Friday 13 Aug 2010 1 comment
When I was a kid,
the
four-color
map theorem was famously unproven; for planar maps, there was a gap between
the number of colors known to be necessary (four) and the number known to be
sufficient (five). When a proof--that four colors sufficed--finally arrived,
it was huge and relied on extensive computer analysis--a victory,
but not a satisfying one.
More recently, there was an ever-narrowing gap between
how
many moves were known to be needed to solve configurations of Rubik's cube
and how many were known to be sufficent in all cases. When that gap was
closed (just recently), the solution came by extensive computer analysis.
(20 moves are necessary and sufficient.)
The team that solved the problem has posted
a description of their method,
along with a table showing how many configurations can be solved in a
minimum of
n moves for 0 <=
n <= 20.
A plot of the configuration counts (some are approximate) appears below.
I fitted a curve to the points not to suggest that the function is
meaningful for non-integer values of
n, but rather to show
how close to exponential (visually linear with a log scale in
y) the
function is until around
n = 18. It's reminiscent
of the growth of a population until it runs out of resources.
Friday 06 Aug 2010 3 comments
This guy assumes curious poses
when he's getting ready to shed.
Wednesday 04 Aug 2010 comment?
I just got back from a short but very pleasant trip to Lone Pine.
I was curious to see how I'd feel upon returning to Colorado;
would I be bummed that I had left the desert again, I wondered.
I wasn't bummed. On the contrary--it felt good to be back here,
and especially good to get back to work. Of all the things I
am grateful for, having work that I enjoy is high on the list.
Having said all that, I must also say that I was loving being in Lone Pine.
There was a rabbit in front of my house to greet
me when I drove up,
the
Caesalpinia was
in bloom, and a (live) scorpion was in my bathtub. Does the desert rock,
or what.