit of the famous Plouffe’s inverter which, given a floating point number, will give you a matching mathematical constant.

This is better documented elsewhere, but I could not find a quick reference on the web as to when you’d want to use the geometric mean instead of the arithmetic (usual) mean.

• Suppose that I’m 30% richer than last year, but last year I was 20% richer than the year before… what is the average growth? Well, my current wealth is 1.3 * 1.2 * w if w is my wealth two years ago. I can expect that if t is the average growth factor over the last two years, then my current wealth is t * t * w. Setting t = 1.25 is the wrong answer. In such a case, choosing t = sqrt(1.3 * 1.2) solves the problem.
• Another case where the geometric mean makes sense is when you are stuck averaging numbers that are not comparable like the time necessary to build a data cube, versus the average query time. Indeed, if a and b are two numbers and a is much smaller than b, then (2a +b)/2 is about the same as (a+b)/2. One component of your system is significantly worse and yet, you get the same average performance? That’s wrong. Computing sqrt (2ab) seems to make much more sense.

Here’s one reason to blog: so that you can belong to a very large network.

The utility of large networks, particularly social networks, can scale exponentially with the size of the network. (Wikipedia entry.)