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    Mahina Kaholokula

    Math 7: Mathematical Puzzles

    23 May 2013

    The Art of Chess Puzzles

    Chess is one of few things that has stood the test of time. Originating as far back

    as the 6th

    century, it has developed over time to become the worldwide sport that you

    see today. There are clubs and competitions formed around chess; there are books and

    movies dedicated to its history and its strategies; and there are hundreds of different

    variations of the game now propagating around the world, from using non-standard

    boards (ex. in dimension), to non-standard pieces (fairy chess, for instance, may use a

    piece called the nightrider1

    which may move any length in the straight direction given

    by a standard knight move i.e. 1 cell up, 2 cells to the side), or with non-standard rules

    (suicide chess has the surprising objective of wanting to lose your king first in the

    game).

    One twist to chess is to create puzzles and problems involving the general game

    rules for others to then solve. The scope and variety of the puzzles is enormous: some

    start out with a pre-configured chessboard and task the puzzler with having one side

    checkmate the other within a certain number of moves. Others start with the endgame

    and have the puzzler work backwards to figure out what the last move played might

    have been. There are tour puzzles that focus on the movement of pieces and on paths

    that could (and could not) be taken. While others still are domination problems,

    1Fairy Chess Piece.

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    interested in the attack range of the pieces and how to set up specific pieces to attack

    specific squares. The list is extensive, with a range of themes and difficulty levels, but

    they all share a couple of key qualities, including the fact that they all embody a certain

    elegance, and they are all enjoyable to both create and especially solve.

    The most common chess problems by far are those in which the board is set up

    in a pre-determined configuration and it is the duty of the puzzler to complete some

    task, often a checkmate for one side or the other, in a given number of moves. This type

    of puzzle tends to advantage those who actually play a bit of chess, as they will have

    more experience with seeing connections and interactions between the pieces. (In fact,

    chess players often use this type of puzzle to practice and further develop these skills of

    perspective and observation.) Still, chess puzzles are meant to be accessible to

    everyone, no matter the skill level, and the only information the puzzler really needs in

    order to solve these problems is knowledge of how the pieces move. Thus we come to

    the classic chess puzzle, consisting of the chessboard, a configuration of pieces, and the

    sentence Mate in Y moves. X to move (X representing a color and Y a number). For

    example, here is one:

    Figure 1. Mate in 2 moves. White to move.

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    Now, if we are to believe that the puzzle is indeed possible, then one conclusion you

    must come to is that there is a unique set of moves that will get White to mate in 2

    moves. This means that once White moves, Black should not have any choice of which

    piece to move where because this is the only way to ensure that White will win every

    time no matter how Black tries to stop it. Thus we are looking for a move that forces

    Blacks next move. Often this means putting the king in check in a way that only allows

    one option for Black getting out of check. This will then lead to Whites winning move to

    checkmate. Here, there are three options for how White can check the Black king: move

    queen to g5, move bishop from c4 to f7, or move queen to f7. Lets consider these

    choices. We can reject the first, since a trade of queens will likely ensue, and White

    wont shake Blacks defense at all. The second seems plausible, but we lose what is

    called a pin in chess, where right now the bishop at c4 is poised to check the Black

    king as soon as the space in f7 is clear. This means that Black has no other option but to

    keep that space occupied. If we move bishop to f7 and the bishop is taken by Blacks

    rook, then White loses the pressure that this pin strategy is useful for. Thus, we come to

    option 3, moving the White queen to f7. It seems counterintuitive to move the queen to

    such a vulnerable spot, but, in fact, sacrifice is often necessary in chess and can have a

    winning payoff in the end. By moving queen to f7, we force Blacks next move: rook

    takes queen (Black can do nothing else to get out of check since the King is not able to

    move into any square out of check). The winning move here for White is then clear:

    when the Black rook takes the White queen at f7, the rook is not only pinned in this

    space by the White bishop, but a new vulnerability to the Black king has opened up via

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    the square f8. White can now take advantage of this and move his rook to e8, thus

    checkmating the Black king.

    This was more of a dense example to use with some formal tactics involved, but

    it illustrates well the strategy and excitement of the game. The strategy obviously comes

    from looking ahead in the game, finding that first move to force the opposition into a

    vulnerable position, and then taking advantage of that vulnerability. The excitement is

    closely related, watching as the drama of finding the right moves and exploiting

    vulnerabilities unfold in exactly the right fashion. Furthermore, by doing these puzzles

    more and more, you learn different strategies and plays, start noticing different

    connections between pieces and placements, and all the while enjoying the challenge

    that the puzzle sets out for you.

    Figure 2: Examples of Knight's Tours.

    The lines represent each jump the

    knight makes.

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    One very popular chess puzzle, which has been around for over 1000 years, is

    the Knights Tour2. The object of the game is to find a path around a chessboard that

    allows a knight to move to each square on the board exactly once. A common variation

    on the puzzle is to distinguish a closed path from an open one (a closed path ends

    on the same square it started on where an open one doesnt have this restriction). Over

    the years, many chess masters, mathematicians, computer scientists, and other curious

    puzzlers have dedicated time and effort to solving this problem, and it was quickly

    observed that there was no one unique solution to find. In fact, a fellow named Brendan

    McKay created a computer program to calculate the total number of Knights Tours

    possible on the standard 8 x 8 chessboard and the result was an astounding

    13,267,364,410,5323! Mapping out these journeys for the purpose of a puzzle, however,

    requires a little more personal investment in the game. Although there are multiple

    methods that have been proven successful, one of the simpler ones is de Moivres

    technique, named after the man who discovered it in the early 18th century. His idea

    was this: start in the outer ring of the board (the outer 2 loops since a knight must

    always move in an L shape) and stay as close as possible to this outer ring before

    working your way in to cover the center squares4.

    2Jelliss, George3McKay, Brendan D.4 Watkins, John J.

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    This strategy, easy to remember and implement, is always a good place to start

    when trying to find a Knights Tour around a new board. Although I did not delve into

    the details behind the technique, part of its success must be based on the fact that by

    touring the outer ring first, you are touching on those squares with the fewest options

    for further movement first, which allows for more flexibility among the inner rings later

    on in the game. By this I simply refer to the fact that a knight can only move to a certain

    number of squares from any given square, and this number is obviously smaller on the

    edges of a board, since a knight cannot move outside the board.

    A very important outcome of this puzzle has been its connection to the

    mathematical field of graph theory. The idea is to represent the Knights Tour as a

    graph, so that the vertices of the graph represent the squares of the board and the

    edges connect the vertices that a knight could legally move between. For example,

    Figure 2 above is a Knights Graph representing an 8 x 8 board. This representation

    Figure 2: De Moivres technique. By starting on square 1

    and moving around the outer ring of the board, onlymoving into the inner rings briefly when necessary, the

    tour can end on square 64. Interestingly, square 64 can

    also connect back to square 1, creating a closed pathsolution.

    Figure 3: Number of Possible Moves. This graph

    shows the number of possible moves a knight can

    make from a given square. Clearly there are moreoptions amongst the inner squares.

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    becomes analogous to a Hamiltonian path, which is simply a path that touches every

    vertex in a graph exactly once. (A Hamiltonian cycle is similarly paired with a closed

    Knights Tour