♡ 34 ( +1 | -1 ) definitely!This is one endgame technique everyone should learn, as you may come into such a situation over the board and if you have no idea how to do it, it may be difficult to work it out under the time control. The same with bishop+knight+king vs. king which is the hardest checkmate to force in the game of chess, but it can be forced if the technique is perfect.
♡ 58 ( +1 | -1 ) Basic MatesB+N+K vs. K is nowhere near the hardest mate to force in the game, and the technique does not even need to be close to perfect (although it usually is). Once you have learned how to play this endgame, it usually becomes trivial, because there's simply nothing the weaker side can do to stop you, regardless of skill. I find other endgames (Q vs. R, or even worse, Q vs. R + P) much more difficult to play, and I would say the hardest basic mate in the game is K + N + N vs. K + P, provided the weaker king is in the center.
♡ 4 ( +1 | -1 ) what about 3 N vs king?there are many little endgames,Q vs B,R vs B,and so on.
♡ 19 ( +1 | -1 ) atrifixthe mates you mention while being more difficult are not always forced, sometimes the defense can force a draw.
While, K+B+N vs. K is ALWAYS forced, excuse me for not being clear enough.
♡ 115 ( +1 | -1 ) These endgamesI have mentioned are almost always won for the stronger side; only in extremely rare cases are they drawn (usually if the weaker side's pawn is far advanced or the weaker side has an immediate stalemate). Even KBB vs. K can be drawn if a certain position arises (for example: White Kf6, Bh8, Bg6; Black Kg8); but these positions are so rare that KBB vs. K is usually classified as a win, along with Q vs. R, Q vs. R+P, NN vs. P, etc. Sometimes, specifically in the case of NN vs. P, the 50 move rule must be suspended or extended to a 100 move rule, but it is ALWAYS a win provided the Pawn is behind Troitsky's line. Whether or not the win can be proven by the stronger side is a question of endgame technique.
As tulkos mentions, there are a few mini-endgames, but most of these are trivial; for example, KNNN vs. K arises so infrequently that it is practically superflous, and the winning technique is virtually the same of KNN vs. KP, only easier. R vs. B is almost always a draw, so it should not be counted as a 'mate' (but is worthy of study).
On a side note, one of the hardest drawn endings to draw is that of KRB vs. KR. This is usually a draw, but there are plenty of opportunities to go wrong, and can sometimes be won against weaker opposition.
♡ 32 ( +1 | -1 ) NNvsP"specifically in the case of NN vs. P, the 50 move rule must be suspended or extended to a 100 move rule, but it is ALWAYS a win provided the Pawn is behind Troitsky's line."
The 50 move rule CANNOT be suspended or extended, in regular chess; and your ALWAYS+provided is subtle: I can write: the KR vs. KB ending is ALWAYS a win, provided that... etc... :)
♡ 51 ( +1 | -1 ) GrantedI made a few assumptions with NN vs. P--if the pawn is behind Troitsky's line, it is a win; if not, it is usually drawn.
Perhaps I am not fully aware of current tournament rules, I'm certainly not up to speed on the latest developments. My understanding was that in very rare situations (for example, some NN vs. P situations require 60-70 moves to win with perfect technique), FIDE would extend the 50 move rule to 100 moves. Perhaps this rule has been changed?
♡ 8 ( +1 | -1 ) YesIt was applied during 80's: 75 moves for some positions; now it's 50 for any kind of position.