Online Guide to Progressive Chess

What is a parity position?

The parity position with the fewest number of pieces is the following:

Black to play in this position can hold easily by shuffling back and forth between g8 and h7, ending finally on g8. If white had to defend this position, however, he would LOSE, because his king would be forced to end on h7.

Now you may be thinking that parity positions are unusual animals that only occur once in a blue moon. However, the theoretical assessment of many positions is dependent on which color is defending the position! In fact, as we shall soon see, king, bishop, and knight against king is a parity position! Black can always force mate with bishop and knight, whereas white cannot!

What material is sufficient to defeat a lone king?

• K+Q
• K+2B, except 2 B of the same color
• K+R+(any piece)
• K+(any 3 of B,N,R), except 3 B of the same color

• K+R
• K+B
• K+N

• K+B+N
• K+2N

K+B+N versus K

The author would like to acknowledge Russell Linnemann, who did all the work related to the theory of this endgame and the K+2N vs. K ending.

• Black always forces mate of the white king.
• White can force mate of the black king unless the black king is on c4, c5, d3, d4, d5, d6, e3, e4, e5, e6, f4, f5, or one of the squares among b4, b5, d2, d7, e2, e7, g4, g5 that is the opposite color of the bishop

Scenario 1: The King in the Center (d4/e4/d5/e5)

In the first stage, the white king is merely forced out of the center 4 squares (there are numerous other ways to do this). Note that if the king had begun on d4 or e5 (assuming the bishop were somewhere else), the same setup would now force the king into the second layer.

Scenario 2: The King in the Second Layer (c4/c5/d3/d6/e3/e6/f4/f5)

In this position, the king is forced onto one of the corner squares of the second layer (c3/c6/f3/f6) or to the outer layer. Alternatively, the king could have gone to a dark square (e.g. c5), but the same exact setup would drive the king to f3, c6, or the outer layers.

We examine what happens if the king tries to stay in the second layer (i.e. goes to c3, c6, f3, or f6) in scenario 3. We examine the consequences of the king going to the squares in the third layer of the same color as the bishop in scenario 5. Finally, we examine the consequences of the king going to the squares in the third layer of the opposite color as the bishop in scenario 6. Because scenario 6 represents a parity position, this is the only way for black to draw the K+B+N v. K ending.

Scenario 3: The King in the Second Layer Corner (c3/c6/f3/f6)

After this check, the king is forced into the third layer. Notice how the bishop on e7 covers d8 and a3 and keeps the enemy king imprisoned. Clearly, the king cannot go to the back row or to the corner, else he will be trivially mated. This leaves only the squares b6 (c7) and b7 to be examined (see scenario 4).

The second layer corner square of the same color as the bishop is also a trivial win (e.g. W: Kc3, B: Ke3, Ba5, Nb6).

Scenario 4: The King in the Third Layer Corner (b2/b3/b6/b7/c2/c7/f2/f7/g2/g3/g6/g7)

Now the king is forced onto the back rank where it is easily mated. Note that putting the king on b7 would not change the position.

Scenario 5: The King in the Third Layer Center, Same Color as Bishop (b4/d2/e7/g5)

Here the king is forced onto one of the squares we have already examined and determined to be a win (see scenario 4).

Scenario 6: Parity Position -- The King in the Third Layer Center, Opposite Color as Bishop (b5/d7/e2/g4)

Finally, we reach the parity position--the key to the whole ending of B+N. The lone king has only two squares to shuffle between (e1 and e2). If black plays the lone king, he can remain safely on the same square. White, however, is forced to walk onto the back row and be mated. There is no checking configuration that can prevent the lone king from escaping to another safe square, i.e. there is simply no way for white to make progress from this position. The only way to win is for black to zugzwang a white king off of the safe square.

The third layer center squares of the same color as the bishop could also be won by zugzwang (W: Kd2, B: Kb2 Bf2 Nf4).

K+2N versus K

• Black can always force mate with K+2N against K
• White can never mate with K+2N against K unless the enemy king is already on the mating square

Scenario 1: Forcing the King out of the Center Two Layers

The above three diagrams show how to force the king out of the center two layers and into the outer two layers of the board.

Scenario 2: Parity Positions

The above three diagrams show the key parity positions for the K+2N versus K ending. Only black can make progress from these positions, forcing white onto the back rank or into the corner. It is worth nothing the "safe" nature of the squares a2, a7, b1, b8, g1, g8, h2, and h7, which are the 8 squares in the outermost layer on which there is no mating configuration.

Scenario 3: Odds and Ends

Finally, we show the two key mating configurations plus the method for forcing the king out of the corner of the next to outermost layer. Even from b2/b7/g2/g7, white could still not force checkmate of a black king, since the black king could simply sit on one of the "safe" squares (a2/a7/etc).

Now suppose one side has a pawn or two? The assessment of most of these positions is fairly obvious. We can make the following generalizations about the lone king:

• K+P versus K is always a draw unless the side with the pawn defends the pawn and controls the queening square
• K+B+P versus K is likewise always a draw (if the defending king can sit in front of the pawn on a square opposite colo from the bishop)
• K+N+P versus K is always a win (Kd4, Pd5, Ne4, e.g. and the pawn moves forward one step each series)
• K+R+P versus K is always a win
• K+2P versus K is always a win if the king can defend both pawns; otherwise, it is a draw, assuming the lone king has time to take the undefended pawn and return to a square in front of the defended pawn

This concludes our discussion of endgames where one side only has a king. Now let's look at some more complicated positions.

What is a fortress?