foot wrote:
Quick! What's the minimum number of moves a knight on g6 needs to get
to e4 ?
Four. (Bad example -- doesn't everybody know that it takes three
moves to get a knight to a horizontally/vertically adjacent square and
four moves to get to the opposite corner of a 3x3 square, anywhere on
the board?)
To answer this question, most chess players will simply start to
mentally move their knight around, counting the moves, until it lands
on the desired square. Let's say you counted 4 moves: e5-g4-f6-e4
That's one way to get there, and so is: f8-e6-g5-e4. But is there a
shorter route?
Obviously not. e4 and g6 are both white squares so it must take an
even number of moves to get between them. There's clearly no two-move
route so four must be the shortest. I realise this is a specific case
but the same kind of reasoning will almost always tell you that
there's no shorter route.
Nonetheless, Alexander's technique sounds like a total waste of time,
to me. The fact that a shorter path must be at least two moves
shorter makes it very easy to count the number of moves on an empty
board: your first guess is almost certainly right. And the technique
doesn't work when there are other pieces on the board. Which is to
say, all the time.
It only takes a moment to see that corner-to-corner can be done in
six moves. Can it be done in fewer? No, because a knight moves a
distance sqrt(5) (about 2.24 squares) on each turn, so can move at
most 8.94 squares in four turns. By Pythagoras, the distance
corner-to-corner is sqrt(49+49), about 9.90 squares. Obviously,
you're not going to work that out over the board but, even heading as
directly as possible for the opposite corner (e.g., a1-c2-d4-f5-g7)
leaves you a square short. And how likely are you to find an
eight-move corner-to-corner path? a1-c2-b4-d3-c5-e4-d6-f7-h8 is
hardly the first thing you'd try.
Dave.
--
David Richerby Mouldy Zen Widget (TM): it's like a
www.chiark.greenend.org.uk/~davidr/ thingy that puts you in touch with
the universe but it's starting to
grow mushrooms!