MATHEMATICS 177 CHESS ENDGAMES (PART I) ENDGAMES WITHOUT PAWNS
 Definition: The phrase "best play" is used in chess to refer to a sequence of moves which maximize the chance of winning, or minimize the chance of losing, or preferably both.
 Definition: The ENDGAME begins at the move in a game (of chess) after which one can calculate or prove (with best play on both sides) who will win the game, or that the game will result in a draw.
 Note 1: BEST PLAY assumes that the opponent has both full knowledge of your strategy. Best play does NOT require the opposition to make an error or fail in some psychological way. BEST PLAY is what GOD would do.
 [Table A]: Under best play by both sides, the following situations always result in a win for the left-hand side:
 [A1] K + Q vs K [A2] K + R vs K (1st of two done in class) [A3] K + B + B vs K (2nd of two done in class) [A4] K + B + N vs K (difficult: may take 4O moves) [A5] K + Q vs K + R [A6] K + R + B vs K + N + N (difficult: analysis by computer) [A7] K + (any piece) + p vs K [A8] K + N + N vs K + p (difficult: may take more than 5O moves, and if the pawn is too far advanced, the game is drawn. Compare [B3] below).
 Note 2: If both sides in [A] above are given an additional single B, N, or R, the stronger side will usually win.
 Note 3: If both sides in [A] above are given an equal number of pawns, the "stronger" (left) side might no longer win, or even draw, but might actually lose!
 Note 4: (DRAWING RULE) If 5O moves are made (i.e., 5O by white AND 5O by black), and no capture or pawn move has occurred, the game is DRAWN. The tournament director may increase the number 5O if he/she deems it best. For example, [A8] has occurred twice in tournaments: one director allowed extra moves permitting a win; the other judged the game a draw.

[Table B] Under BEST PLAY by both players, the following are draws:

 [B1] K vs K [B2] K + B vs K [B3] K + N + N vs K (Compare with [A8] above) [B4] K + R vs K + B [B5] K + R vs K + N
 Note 5: If both sides in [B2] through [B5] above are given an equal number of added rooks, knights, or bishops (but not pawns or queens), the stronger left side then usually WINS by the eventual capture of a piece from the weaker side.
 Note 6: Recent computer advances have enabled chess stategists to examine ALL possible alternative moves in many endgames. The version of this page handed out in class reproduces an article describing the computer analysis of endgame A6 above.