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Is a knight really insufficient material?

If your Opponent has a Knight or pawn, with you only having a knight, you can checkmate. Ex 1) lichess.org/study/Pi7K9Xrd. Ex 2) lichess.org/387Vns55

So in a tourney, if your opponent had a knight or pawn vs ur knight and u ended up flagging, if they flagged you would win no? For the above reasons. Same question for OTB tourneys
For example 1, replace the pawn with a bishop, a scenario in which a bishop also works
You are right. If you lose on time, then you lose, except when there exists no series of legal moves leading to checkmate, in which case it is a draw. This applies to all games with time + increment < 60 minutes per player per game, on-line as well as over the board. At slower time controls a player can call the arbiter and claim a draw when judged by the arbiter the opponent does not try to win.
A lot of tournaments excluded that "referee claim" in their rules and even the arbiter can be like: "wanna see some more moves"...

So basically it could be a loss in a tournament game with classic time control as well.

PS: the "close-to-official" rule has been implemented here for about 1 year, you should adress it to the USCF, chess.com or - I don't know.
It would have been better to loose the knight or move it away from his own king. Without assistance from the opponent a checkmate is not possible. So a knight is considered insufficient material to checkmate without assistance. The assistance comes from the opponent playing the worst moves possible until their king is checkmated. On move 44 the knight could have been captured by either the king or the knight.
It is usually given draw in OTB after a few more moves are made so that the referee can see that you are able to hold the position. I had this once with rook versus rook when my opp didnt accept the draw offer, i claimed, referee said, 'play some more moves' and ten moves later he declared it to be draw.

In online chess there are no referees and if you play on sites which are aware of that fact you will get draw always if he has just a knight. Which is not optimal, it should just be draw if you have more than just pawns.

Lichess is very strict in this regard, and you will lose if a mate is possible in theory. So, he has 8 rooks, you have just knight, he loses on time, you win. This will be draw on other sites like ICC or chess.com.
Knight versus 8 or 1 rook can never checkmate by a series of legal moves.
that is why N vs R is a win while B vs R is draw on lichess. The bishop indeed can not mate.

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