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(From Queen's Mathematical Communicator, 1991) STRUCTURAL ASPECTS OF CHESS PROBLEMSBy Andrew Kalotay Andrew Kalotay (Arts '64, M. Sc. '66, Ph.D. Toronto) represented Canada in the 1966 Chess Olympiad in Cuba. Although the game of chess has many mathematical elements, these tend to be more pronounced in composed problems. Unlike their close relatives, endgames ("white to play and win"), problems ("white to play and mate in two moves") are void of any practical value (whatever practical value playing chess might have). The goal of composers is to express artistic ideas and to discover new structures. Although I have always enjoyed the challenge of solving problems, I became interested in composing only relatively recently. I have been specializing in problems employing relatively few pieces. A problem with at most 7 pieces is called a miniature, a problem with at most 12 pieces is a Meredith; most modern problems express complicated themes that require more than 12 pieces. Several computer programs for solving problems are commercially available. The more sophisticated of these do not rely on brute force methods, but instead exploit the existence of a limited number of terminal positions, which may further be reduced by symmetry considerations. These programs are used by composers for validation, i.e., to check that the intended solution exists and that it is unique. I have been particularly interested in so-called helpmates, where the two sides cooperate to create a terminal position in which white checkmates black. The inventor of helpmates was the 19th century American problemist Sam Loyd. His original composition (Fig. 1) appeared in the Chess Monthly in 1860. In the solution, white builds a battery by placing the bishop in front of the rook, while the black king cooperates by marching to the line of the rook. The mate results when the white's bishop fires the battery, producing a discovered check. Figure 1. Helpmate in 3 moves (S. Loyd, Chess Monthly, 1860) 
Incidentally, Loyd is also believed to be the inventor of the puzzle with a 4 by 4 square with 15 numbers to be rearranged sequentially. I recall discussing this puzzle with respect to the notion of parity in our linear algebra course. Most modern helpmates attempt to express a common theme several times. Often a problem is in fact a series of problems obtained by twinning, i.e., by changing the original position slightly. Since 1978 a jury of international experts has been awarding a prize for the best problem published during the year that employs at most four pieces, of which teo are normally kings. Because of the limited material, it is difficult to express profound ideas; a mathematical analogy might be the proof of a theorem based upon two well-chosen axioms. The winning composition in 1989 in this "Four Men Only" tourney was a joint work by Pal Benko and me. This helpmate in 3½ moves had appeared in the July 1989 issue of the British publication called The Problemist, where it also won fourth prize in the helpmate category. Benko, by the way, is unique among chessplayers in being both an international grandmaster and a first-rate composer of both endgames and problems. The problem (Fig. 2) is an extension of Loyd's original idea. Each solution consists of the building and the firing of a battery. In the original position the rook discovers the bishop, in the second position the bishop discovers the rook, and in the final position the king discovers the rook. The interested reader may wish to show that this effect cannot be achieved by the king discovering the bishop. Figure 2. Helpmate in 3½ moves (P. Benko and A. Kalotay, The Problemist, July 1989) 
Comment by judge for Problemist: "Three different battery formations with only four pieces! Good twinning, antiduals." Comment by "Four Men Only" Jury: "Three time battery with discovered mate: rook-bishop, bishop-rook, king-rook!"
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