Mathematics and Chess

John MacQuarrie


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Chess problems take many different forms. The most common form is given by the specification of a board position, the specification of the state of play, and a statement of the condition of solution. Note that this is an extremely general form, and can result in a great many different types of problem. The board position given in a Chess problem will generally be far simpler than the initial position, with far fewer options available to the players.

The potential lack of available pieces makes the following rather odd statement possible: Chess problems are older than Chess. As has been said, Chess as we think of it now did not exist six hundred years ago, though variants of it did. In Arab countries from the sixth Century, a variation of Chaturanga, called Shatranj was played. As with Chaturanga, there were similarities and differences to modern Chess. Regarding the back row of pieces, the King, the Rook and the Knight move in the same way as their modern counterparts, while the Elephant (later becoming a Bishop) jumped two squares diagonally, and the General (later becoming the Queen) moved one square diagonally. It is from Shatranj that, we believe, is found the origin of the Chess Problem. Manuscripts have been found containing a great many Shatranj problems, known as Mansubat. Amongst the problems found in the manuscripts were several involving only those pieces that still remain in chess today. An example of such a 'Chess' problem is given below; it is many hundreds of years older than modern Chess. (Murray, 1913)
Black to Play and Win    Projects MacQuarrie Diagrams ch2 1

Thinking of the board as an array, the board will be indexed as shown above. In this way, every square is uniquely indexed.

Note that the problem has been completely specified by the board position shown in the diagram, the specification of the state of play (Black to Play), and the condition of solution (Black to Win).

In an attempt to give an idea of the logical, deductive reasoning that must permeate the thoughts of the Chess problem solver, we shall work through the main steps in this simple problem.

Black, to play, has 29 legal moves. To go through each of these one by one would be time-consuming and frustrating. However, a consideration of the board will impose an extra condition upon the first move, and every Black move thereafter. The position in the bottom left corner of the board is such that given the option, White will move the Rook d2 - d1, checkmating Black's King. In order that this does not happen, Black must ensure that White is always forced into making a different move. This can only be done if after every Black move, the White King is in Check. Thus, this condition limits the 29 possible first moves by Black to only 4, all of which are Rook moves (f6 - f7, f6 - g6, h6 - g6, h6 - h7). Note next that the Knight protects neither Rook, and both are under threat from the White King. If either Rook were to change row, the King could certainly take one of them without benefit for Black. Thus, neither of these two options is sensible, it will be impossible to Checkmate the King with only a Rook and a Knight without at any stage leaving the King in a position out of Check. We have thus reduced the possibilities to 2 (f6 - g6, h6 - g6), and from here it is easy to consider the possibilities one by one. On doing this, it will be quickly discovered that the move h6 - g6 leads to difficulties, and will not produce a Mate for Black, while after performing the move f6 - g6, the Mate follows naturally, by moving the Rooks appropriately. The complete solution, where White moves are on the left, Black moves on the right, K stands for King and R stands for Rook, is given below. + means that the move results in a Check.
1. ... , R f6 - g6 +
2. K g7 - f8 , R h6 - h8 +
3. K f8 - e7 , R h8 - h7 +
4. K e7 - f8 , ... (If 4 K e7 - e8 or d8, R g6 - g8 is mate.)
4. ... , R h7 - f7 +
5. K f8 - e8 , R g6 - g8 Mate

By far the most common type of Chess problems are those whose solution involves checkmating the opposition. Further, the most common type of said problems are stated 'Mate in N moves' where N is usually (though far from exclusively) two, three of four. These problems require that from the position given, a method is found that will Checkmate the opposing King within the allocated number of moves, no matter what moves are made by the opposition. Problems of this type are extremely varied and, even when N is as little as two, are often far more difficult and interesting than the example provided. As demonstration of this, attempt the following problem, composed by A.F. Mackenzie in 1904 and published in the Sydney Morning Herald (Quoted in White, 1911).
White to Play and Mate in Two    Projects MacQuarrie Diagrams ch2 2

While postponing the solution, it is worth noting that in the 20th Century work went into the task of classifying all such problems. Alain C. White pioneered this formal pursuit in his publication First Steps in the Classification of Two-Movers (1911). Though the process of categorising and analysing Chess problems, in order that they be classified, certainly has mathematical worth, the details and intricacies of the process will be of little interest to any but the Chess enthusiast. Further discussion of this topic has therefore been omitted.

The first move in the previous problem is Knight c4 - a3. The most obvious (though not the only) return move by Black is for the King to take the Rook on e3. Even from here, it is not immediately clear what move must be done to Checkmate the King. In fact, the correct move is Knight c3 - e4.

There are many other different types of Chess problem, involving different types of mathematical reasoning. While it would be fruitless to go through all of them, one more example is included, since the inductive logic required is interesting.

Colouring problems give a board position in which the pieces are uncoloured. Working only with the assumption that the position has been reached through a legal game of Chess, the problem asks that the pieces be coloured in correctly. The solution is obtained using a process called 'Retrograde Analysis' where, based on conditions required by the position, the game must be played in reverse so that the necessary deductions can be made. Such problems can become incredibly complex, with tens of pieces requiring to be coloured, but as a simple demonstration consider the following, composed by Gideon Husserl and printed in Feenschach in 1986 (Quoted in Schnoebelen, n.d.):

Colour the Pieces    Projects MacQuarrie Diagrams ch2 3

Note that both Kings are potentially in Check from both the Queen and the Rook. The first deduction we can make is that the lower two pieces (on row 6) must be of the same colour, since, if they were not, the previous move must have been a row shift of the Rook, which it could not have been since the Queen would be checking the corresponding King. Also, the Rook and the Queen must be of the same colour, since the two Kings can never simultaneously be in Check. Now we must consider what the last move must have been in order to bring about the position shown. Since the opposing King can only be in Check during his move, the pieces must have been such that this was not the case. A piece must have stood on g7, obstructing the Rook, but this piece could not be the King or a Queen. Thus, the only possibility is that the piece was a pawn, and that it was promoted to a Queen by taking a piece on h8 (which must have been a Knight or a Bishop). Thus the Queen is White, as are the pieces on row 6. The King on g8 is Black.

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