King and two Knights vs King Puzzle One: Black to move, helpmate in two

 A B C D E F G H 8 8 7 7 6 6 5 ♘ 5 4 4 3 3 2 ♔ 2 1 1 A B C D E F G H

The Knight is the most fascinating piece in all of chess because no other piece in the game shares its movement affinity. The King, Queen, and Rook all traverse the ranks and files, while the Queen and Bishop command the diagonals of the chessboard. The Knight is the only piece to not fall into these categories, instead having the ability to move to the set of all squares which are closest to it while being neither diagonally nor orthogonally related to the square the knight is currently on.

Said another way: the knight moves two squares along a rank or file, and then one square in either direction along the other axis.

I prefer to explain it as the set of all closest squares which are neither diagonal nor orthogonal, but one must play the room.

A King and a single Knight versus a King is a draw due to insufficient material to execute checkmate. Neither King may make a move to put themselves in check, and it isn't possible for the Knight to both deliver check and prevent escape. In the best case scenario, where the stronger side has the opposition, forces the king to the corner, and is able to deliver check by the Knight on the subsequent move, the weaker side King will have 1 free square to move to. Along the edge of the board: there will be at least 2 free squares. Any set of squares near the middle: 4 squares. There is no way to attain checkmate with only one Knight. As you may well observe from your own games, anytime a piece is one square away and diagonal to a Knight, the Knight can attack that piece on its very next turn. However, the inverse is also true. If a piece is one square away and orthogonal to a Knight, the Knight cannot attack that piece on its next turn. In fact, if we limit the example to just a single Knight and King of opposite color, we see that if the King ever closes to an immediately adjacent and orthogonal square of the Knight's, then the Knight will be hard pressed to escape the King!

Knight on e3, King on f3
MoveWhiteBlackAnnotation
1Nf1Kf2If the King didn't move, the Knight could attach with 2.Nd2 or 2.Nh2
2Nh2Kg2The Knight achieves nothing
3Ng4Kg3The Knight is farther from the corner at least...
4Nf60.5 - 0.5The Knight runs away and White now believes when Black said "You have insufficient material to checkmate me."

Now, two Knights, you say? Checkmate is, as an academic exercise, possible.

But it's going to take some help.

A King and any major piece can force checkmate. So to can a King and two Bishops, or a King and a Bishop and Knight. A King and two Knights cannot force checkmate. The two diagrams below show the total board control exhibited by a pair of adjacent Knights. What is, during the midgame, a force to be reckoned, becomes instead by the endgame quite underwhelming. The strongest possible deployment is to have the pair of Knights sharing a rank or file among the middle four squares of the board. Ideally, this position will have forced the weaker side's King to traverse a long edge of the board without being able to approach the middle.

As illustrated in Puzzles One and Two, this is the only position from which checkmate is academically possible (depending upon the stronger side's King placement, either in the corner or along the edge). However: it is important to note and remember that as the stronger side you are hoping for a blunder, preying on a fledgling player, or expecting your opponent to lose intentionally. Not the most sporting options. As the weaker side's player, it is most important to move closer to d5, e5, d4, or e4 whenever possible, and to track exactly how many moves have passed since the last capture or pawn move to claim a draw by the Fifty-move rule.

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                        ¤ ¤      ¤ A B C D E F G H 8 8 7 7 6 ¤ ¤ 6 5 ¤ ♘ ¤ 5 4 4 3 ¤ ¤ 3 2 ¤ ¤ 2 1 1 A B C D E F G H

It does no better to have a space between the Knights. If they share a rank or file, then there are many similar holes in protection as we see in the diagonal-adjacent example above. If they share a diagonal, then there is virtually no support between the pieces (it will take three moves for one Knight to protect the other) due to the lack of cohesion with respect to the consecutive squares the Knights control.

All said, there are only two possible ways to helpmate versus a King and two Knights. By avoiding these precise solutions in a live game, you'll easily be able to avoid checkmate and collect your draw by the fifty-move rule versus even the most obstinate of players.

King and two Knights vs King Puzzle Two: Black to move, helpmate in two

 A B C D E F G H 8 8 7 7 6 6 5 ♘ 5 4 ♚ 4 3 3 2 2 1 1 A B C D E F G H

Chess

Sources

Solutions to puzzles

Solution to King and two Knights vs King Puzzle One
MoveWhiteBlackAnnotation
1...Ka2Versus two Knights the middle is life. Helpmate? Seek a corner!
2Nc3+Ka1Versus two Knights the center is safe. The edges, less so. The corner is the only square you can be checkmated on!
3Nb3#1-0Helpmate

Solution to King and two Knights vs King Puzzle Two
MoveWhiteBlackAnnotation
1...Ka5This is the only legal move
2Nc6Ka4The king has three options. In a live game, move towards the center. There is only one correct move for a helpmate
3Nc3#1-0Helpmate