If white has only one minor piece against a lone king, it is not possible to arrange the pieces so that the black king is checkmated. If white has a king and minor piece and so does black, checkmate can occasionally occur (see below left,) although it cannot be forced. If white has three pieces, the mate is easy. As we shall see, with two bishops white's job is relatively easy, with a bishop and knight it is somewhat harder, and with two knights it is typically impossible (although it can happen against a very bad opponent.)
The diagram at left shows a mate in 6 with two bishops.
1. Bc2 Kf1 2. Bd2 Kg1 3. Kg3 Kf1 4. Bd3+ Kg1 5. Be3+ Kh1 6. Be4#
If black plays 3. ... Kh1, instead of Kf1, white responds with the line: 4. Be1 Kg1 5. Bf2+ Kh1 6. Be4#, or 5...Kf1 6. Bd3#
The bishops coordinate by taking adjacent diagonals, thus cutting off the opposing king.
In the diagram position with Black to move, White can "gain a move" by triangulating his king:
1...Kd1 2. Kf4 Ke1 3. Ke4 Kd1 4. Kf3 Ke1. Now the diagram position is reached with White to move, and play proceeds as above. Black's king, hemmed in by the two bishops, cannot escape from the d1-e1 box.
In order to mate with a bishop and knight, white must force black's king to the corner that is the same color as his bishop - in this case the a8 square. This is not so easy to do, because the bishop and knight do not coordinate well in cutting off the opponent's king. Mate can always be forced unless black can win a white piece, but, even with best play, the mate may take over 30 moves.
In the position at right, much of the hard work has already been done. Black is being pressed in the direction of the a8 corner. Nevertheless, nine more moves are required to mate.
1. Kd6 Kc8 2. Ke7 Kb7 3. Kd7 Kb8 4. Ba6 Ka7 5. Bc8 Kb8 6. Ne7 Ka7 (or 6. ... Ka8 7. Kc7 Ka7 8. Nc6+ Ka8 9. Bb7# ) 7. Kc7 Ka8 8. Bb7+ Ka7 9. Nc6#
When white has two knights against a bare king, it is impossible to mate without cooperation from black. The diagram at left illustrates his predicament. The black king is cornered, but white has no way to force checkmate. Moving either Nf3, or moving Nh3 all lead to stalemate. White can try for 1. Ne4 Kg1 2. Nf3+ Kh1?? 3. Nf2#, but instead black plays 2. ... Kf1 3. Nd4 (otherwise black can play Ke2, and white must corner him again) 3. ... Ke1 and white still has a challenge to confine the black king.
However, checkmate can happen with king and two knights against king if the opponent makes enough foolish moves, ending with a checkmate position like those to the left. It is easier if the losing side's king is hemmed in by his own pieces, like that to the near left.
If black has one pawn, white may be able to win by cornering the black king with a king and one knight, and avoiding stalemate by letting black to advance his pawn, while the other knight administers mate. This is more likely to succeed if the black pawn is not far advanced, and needs at least three moves to become a queen. Alexey Troitsky, the famous endgame study composer, produced an extensive analysis of this endgame and composed several endgame studies with two knights against one pawn. Some positions require as many 70 moves without a pawn move or capture. This endgame has some historic interest - it was included in the amendment of the "50 move draw rule" to exclude those positions in which it could be demonstrated that more moves were required. (The exception to this rule has since been rescinded; the rule now dictates that either player may claim a draw after 50 moves without a capture or pawn move, regardless of the position on the chessboard. This means that the definition of the fifty-move rule now specifically excludes king-and-two-knights-versus-king-and-pawn positions.) The ending of two knights versus one pawn has occasionally occurred in practical tournament play.