Stalemate is a position where the player in turn has no possible move AND his king is not under an attack. This confuses players often since it’s very similar to checkmate, but must not be mistaken. In a checkmate, the king is under a check.
If an identical position has occured at least three times with the same player to move on each turn, the player in turn can claim a draw.
If during the last 50 moves by each player, no pawn has been moved and no piece capture has been made, a draw can be claimed by either player. This is another important reason why keeping notation comes in handy.
In chess, players are always allowed to offer a draw. These days some tournaments restrict the draw offers since they happen quite often in high level tournaments and agreeing to a draw in 5 moves or less is not exactly great PR. Offering a draw in chess must happen right after a player makes a move. You make a move, press the clock and offer a draw. The other player is able to accept the draw until he makes his own move. After this the offer is off the table.
If neither side have enough pieces to checkmate each other, the game ends in a draw. This is the case for example when only the kings are left on the board.
Even having one pawn on board is enough for the player to win since it is technically possible to checkmate with a pawn that is promoted to a queen. On the other hand, having a single knight is not enough.
You might ask how this is possible, especially since this extremely rarely happens in online chess and requires some sort of lag. But in live chess it occurs quite often.
For more information read the FIDE official rule handbook.
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