Comments/Ratings for a Single Item

There cannot be any checkmate when only King, Knight, and opposing King are
left on the board.  This is also true when only King, Bishop, and opposing
King are left on the board.

If all you have is a King and a Knight, I don't think you've got enough
material to mate a solitary King using traditional chess rules, even if
your opponent were to help you out and move his King where you want it to
be moved.

However, if you are playing chess on a nonstandard board, it might be
possible.  For instance, if the traditional 8x8 board had extra corners
that were squares that a King could stumble into, a 'mate' of some kind
might be possible.  Naturally, we'd be  talking about a pretty strange
chessboard - for instance, a non-euclidean board - perhaps with an extra
square just beyond each corner square - reachable by King or Knight, and
entered as though it were just another square on the diagonal.

But short of that, the simple answer to your question is: 'No - a King
and Knight cannot mate a solitary enemy King, on a traditional 8x8 board.,
with no other pieces to assist.'

King, Knight and Bishop CAN win against the lonely King, but I´m not sure
it is mandatory in ANY initial position. I have my doubts about if there
are certain positions which require more than 50 moves for a forced
victory, and in such cases the game is draw. I suspect, indeed, that these
initial positions exist. If someone has the answer, it should be
interesting to see it here.

My understanding is that KBN vs. K can win in under 50 moves from any
starting position except those where the lone King can immediately capture
the Knight or Bishop.  If my memory's not too crazy, I think the maximum
number of moves required is somewhere around 30-35.  It can be done in 50,
but there's often not much room for error.

K-B-N vs. K : I have read that the maximum number of moves needed is 33,
but the sequence must be extremely precise, an slight error can be enough
for surpasing the 50th. move, so it is difficult win the game in every
case.
I have also read an old article that K-Q vs. K-R is a victory for the
superior band in at most 36-40 moves, but the analysis is too complicated,
even with computers, to be enterely sure, by cause of the high branching
factor...And the article conclusion was based on a particular strategy by
part of the K-R team. It is also certain that slight errors are enough for
a draw by the 50 moves rule, playing under optimal defensive sequence. In
the game Svidler-Gelfand 2001 world championship eliminatories, Svidler
commited a slight error, but also Gelfand essayed a novel strategy, with
the Rook close to his King amost all the time as theory suggests, but
moving the Rook sometimes far from his King to precise squares in very
precise moments. By curiosity, I made an experiment with Chessmaster 9000
playing against itself, with a far-from-trivial initial position I have
found in Internet. The result: draw, by the 50 moves rule. This is not a
conclusive remark, but may be an indicative of the difficulties for the
superior band. This problem is a good exercise to be tested using
high-level programs running on powerful computers.

Hmmm... Chessmaster 9000 should never lose or draw any winable endgame with
5 or less pieces; it has an end-game database which contains all possible
positions with 5 or less pieces and can lookup the perfect responce and
number of moves to mate.

I just did a quick websearch and saw something about a flaw with the
endgame database originally shipped with CM9000, so I assume that's the
problem.

Thanks, Greg, I have seen the reported bug, but I am not very clear about
what caused it. The CM9000 version I used is of a friend of mine, and it
was downloaded from CD on his computer. I am not sure if the endgame in
question was reported as a flaw. On the other hand. I don´t know whether
it has beeen demostrated it is a victory for the K-Q team, I only know
that it is known a set of positions from which the victory for the K-Q is
guaranteed, but I am not sure how many moves are needed, if possible, to
reach any of these positions if you begin from any other. It is possible
that the problem for decide if it is a win for the superior team in this
case (KQ vs KR, any initial position) is not so trivial, and in fact,
there was a constest for a computer solution in 2002 (Regionals 2002
Warm-up Contest, Problem setter: Jimmy Mårdell). The most known
problem-position in the literature is the following:
Black: King on b3, Rook on e2; White: King on b1, Queen on a1, white moves
first (of course, if not, Black wins in 1)