New Directions in Enumerative Chess Problems

Normally a chess problem must have a unique solution, and is deemed unsound even if there are alternatives that differ only in the order in which the same moves are played. In an enumerative chess problem, the set of moves in the solution is (usually) unique but the order is not, and the task is to count the feasible permutations via an isomorphic problem in enumerative combinatorics. Almost all enumerative chess problems have been “series-movers”, in which one side plays an uninterrupted series of moves, unanswered except possibly for one move by the opponent at the end. This can be convenient for setting up enumeration problems, but we show that other problem genres also lend themselves to composing enumerative problems. Some of the resulting enumerations cannot be shown (or have not yet been shown) in series-movers. This article is based on a presentation given at the banquet in honor of Richard Stanley’s 60th birthday, and is dedicated to Stanley on this occasion. 1 Motivation and overview Normally a chess problem must have a unique solution, and is deemed unsound even if there are alternatives that differ only in the order in which the same moves are played. In an enumerative chess problem, the set of moves in the solution is (usually) unique but the order is not, and the task is to count the feasible permutations via an isomorphic problem in enumerative combinatorics. Quite a few such problems have been composed and published since about 1980 (see for instance [Puu, St4]). As Stanley notes in [St4], almost all such problems have been of a special type known as “series-movers”. In this article we give examples showing how several other kinds of problems, including the familiar “mate in nmoves”, can be used in the construction of enumerative chess problems. the electronic journal of combinatorics 11(2) (2004–2005), #A4 1 We also extend the range of enumeration problems shown. For instance, we give a problem whose number of solutions in n moves is the n-th Fibonacci number, and another problem that has exactly 10 solutions. This article is organized as follows. After the above introductory paragraph and the following Acknowledgements, we give some general discussion of enumerative chess problems and of how a problem might meaningfully combine mathematical content and chess interest. We then introduce some more specific considerations with two actual problems: one of the earliest enumerative chess problems, by Bonsdorff and Väisänen, and a recently composed problem by Richard Stanley. We then challenge the reader with ten further problems: another one by Stanley, and nine that we composed and are published here for the first time. We conclude by explaining the solution and mathematical context for each of those ten problems. Acknowledgements. This article is based on a presentation titled “How do I mate thee? Let me count the ways” that I gave the banquet of the conference in honor of Richard Stanley’s 60th birthday; the article is dedicated to him on this occasion. I thank Richard for introducing me to queue problems and to many other kinds of mathematical chess problems. Thanks too to Tim Chow, one of the organizers of the conference, for soliciting the presentation and proofreading a draft of this article; to Tim Chow and Bruce Sagan, for encouraging me to write it up for the present Festschrift; and to the referee, for carefully reading the manuscript and in particular for finding a flaw in the first version of Problem 8. This paper was typeset in LTEX, using Piet Tutelaers’ chess font for the diagrams. Several of the problems were checked for soundness with Popeye, a program created by Elmar Bartel, Norbert Geissler, and Torsten Linss to solve chess problems. The research was made possible in part by funding from the National Science Foundation. General considerations. All enumerative chess problems of the kind we are considering lead to questions of the form “in how many ways can one get from position X to position Y in n moves?”. But in general they are not explicitly formulated in this way, because this would be too trivial in several ways. It would be too easy for the composer to pose an enumerative problem in this form; it would be too easy for the solver to translate the problem back to pure combinatorics; and the problem would have so little chess content that one could more properly regard it as an enumerative combinatorics problem in a transparent chess disguise than as an enumerative chess problem. Instead, the composer It would be interesting to have enumerative chess problems not of this form, which would thus connect chess and enumerative combinatorics in an essentially new way. To be sure, there are other known types of enumerative problems using the chessboard or chess pieces, but these are all chess puzzles rather than chess problems, in that they use the board or pieces without reference to the game of chess. The most familiar examples are the enumeration of solutions to the Eight Queens problem (combinatorially, maximal Queen co-cliques on the 8 × 8 board) and of Knight’s tours, and their generalizations to other rectangular board sizes. Of even greater mathematical significance are “Rook numbers” (which count Rook co-cliques of size n on a given subset of an N ×N board, see [St1, p. 71ff.]) and the enumeration of tilings by dominos (a.k.a. matchings) of the board and various subsets (as in [St1, pp. 273–4 and 291–2, Ex.36]; see also [EKLP]). the electronic journal of combinatorics 11(2) (2004–2005), #A4 2 usually specifies only position X, and requires that Y be checkmate or stalemate. (These are the most common goals in chess problems, though one occasionally sees chess problems with other goals such as double check or pawn promotion.) The composer must then ensure that Y is the only such position reachable within the stated number of moves, and the solver must first find the target position Y using the solver’s knowledge or intuition of chess before unraveling the problem’s combinatorial structure. This also means that one diagram suffices to specify the problem. Another way to attain these goals is to exhibit only position Y and declare that X is the initial position where all 16 men of one or both sides stand at the beginning of a chess game. Most of the new problems in this article are of this type, known in the chess problem literature as “proof games” or “help-games” (we explain this terminology later). Two illustrative problems: Bonsdorff-Väisänen and Stanley A: Bonsdorff-Väisänen, 1983 kZ Z Z Z Z Z Z Z pOKZ Z Z o Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Series helpmate in 14. How many solutions? B: Richard Stanley, 2003 Z Z Z Z ZbZ Z Z spZ o j Z Z Z M ZKo Z Z Z a ZpZ Z Z Z Z Z S Z ZN Series helpmate in 7. How many solutions? An early example of an enumerative chess problem is Diagram A, composed by Bonsdorff and Väisänen and published in 1983 in the Finnish problem periodical Suomen Tehtäväniekat. This problem, and Stanley’s Diagram B, are examples of the “series helpmate”, an unorthodox genre of chess problems that is particularly well suited to the construction of enumerative problems. Black makes an uninterrupted series of moves, at the end of which White has a (unique) mate in one. The moves must be legal, and Black may not give check, except possibly on the final move of the series (in which case White’s mating move must also parry the check). Problems that require one side to make a series of moves are known as “series-movers”. Series stipulations appear regularly in the problem literature, though they are regarded as unorthodox compared to stipulations in which White and Black alternate moves as in normal chess-play. Such alternation is not a common element in enumerative combinatorics, and most enumerative chess problems avoid it, either explicitly by using a series stipulation, or implicitly by ensuring that the combinatorial structure involves only one player’s moves. This is the case for almost all problems in this article. A notable exception is Diagram 3, where (as in [CEF]) the comthe electronic journal of combinatorics 11(2) (2004–2005), #A4 3 binatorial problem is chosen to be expressible in terms of move alternation. In one of the other problems, both White and Black moves figure in the enumeration but do not interact, so that the problem reduces to a pair of series-movers. Likewise, enumerative problems usually do not involve struggle between Black and White: indispensable though it is to the game of chess, this struggle does not easily fit into a combinatorial problem. Usually the stipulation simply requires both sides to cooperate, or one side not to play at all, thus pre-empting any struggle. Our Diagram 4 is presented as a “mate in n” problem, which usually presupposes that Black strives to prevent this mate; but here Black has no choice, so again there is no real struggle. In Diagram 5, also a “mate in n”, Black again can do nothing to hinder White, but does have some choices, which the solver must account for. A (again): Bonsdorff-Väisänen, 1983 kZ Z Z Z Z Z Z Z pOKZ Z Z o Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Series helpmate in 14. How many solutions? A: Bonsdorff-Väisänen, 1983 ka Z Z Z aPZ Z Z ZKZ Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z The target position for Diagram A In the Bonsdorff-Väisänen problem, Black has 14 moves to reach a position where White can give checkmate. The only such checkmate reachable in as few as 14 Black moves is A, after both pawns have promoted to Bishops and moved to b8 and a7 via e5, blocking the King’s escape so that White’s move b7 gives checkmate. Thus the pawns/Bishops must travel along the following route: a6—a5—a4—a3—a2—a1(B)—e5—b8—a7 starting at a6 and a5, moving one space at a time, and ending at b8 and a7, with the pawn that starts on a5 (and the Bishop it promotes to) always in the lead. Enumerative chess problems such as this, where all the relevant chessmen move in one direction along a single path, are known as “queue problems”: the chessmen are imagined to be waiting in a queue and must maintain their relative order. The number of feasible move-orders is given by a known but nontrivial formula, making such queues appropriate for an enumerative chess problem. Here the queue contains just two units, which begin at the first two squares of the path and end in its last two squares. In this case the formula yields Cn = (2n)!/(n!(n+1)!), where n is the number of moves played by each unit in the queue. Hence the number of solutions of Diagram A is C7 = 14!/7!8! = 429. the electronic journal of combinatorics 11(2) (2004–2005), #A4 4 The Cn are the celebrated Catalan numbers, which enumerate a remarkable variety of combinatorial structures; see [St2, pages 221–231] and Sequence A000108 in [Slo]. In the setting of enumerative chess problems, a particularly useful way to see that Diagram 1 has C7 solutions is to organize Black’s moves as follows: a4 a3 a2 a1B Be5 Bb8 Ba7 a5 a4 a3 a2 a1B Be5 Bb8 The top (resp. bottom) row contains the lead (rear) pawn’s moves; a move order is feasible if and only if each move occurs before any other move(s) appearing in the quarter-plane extending down and to the right from it. These constraints amount to a structure of a poset (partially ordered set) on the set of Black’s moves, with x ≺ y if and only if x 6= y and move y appears in or below the row of x, and in or to the right of the column of x. In the problem, x ≺ y means x must be played before y, and a solution amounts to a linear extension of ≺, that is, a total order consistent with ≺. This kind of analysis applies to many enumerative chess problems. There is no general formula for counting linear extensions of an arbitrary partial order, but in many important cases a nontrivial closed form is known. For a queue problem such as Diagram A, with two chessmen that start next to each other at one end of the route and finish next to each other at the other end, the poset is the Young diagram corresponding to the partition (n, n) of 2n, and a linear extension corresponds to a standard Young tableau of shape (n, n). Therefore the number Cn of extensions can be obtained from the hook-length formula. The hook-length formula also answers any queue problem with k chessmen that start at the first k squares of the queue line, or equivalently end on the last k squares. Many such problems have been composed (see for instance [Puu]). Even the special case of k = 2 queues that lead to Catalan numbers has appeared in several published problems besides the Bonsdorff-Väisänen problem analyzed here. One example is a Väisänen problem that appears as Exercise 6.23 in [St2, p.232]. Another is the problem cited as Diagram 0 in the next section. B (again): Richard Stanley, 2003 Z Z Z Z ZbZ Z Z spZ o j Z Z Z M ZKo Z Z Z a ZpZ Z Z Z Z Z S Z ZN Series helpmate in 7. How many solutions? B: Richard Stanley, 2003 Z Z Z Z Z Z Z a Z Z Zrj Z o Z o ZKZ Z Z Z ZpZ Z Z Z o Z Z S Z Zb The position after Black’s series in Diagram B Also available and updated online from Richard Stanley’s website, see [St3]. the electronic journal of combinatorics 11(2) (2004–2005), #A4 5 Diagram B is a problem by Stanley [St4, pp.7–8] that also leads to an enumeration of linear extensions of a partial order, but one of a rather different flavor. Black must play the four pawn moves c6, d3, f2, fxg5, opening lines for Black’s Rook and two Bishops to play Rg6, Bg7, Bxh1 to reach position B, after which White mates with Rxh1. In a feasible permutation, each Rook or Bishop move must be played after its two line-opening pawn moves. We write these constraints as f2 < Bxh1 > c5 < Rg6 > fxg5 < Bg7 > d3. This means that in any feasible order of Black’s moves, such as 1 c5 2 fxg5 3 f2 4 Rg6 5 d3 6 Bxh1 7 Bg7, the moves f2, Bxh1, . . . , d3 must be numbered by integers that constitute a permutation of {1, 2, . . . , 7} satisfying those inequalities (such as 3 < 6 > 1 < 4 > 2 < 7 > 5 in our example). Therefore the solutions of Diagram B correspond bijectively with updown permutations of order 7. It is known that the number of up-down permutations of order n is the n-th Euler number En, which may be defined by the generating function