Here is a puzzle composed by Richard Réti in 1921. It looks impossible.
Here is a puzzle composed by Johann Sebastian Bach in 1747. It also looks impossible.
What is the connection between these two puzzles? What makes them puzzles, and what are their solutions?
In Richard Réti’s puzzle, there is a race between two pawns: the white pawn is racing up the board, the black pawn is racing down. If either of the pawns reaches its final square, it becomes a queen and that side wins the game; if both become queens simultaneously, the game is drawn.
The puzzle requires the White player, who moves first, to draw the game. What makes the puzzle seem impossible is that the black king is easily able to stop the white pawn from queening, whereas the white king seems to have no hope at all of stopping the black pawn. This is best shown by a concept taught to all beginning chess players, “the square of the pawn”.
When you have no pieces left except your king, and your opponent is racing a pawn towards its queening square, you can stop it if — and only if — your king can enter “the square of the pawn”. As you can see, the black king is already in the white pawn’s “square”, while the white king is three moves away from the black pawn’s “square”. The white king also seems hopelessly far away from protecting its own pawn if the black king approaches and captures it.
However, the puzzle does have a solution. The study consists of three elements: the square of the white pawn; its upside-down counterpart, the square of the black pawn; and the “Royal Piece”, the white king. The movement of the white king is the key: it has to move in relation to both the “squares of the pawns” simultaneously. As if the Royal Piece has the job of harmonising both other elements in a kind of counterpoint.
Now back to Bach and his impossible puzzle.
Bach’s puzzle is from one of his last pieces, the Musical Offering [Musikalisches Opfer]. The story of the piece is that Bach visited the court of Frederick the Great of Prussia in 1746. The king played Bach a theme of his own composition (the king was a keen and accomplished flute player):
At the king’s request, Bach proceeded to improvise a three-part fugue on the theme. He then apologised that he could not improvise something more worthy of such a wonderful theme, and promised to send King Frederick a finished piece once he had returned home to Leipzig. The result was the Musical Offering, which consists of three groups of pieces, all based on the “Royal Theme”. One of the groups is a collection of ten canons, which Bach presents in the manuscript as puzzles.
A canon is a piece in which a musical theme plays in one voice and is repeated by a second voice before the first one has finished: the simplest kinds of canons are rounds, such as London’s Burning or Frère Jacques. Puzzle canons, which were very popular in Bach’s day, provide just the melody of the canon, with cryptic instructions for constructing the second (and possibly third or further) voices. The solver has to work out how to create a harmonious result while obeying the instructions.
The canons of the Musical Offering all harmonise the “Royal Theme”, a meandering and chromatic melody which is hard enough to harmonise without making the accompaniment work as a canon at the same time. In the canon I am discussing here, the instructions are that it must work “Per augmentationem, contrario motu”, which means that the second part must play the melody in notes twice as long as they were originally (augmented, or “Per augmentationem”), and also upside-down (in contrary motion, or “contrario motu”). At the same time, the result of these two melodies playing together must harmonise with the notes of the “Royal Theme”. Quite a task.
By now, the similarity that I find between Bach’s puzzle canon and Réti’s chess puzzle might be clear. In Bach’s canon, the “Royal Theme” must harmonise simultaneously two other themes, which are upside-down versions of each other and proceed at different speeds. In Réti’s puzzle, the “Royal Piece” must coordinate simultaneously with two areas of the chess board, which are upside-down versions of each other, featuring pawns racing at different speeds. Réti’s puzzle is a study in counterpoint, as is Bach’s.
The solution to Réti’s puzzle
The first move in Réti’s solution is not hard to see — the white king advances in chase of the black pawn:
But the king does not just chase after the black pawn: the Royal Piece moves not to h7 (KR7) but to g7 (KN7). He moves towards the “square of the black pawn”, but also towards the “square of the white pawn”. By harmonising his move with both squares, the white king can, contrapuntally, achieve the harmonious equilibrium of a draw.
If the black pawn simply races to become a queen, the white king supports and advances his own pawn:
In the resulting position, black has the choice of queening the pawn and allowing White to do the same, or attacking the white pawn, when the Royal Piece harmoniously supports it:
Either way, a draw is achieved. And astonishingly, Black cannot disrupt this contrapuntal harmony by first advancing the black pawn and then attacking the white one:
By continuing to approach both “squares of the pawns”, the Royal Piece keeps them harmonised in counterpoint. In the position above, Black can take the white pawn, but then the Royal Piece will enter the “square of the black pawn”; or Black can race the pawn towards queening, but then the Royal Piece will save His own pawn.
The solution to Bach’s puzzle
The following realisation of Bach’s puzzle features Frederick of Prussia’s own instrument, the baroque flute, which is entirely appropriate. The cool animation shows the musical lines as blobs: you can see the Royal Theme (played twice, with orange blobs when the viola da gamba plays it, and red blobs when the flute does), accompanied in counterpoint by the canon, played by the harpsichord (green blobs) and the violin (blue blobs). You can see that the shape of the blue blobs is an upside-down version of the green blobs (contrario motu); you can hear that it is going at half the speed (per augmentationem), which is why the line of blue blobs is only half the length of the line of green blobs.
This particular solution to the puzzle was arranged by Silas Wollston, whom I happen to know: he studied with my colleagues at the Music Department of The Open University, gaining his PhD in 2009. Today he is a Fellow of Queens’ College, Cambridge. I don’t know if Silas plays chess; he certainly plays Bach as few people can.
Conclusion: chess, music, counterpoint
The intellectual pleasure to be had from solving each of these two puzzles (or, in my case, marvelling at the solutions without being able to solve them) seems to me to be of the same kind in each case. Both studies, when solved, produce a result that is elegant, sophisticated and deeply satisfying to witness. On one level, there is the technical brilliance of manipulating a restricted range of materials: just four pieces on Réti’s chessboard; the constraints of harmonising the Royal Theme in Bach’s canon. On another level, though, each study demonstrates the challenge set by all counterpoint in music, and by all games of chess as well: that of holding in the mind simultaneously several different fields of action, which behave independently of each other and yet interact and may alter each other at any moment. This, I believe, is why so many musicians are also lovers of the Royal Game, and vice versa. Certainly Richard Réti was, as I discuss in Réti the Modernist. Chess is a contrapuntal art.