The Mathematical Knight
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IIh',l~--"El,t=+n,,~ut+l[,,.-~+am=:l'i"l<=4n+,~nnu,,~+.-1,+,<.--~ Michael Kleber and Ravi Vakil, Editors ! uch has been said of the affinity understood using the terminology and M between mathematics and chess: techniques of combinatorics. We also The two domains of human thought where relate a few of these ideas with practi- very limited sets of rules yield inex- cal endgame technique (see Diagrams haustible depths, challenges, frustra- lff., 10, 11). Mathematical tions and beauty. Both fields support a The latter half of the article shows venerable and burgeoning technical lit- some remarkable chess problems fea- erature and attract much more than turing the knight or knights. Most Knioht their share of child prodigies. For all "practical" chess players have little pa- that, the intersection of the two do- tience for the art of chess problems, Noam D. Elkies mains is not large. While chess and which has evolved a long way from its Richard P. Stanley mathematics may favor similar mind- origins in instructive exercises. But the sets, there are few places where a same formal concerns that may deter chess player or analyst can benefit the over-the-board player give some from a specific mathematical idea, problems a particular appeal to math- such as the symmetl7 of the board and ematicians. For instance, we will of most pieces' moves (see for instance exhibit a position, constructed by This column is a place for those bits of [24]) or the combinatorial game theory P. O'Shea and published in 1989, where of Berlekamp, Conway, and Guy (as in White, with only king and knight, has contagious mathematics that travel [4]). Still, when mathematics does find just one way to force mate in 48 (the from person to person in the applications in chess, striking and in- current record). We also show the community, because they are so structive results often arise. longest known legal game of chess that elegant, suprising, or appealing that is determined completely by its last Introduction move (discovered by R0sler in 1994)- one has an urge to pass them on. This article shows several mathemati- which happens to be checkmate by Contributions are most welcome. cal applications that feature the knight promotion to a knight. and its characteristic (2,1) leap. It is based on portions of a book tentatively Algebraic notation titled Chess and Mathematics, cur- We assume that the reader is familiar rently in preparation by the two au- with the rules of chess, but we assume thors of this article, which will cover very little knowledge of chess strategy. all aspects of the interactions between (The reader who knows, or is willing chess and mathematics. Mathemati- to accept as intuitively obvious, that cally, the choice of (2,1) and of the 8 • king and queen win against king, or 8 board may seem to be a special case even against king and knight if there is of no particular interest, and indeed we no immediate draw, will have no diffi- shall on occasion indicate variations culty following the analysis.) The and generalizations involving other reader will, however, have to follow leap parameters and board sizes. But the notation for chess moves, either by long experience points to the standard visualizing the moves on the diagram knight's move and chessboard size as or by setting up the position on the felicitous choices not only for the game board. Several notation systems have of chess but also for puzzles and prob- been used; the most common one lems involving the board and pieces, in- nowadays, and the one we use here, is cluding several of our examples. "algebraic notation," so called because We will begin by concentrating on of the coordinate system used to name Please send all submissions to the puzzles such as the knight's tour. Many the squares of the board. In the re- Mathematical Entertainments Editor, of these are clearly mathematical prob- maining paragraphs of this introduc- Ravi Vakil, Stanford University, lems in a very thin disguise (for in- tory section we outline this notation Department of Mathematics, Bidg, 380, stance, a closed knight's tour is a system. Readers already fluent in alge- Stanford, CA 94305-2125, USA Hamiltonian circuit on a certain graph braic notation may safely skip to the e-mail: [email protected],edu ~), and can be solved or at least better next section, A Chess Endgame. 22 TIlE MATHEMATICAL INTELLIGENCER 2003 SPRINGER VERLAG NEW YORK Each square on the 8 x 8 board is analysis begins with a Black move, we natural try is 1.KXc2, eliminating one uniquely determined by its row and use " . " to represent the previous pawn and imprisoning two of Black's column, called "rank" and "file" re- White move; thus "1 . Nd3!" is the remaining three men in the corner. But spectively. The ranks are numbered same first Black move. 1 . Nd3! breaks the blockade (Dia- from 1 to 8, the files named by letters A few further refinements are gram 2a). Black threatens nothing but a through h. In the initial array, ranks needed to subsume promotion and controls the key square el. The rules 1 and 2 are occupied by White's pieces castling, and to ensure that every move of chess do not allow White to pass the and pawns, ranks 8 and 7 by Black's; is uniquely specified by its notation. move; unable to go to el, the king must both queens are on the d-file, and both For instance, if Black were to move move elsewhere and release Black's kings on the e-file. Thus, viewed from first in Diagram 1 and promoted his c2- men. After 2.K• d3 (or any other move) white's side of the board (as are all the pawn to a queen (giving check), we Kbl followed by 3... alQ, Black wins diagrams in this article), the ranks are would write this as 1... clQ+, or more easily. numbered front bottom to top, the files likely 1 . clQ+?, because we shall from left to right. We name a square by see that after 2.Kxcl White can draw. Diagram 2a its column followed by the row; for in- Short and long castling are notated 0-0 stance, the White king in Diagram 1 be- and 0-0-0 respectively. If the piece and low is at d2. Each of the six kinds of destination square do not specify the chessmen is referred to by a single let- move uniquely, we also give the de- ter, usually its initial: K, Q, R, B, P are parture square's file, rank, or both. An king, queen, rook, bishop, and pawn extreme example: Starting from Dia- (often lowercase p is seen for pawn). gram 9, "Nbl" uniquely specifies a We cannot use the initial letter for the move of the c3 knight. But to move it knight because K is already the king, to d5 we would write "Ncdh" (because so we use its phonetic initial, N for other knights on the b- and f-files could kNight. For instance, Diagram 1 can be also reach dh); to a4, "N3a4" (not described as: White Kd2, Black Kal, "Nca4" because of the knight on c5); Nf2, Pa2, Pc2. and to e4, "Nc3e4" (why?). To notate a chess move we name the piece and its destination square, inter- A Chess Endgame polating "x" if the move is a capture. We begin by analyzing a relatively sim- Diagram 2b For pawn moves the P is usually sup- ple chess position (Diagram 1). This pressed; for pawn captures, it is re- may look like an endgame from actual placed by the pawn's file. Thus in Dia- play, but is a composed position--an grain 11, Black's pawn moves are "endgame study"--created (by NDE) notated a2 and a • b2 rather than Pa2 to bring the key point into sharper and P • b2. We follow a move by "+" focus. if it gives check, and by 'T' or "?" if we regard it as particularly strong or weak. Diagram 1 In some cases 'T' is used to indicate a thematic move, i.e., a move that is es- sential to the "theme" or main point of the problem. As an aid to following the analysis, moves are numbered consecutively, from the start of the game or from the diagram. For instance, we shall begin Returning to Diagram 1, let us try in- the discussion of Diagram 1 by con- stead 1.Kcl! This still locks in the Black sidering the possibility 1.KXc2 Nd3!. Kal and Pa2, and prepares to capture Here "1" indicates that these are the Pc2 next move, for instance 1 . .. White's and Black's first moves from Nd3+ 2.K• arriving at Diagram 2a the diagram; "K• means that the with Black to move. White has in effect White king captures the unit on c2; and succeeded in passing the move to "Nd3!" means that the Black knight White to move Black by taking a detour from d2 to c2. moves to the unoccupied square d3, Now it is Black who cannot pass, and and that this is regarded as a strong White, reduced to bare king, can do any move restores the White king's ac- move (the point here being that Black no better than draw, and even that with cess to el. For instance, play may con- prevents 2.Kcl even at the cost of let- difficulty: Black will surely win if either tinue 2... Nb4+ 3.Kcl, reaching Dia- ting White capture the knight), when pawn safely promotes to a queen.