Quick! What's the minimum number of moves a knight on g6 needs to get
to e4 ?

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?

To answer that question with some measure of certainty (as opposed to
just guessing), most chess players would need to try a number of
alternate paths to e4. This might take an inordinate amount of time in
a game, and when the player has the answer, they might even need to go
through the moves again to double-check their answer, taking even more
time.

And what if the player was interested in more than one destination
square for the knight, or in the minimum number of moves it would take
for knights starting from two different squares to reach the same
square? Depending on where that square was, coming up with an answer
you felt confident in could really take a long time and be very
error-prone using the ordinary method of counting knight moves.

Fortunately, there's a better way. I'm a fan of obscure chess books, so
when I was at a local chess store the other day, the book "Knight
Moves" by Charles Alexander caught my eye. In it he describes a
relatively simple method, which he calls "Alexander's Technique", to
count the minimum number of knight moves it would take a knight to move
from any given square to any other square.

Alexander's Technique consists of a number of rules-of-thumb which the
player would need to memorize and apply. The rules range from the
incredibly simple -- such as learning that the only times it would take
a minimum of 6 moves to get a knight from one square to another is if
the knight starts on one of the corner squares (a1, a8, h1, h8) and
wishes to go to the diagonally-opposite square (ie. from a1 to h8, h1
to a8, etc...) -- to a number of more complex (but quite easily
learnable) rules.

It only took me a few minutes to start applying the rules after I'd
read the book, and maybe another ten minutes to half an hour to
completely internalize them... and after an hour or so of practice, I
feel completely confident I'd be able to quickly (certainly under about
15 seconds, and often as little as a second or two) to figure out what
the minimum number of moves it would take for a knight to get from any
one given square to any other.

The method can also be used to determine the minimum number of moves it
would take a knight to get to groups of squares. For example, if the
knight is **not** on one of the corner squares, you immediately know
that the minumum number of moves it would take to get to any other
square can not be greater than 5. The minimum number of moves it would
take a knight to move to other groupings of squares, such as any black
or any white square, could also be easily counted.

Alexander's Technique does have its limitations, the most obvious one
being that the presence of other pieces on the board might delay or
completely stop the knight from reaching a given square in the minimum
number of moves. For example, some moves might be illegal given the
situation on the board at a given time (such as having a piece of the
same color as the knight block one of the squares the knight needs to
move to), or if the knight gets captured on its way, etc... In this
case, the player is pretty much on his own in taking account of these
possibilities. Still, he can be certain that the knight will not reach
the destination square in less than the number of moves Alexander's
Technique tells him it will.

Another limitation is that the player could still make a mistake in
applying Alexander's Technique, perhaps requiring a double check of
some sort (by either re-applying Alexander's Technique, or manually
counting out the moves).

Still, given what Mr. Alexander set out to achieve with his method, I'd
say he admirably reached his goal.

One other thing that I should mention is that this is a really thin
book. All together it's only 82 pages, about 20 of which are exercises,
answers, and the index. Also, if you're just interested in learning the
technique itself, without knowing why it works, you could probably skip
the first 40 pages. And, if we also omit the 7 pages of examples of the
technique in action, that only leaves about 8 pages that you'd have to
read to learn the technique.

Of course, mastering the technique will require practice, so I advise
going through the examples, and doing at least 6 to 12 of the exercises
(which shouldn't take you more than an hour or so). After that, you'll
be counting knight moves like a pro!

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!

On 08 Oct 2007 14:13:25 +0100 (BST)
David Richerby wrote:

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?)

No. I don't think everyone knows that. In fact, I'd bet that most
chess players don't know that. And they probably don't know any other
techniques for determining the minimum number of moves required for
a knight to get from one given square to another.

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.

How do you know there's no two-move route? You probably tried a
number of routes to move your knight from g6 to e4 and found it
couldn't get there in two moves. That's exactly the kind of trial and
error approach that the technique taught in this book is designed to
improve on.

I realise this is a specific case but the same kind of reasoning
will almost always tell you that there's no shorter route.

It's good that you are able to reason about the moves a knight makes,
at least to a certain extent. You might want to try doing the same
with 5 and 3 move squares before you write this method off, though.

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.

Whether you're able to guess right without using the techniques in this
book really depends on where the knight is and where your desired
square is. Try guessing the number of knight moves required to go from
h1 to g2, or g1 to h6. You might be surprised. Or not. In which case,
this book is not for you. But I'd bet most people would take some time
to figure these out with any amount of certainty. And if you're
comparing how long it would take to get to more than one or two squares
then the time spent on these types of calculations can really add up...
especially if you want to do more than just guess.

And the technique doesn't work when there are other pieces on the board.
Which is to say, all the time.

Actually, it does work even when there are other pieces on the board,
as what the technique tells you is the **minimum** number of moves that
a knight will take to get to any given square. In other words, using
this technique you'll know that it will never take less than this
number of moves for a knight to get to its destination square. Of
course, it could always take more, depending on the situation on the
board, but even if there are other pieces on the board, that doesn't
mean they're going to necessarily affect the knight's path to where
it's going. It really depends on the position.

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.

Well, the path from one corner to the diagonally opposite corner is
really a "corner case" (pardon the pun), in that this kind of route
doesn't happen very often at all. 2, 3, 4, and 5 move paths are much
more common, and happen to be the ones where this method becomes most
useful.

foot wrote:
David Richerby wrote:
(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?)

No. I don't think everyone knows that. In fact, I'd bet that most
chess players don't know that.

Really? I'll admit that I had to check that the statement is true
around the edges of the board.


foot wrote:
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.

How do you know there's no two-move route?

Because any knight move from g6 either puts the knight on a square
adjacent to e4 (f4 or e5) or on a square a long way from e4.

This really isn't any harder than `How do you know there isn't a
one-move route for a bishop from g2 to d3?'


You probably tried a number of routes to move your knight from g6 to
e4 and found it couldn't get there in two moves.

No. It's instantly obvious to me that there's no two-move route. And
I'm not a strong player; I'm just familiar with how the pieces move.


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.

Whether you're able to guess right without using the techniques in
this book really depends on where the knight is and where your
desired square is.

Disagree. It really isn't at all difficult.


Try guessing the number of knight moves required to go from h1 to g2

Four: that's not a guess but an instant calculation. My reasoning is
that any move from h1 puts the knight on a square horizontally or
vertically adjacent to g2 and it takes three moves to get a knight
between horizontally or vertically adjacent squares.

or g1 to h6.

Four again; calculated in at most two seconds. Reasoning: h6 is too
far away to be able to get there in two moves (it's not possible to
get beyond the fifth rank) and g1-h3-g5-f7-h6 gets there in four.

But I'd bet most people would take some time to figure these out
with any amount of certainty.

Maybe so. But here's how to do it and get it right in seconds almost
every time.

Important knowledge
-------------------
1) Moving between two squares of the same colour must take an even
number of moves.
2) Moving between differently coloured squares must take an odd number
of moves.

Special cases
-------------
3) Moving to a diagonally adjacent square takes two moves, unless the
source or destination is a corner, in which case it takes four.
4) Moving to a horizontally or vertically adjacent square always takes
three moves.
5) Moving to the opposite corner of a 3x3 square always takes four
moves.

Technique
---------
6) To find the fastest route, move towards the destination in as
straight a line as possible and then use the special cases 3-5 to
determine the answer once you get close enough.
7) Avoid putting yourself on the opposite corner of a 3x3 square from
your destination if at all possible.

(For example, if you want to go from a1 to d5, don't move to b3
because it will take four more moves to get to d5. Instead, move to
c2, then b4 and d5 -- three moves.)

If you're wrong, you have to be wrong by at least two moves (points 1
and 2) and that's pretty hard to do.


And if you're comparing how long it would take to get to more than
one or two squares then the time spent on these types of
calculations can really add up...

They really don't. They're so fast that the difference between a
second or two (getting to one square) and two or three seconds
(getting to two squares) is negligible. Why would I ever want to know
how long it takes a knight to get to more than two squares?


And the technique doesn't work when there are other pieces on the
board. Which is to say, all the time.

Actually, it does work even when there are other pieces on the
board, as what the technique tells you is the **minimum** number of
moves that a knight will take to get to any given square.

And so does my technique. Much faster. And actually looking for the
routes will automatically show you which ones are or are not possible.


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 [argument using
Euclidean distance].

Well, the path from one corner to the diagonally opposite corner is
really a "corner case" (pardon the pun), in that this kind of route
doesn't happen very often at all.

Of course. I wouldn't have dared to use such a crazy method as
calculating Euclidean distances if I thought people might need to do
this often. :-)

2, 3, 4, and 5 move paths are much more common, and happen to be the
ones where this method becomes most useful.

If you have difficulty working out that it takes two moves to get a
knight between two squares, I really recommend you spend more time
playing chess and familiarizing yourself with the pieces. I hope that
doesn't sound patronizing; it isn't meant to.

I'll admit to being utterly flabbergasted as to why somebody would
write an eighty-page book explaining how to do something this simple.


Dave.

--
David Richerby Simple Painting (TM): it's like a
www.chiark.greenend.org.uk/~davidr/ Renaissance masterpiece but it has no
moving parts!

On 09 Oct 2007 11:21:14 +0100 (BST)
David Richerby wrote:

foot wrote:
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.

How do you know there's no two-move route?

Because any knight move from g6 either puts the knight on a square
adjacent to e4 (f4 or e5) or on a square a long way from e4.

So, when you say "any knight move", you're pretty much admitting that
you checked to see where every knight move landed. You're essentially
mentally checking to see if there's a two-move route.

From what you describe, it sounds like you thought something like:

"g6t o e7, and it's too far from e7 to make it to e4 in one move.
then, g6 to h4, and it's too far from h4 to make it to e4 in one move.
then, g6 to f8, and it's too far from f8 to make it to e4 in one move.
then, g6 to e5, and it's adjacent to e4, so the knight can't make it
from e5 to e4 in one move. then, g6 to f4, which is adjacent to e4, so
the knight can't make it from f4 to e4 in one move."

Once again, this is exactly the kind of trial and error approach the
book's method is supposed to improve upon.

This really isn't any harder than `How do you know there isn't a
one-move route for a bishop from g2 to d3?'

I think it is harder. Bishops move in straight lines. Seeing that a
given square is on or off the straight line path leading from a bishop
is a lot easier than the process you seem to have used.

By the way, I'm not claiming that the book's method makes recognizing
the minimum number of moves to every single possible destination square
the knight can get to as easily recognizable as a bishop move, but
there are some squares which studying the book will let you recognize
just as easily, and the others you can figure out without the trial and
error of the sort you seem to be using.

You probably tried a number of routes to move your knight from g6 to
e4 and found it couldn't get there in two moves.

No. It's instantly obvious to me that there's no two-move route. And
I'm not a strong player; I'm just familiar with how the pieces move.

Well, "obvious" doesn't mean you didn't try every possible single
move route, and then see if a second move can get you to your
destination square.... which is what you apparently did. It may be
"obvious", but it's not very efficient, and it's easy to make a mistake
when you're trying all these possible moves, or at least wind up
wondering if you really tried them all or might need to go back and
re-try them, just to be sure you got it right... which means more time,
and more opportunity for error.

But I'd bet most people would take some time to figure these out
with any amount of certainty.

Maybe so. But here's how to do it and get it right in seconds almost
every time.

Important knowledge
-------------------
1) Moving between two squares of the same colour must take an even
number of moves.
2) Moving between differently coloured squares must take an odd number
of moves.

Special cases
-------------
3) Moving to a diagonally adjacent square takes two moves, unless the
source or destination is a corner, in which case it takes four.
4) Moving to a horizontally or vertically adjacent square always takes
three moves.
5) Moving to the opposite corner of a 3x3 square always takes four
moves.

Very good. Up to this point, the technique you describe is similar to
what's in the book. However, you still haven't given any rules for
determining if a given square is 3 or 5 moves away (except when the
destination square is a horizontally or vertically adjacent square.

Technique
---------
6) To find the fastest route, move towards the destination in as
straight a line as possible and then use the special cases 3-5 to
determine the answer once you get close enough.

Ok. Well, now you're back to the trial and error method. There is a
better way than this, and the book teaches it.

And if you're comparing how long it would take to get to more than
one or two squares then the time spent on these types of
calculations can really add up...

They really don't. They're so fast that the difference between a
second or two (getting to one square) and two or three seconds
(getting to two squares) is negligible. Why would I ever want to know
how long it takes a knight to get to more than two squares?

Your method will find that moving to a certain square will take a
minimum of 5 moves in a second or two? I don't think so.

And the technique doesn't work when there are other pieces on the
board. Which is to say, all the time.

Actually, it does work even when there are other pieces on the
board, as what the technique tells you is the **minimum** number of
moves that a knight will take to get to any given square.

And so does my technique. Much faster. And actually looking for the
routes will automatically show you which ones are or are not possible.

Try using your method to on a square that's a minimum of 5 moves away,
like g1 to a6 (the example I meant to give in my last message, but said
"h6" for some inexplicable reason). With your method, it sounds like
there will be trial and error involved, and you will be left guessing
as to whether there is actually a shorter route that you might be
missing.

With the method taught in the book there is no guessing or trial and
error. It's just a matter of applying a rule sort of like your
"Moving to the opposite corner of a 3x3 square always takes four
moves," only the rule will apply for squares which are a minimum of 5
moves away.

2, 3, 4, and 5 move paths are much more common, and happen to be the
ones where this method becomes most useful.

If you have difficulty working out that it takes two moves to get a
knight between two squares, I really recommend you spend more time
playing chess and familiarizing yourself with the pieces.

Well, it's not a matter of difficulty. It's a matter of whether you're
figuring out the minimum number of moves it will take a knight to get
to a given square through trial and error and guessing, or whether
you're doing it by applying rules and getting answers you can feel
confident in, and doing it faster than the parts of your method that
involve trial and error.

I'll admit to being utterly flabbergasted as to why somebody would
write an eighty-page book explaining how to do something this simple.

Well, I found the book to be extremely useful, and I bet most other
chess players will too. You seem to already know quite a bit more
about moving knights than the average chess player (certainly much more
than I did before I bought the book), so it sounds like the book may
not be the best investment for you (though even for you, I think
the book could teach you some better methods than the ones you're using,
for certain squares on the board, such as the squares 3 and 5 moves
away).

And, by the way, how did you know that "Moving to the opposite corner
of a 3x3 square always takes four moves." Did you, perhaps, read this
in a book? Did you figure it out yourself? Or did someone teach you?
(someone who themselves probably read it in a book) If it was,
directly, or indirectly, received from a book, how can you maintain
that such a book is useless?

On 09 Oct 2007 11:21:14 +0100 (BST) David Richerby
wrote:

I'll admit to being utterly flabbergasted as to why somebody would
write an eighty-page book explaining how to do something this simple.

I never claimed you had to read the entire eighty-page book to learn
the methods within. In fact, in my review I explicitly said that you'd
only need to read about 8 pages of it to learn the method. The rest of
the book explains why the method works, provides examples, exercises,
answers, reference, and index... with lots of diagrams to boot. All
this makes the method easier to learn.

I, for one, am glad the author didn't try to cram the whole method in
to a single page, without any examples, explanations, exercises,
diagrams, or answers... as I would have probably had a harder time
learning it.

Anyway, no one will force you to read the entire book. You can read as
much or as little of it as you find useful.

foot wrote:
And, by the way, how did you know that "Moving to the opposite
corner of a 3x3 square always takes four moves."

I honestly can't remember. It's something I've known `forever', so to
speak.


If it was, directly, or indirectly, received from a book, how can
you maintain that such a book is useless?

Perhaps it was from a book. But if it was, it was a book that
happened to contain this information and much more besides, not a book
specifically about how to get a knight from A to B.


David Richerby wrote:
I'll admit to being utterly flabbergasted as to why somebody would
write an eighty-page book explaining how to do something this
simple.

I never claimed you had to read the entire eighty-page book to learn
the methods within. In fact, in my review I explicitly said that
you'd only need to read about 8 pages of it to learn the method.

Well, yes. But it's still an eighty page book and it's still eight
pages to describe how to do something trivial.


I, for one, am glad the author didn't try to cram the whole method
in to a single page, without any examples, explanations, exercises,
diagrams, or answers... as I would have probably had a harder time
learning it.

Of course. And it may well be an especially well-written book that
explains the method extremely well. But it still strikes me as being
like a book that teaches you enough rock-climbing skills to let you
get to the second floor of your house by scaling the outside wall and
coming in through the window, rather than just telling you to walk up
the stairs.


Dave.

--
David Richerby Broken Ghost (TM): it's like a
www.chiark.greenend.org.uk/~davidr/ haunting spirit but it doesn't work!

On Oct 9, 11:33 am, foot wrote:
And, by the way, how did you know that "Moving to the opposite corner
of a 3x3 square always takes four moves." Did you, perhaps, read this
in a book? Did you figure it out yourself? Or did someone teach you?
(someone who themselves probably read it in a book) If it was,
directly, or indirectly, received from a book, how can you maintain
that such a book is useless?

On 09 Oct 2007 11:21:14 +0100 (BST) David Richerby

wrote:

I'll admit to being utterly flabbergasted as to why somebody would
write an eighty-page book explaining how to do something this simple.

I never claimed you had to read the entire eighty-page book to learn
the methods within. In fact, in my review I explicitly said that you'd
only need to read about 8 pages of it to learn the method. The rest of
the book explains why the method works, provides examples, exercises,
answers, reference, and index... with lots of diagrams to boot. All
this makes the method easier to learn.

I, for one, am glad the author didn't try to cram the whole method in
to a single page, without any examples, explanations, exercises,
diagrams, or answers... as I would have probably had a harder time
learning it.

Anyway, no one will force you to read the entire book. You can read as
much or as little of it as you find useful.

This may be of interest as an intellectual exercise or parlor trick,
but its utility in practical play seems rather limited. I suppose it
might save some calculating time in an unusual endgame where it was
critical to get a knight to a distant square in the fewest moves, but
in most real-game positions, especially in the middle game, any
moderately experienced player can plot a knight's optimal path with
ease. Does Alexander claim his technique is of much practical use?

On 09 Oct 2007 17:33:33 +0100 (BST)
David Richerby wrote:

If it was, directly, or indirectly, received from a book, how can
you maintain that such a book is useless?

Perhaps it was from a book. But if it was, it was a book that
happened to contain this information and much more besides, not a book
specifically about how to get a knight from A to B.

Why do you object to niche chess books? If they achieve what they set
out to do, what's the problem?

I never claimed you had to read the entire eighty-page book to learn
the methods within. In fact, in my review I explicitly said that
you'd only need to read about 8 pages of it to learn the method.

Well, yes. But it's still an eighty page book and it's still eight
pages to describe how to do something trivial.

I don't think it's trivial. Nor do I think most chess players would
find it trivial. I think the book could have been shorter (by
omitting the first half that describes why the method works), but then
it would have been less interesting. It could have also been shortened
by taking out the examples, exercises, and diagrams, but then it would
have made learning the method harder. So, I think it's really the
perfect length.

I, for one, am glad the author didn't try to cram the whole method
in to a single page, without any examples, explanations, exercises,
diagrams, or answers... as I would have probably had a harder time
learning it.

Of course. And it may well be an especially well-written book that
explains the method extremely well. But it still strikes me as being
like a book that teaches you enough rock-climbing skills to let you
get to the second floor of your house by scaling the outside wall and
coming in through the window, rather than just telling you to walk up
the stairs.

Well, the trial and error parts of the method you describe is not like
"walking up the stairs", but more like trying to solve a maze by
randomly picking directions, instead of always making right hand turns.
But, hey, it works for you... and you seem to be in no hurry to improve
on it. Fine. But I do think that many other chess players will be
interested in learning a better way.

On Tue, 09 Oct 2007 09:42:58 -0700
Taylor Kingston wrote:

This may be of interest as an intellectual exercise or parlor trick,
but its utility in practical play seems rather limited. I suppose it
might save some calculating time in an unusual endgame where it was
critical to get a knight to a distant square in the fewest moves,

I don't think this is unusual at all. I've played many endgames myself
where doing just this was important. Using a knight to try to stop a
pawn from queening is relatively common. Sure, it doesn't happen in
every endgame, but I think it happens often enough that improving one's
technique would be useful, and certainly more than a "parlour trick".

but in most real-game positions, especially in the middle game, any
moderately experienced player can plot a knight's optimal path with
ease.

Most middlegame knight moves aren't very lengthy, so in general I
agree. But sometimes you do want to get a knight from the queenside
to the kingside (or perhaps to an adjacent square, which can sometimes
take up to four moves), and this method will quickly and reliably tell
you what the minimum number of moves such a path must take. I think
this is also useful... though I suppose some people might be satisfied
with just using a trial and error method.

Does Alexander claim his technique is of much practical use?

Well, here's the blurb from the back of the book. (I don't know if
this was written by Alexander or his publisher)

"Knight moves are the most difficult to visualize on the chess board.
In the endgame it is frequently necessary to determine whether a knight
can reach a certain square to intercept a passed pawn or protect one of
your own pieces or pawns. In the time pressure of many tournament
endgames, it is even more difficult to concentrate and feel confident
that you have correctly calculated the possible path of the knight and
the number of moves necessary in the variation. This often leads a
player to repeat the sequence in the mind a number of times to feel
certain that his projected moves will indeed accomplish the objective.
Mr. Alexander has developed a set of tools which make it very precise
and quick to determine the required number of moves, and whether or not
a sequence of knight moves will achieve a goal. The author has
developed a wonderful set of graphs which visualize the possibilities
of moving a knight to a desired goal. Many players will find this
technique to be invaluable in winning endgames involving the knight, or
even combinations in the course of the middlegame."