polar explorer problem

A polar explorer walks 1km South, then 1km East, then 1 km North to end up at the point where he started. Where can he be? How many places (solutions) are there? Can you prove it?

Comments will be screened: spoilers won't be published (at least for a while).

(As usual, my interest here is cognitive: how do people form mathematical beliefs?)

wireless protocols

Why don't the protocols between laptops and the access points adapt to their distance?

The weaker the signal gets, the more redundant the protocol should become, promoting the optimal level of error-correction.

world cup in the USA

Dear Americans,

Just out of curiosity, how many of you are following your team during the World Cup?

Any Pittsburghers planning to watch the final on 9 July?

debugging one's thoughts, a system for

Here's a process for debugging one's thoughts:
* state your beliefs (data) in a formal language
* state hypotheses (i.e. something of the form forall x_1,..x_n, phi(x_1,...,x_n), e.g. explaining patterns in the data). (ignore all statistical connotations of the word "pattern": I'm talking about logical relationships)
* computer checks for consistency between beliefs and hypotheses, by trying to prove contradictions and absurdities from the premises above... also warn if unnecessary concepts are used, etc.
* now the user can revise hypotheses / concept definitions himself, or the machine could suggest a way out.

Go back to the first step to continue your theory-building / exploration for the purposes of understanding the stuff.

I think this would be similar to Socratic dialogue.