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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.) These representations in graph theory led to the ability of

mathematicians to determine whether our not a Knights Tour was even possible on a

chessboard of any given dimension. The bottom line is that if a Hamiltonian path exists,

than a Knights Tour is possible, and if a Hamiltonian path does not exist, than there is

no legal Knights Tour possible on the board5.

This connection between the Knights Tour and graph theory led in 1991 to

Schwenks Theorem, created by Allen Schwenk, which lists out on what chessboards (by

dimension) a Knights Tour exists. While I will not probe deeper into the proofs of these

conditions, I think its interesting enough to note them down here:

Schwenks Theorem: An m x n chessboard with m nhas a Knights Tour unless

one or more of the following three conditions hold:

(a) m and n are both odd;(b)m = 1, 2, or 4; or

5 Watkins, John J.

Figure 4: A Knights Graph representing a 4x4 chessboard. There

is no Hamiltonian path that exists here, so it is impossible to do a

Knights Tour on a 4x4 chessboard.

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(c) m = 3 and n = 4, 6, or 8.6

While the problem of the Knights Tour has been around since the 9th

century, it has

taken on a new vitality in the past couple centuries. Famous minds from Euler to

Babbage7

have toyed with the puzzle; new solutions and strategies have been brought

into existence; and of course there are numerous variations on the puzzle including

straying from the standard 8x8 board or using other pieces instead of the knight (with

various restrictions here as well). The drama and excitement of finding that perfect

journey around a chessboard has thus captured and held the attention and curiosity of

minds all around, and continues to do so even to this day.

Chess has always been considered a pastime of intellectuals. Starting out as a sport

for nobles, chess retained its reputation through the 19th

century as a game purely for

educated and cultured society before attitudes changed and chess became more widely

played across the social ladder8. These stereotypes did not evolve completely without

reason, however, as playing chess does take a great deal of skill that only the noble class

of earlier centuries had the time to develop. Today, many more people have the time to

dedicate to learning such skills, and, indeed, studies have been held that prove that this

is time well spent. A good question raised here is whatskills, specifically, are being

developed by this game? It turns out that there may be more than you might imagine.

For instance, there are the more obvious qualities that chess promotes, such as focus

6Watkins, John J.7Jelliss8The History of Chess

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and patience in choosing a next move. This choice of which piece to move to what spot

also causes the player to think ahead, to anticipate how that move will affect future

play, etc. These skills are crucial not only to chess but to life in general, where

concentration, patience, looking forward, and strategizing can all help advance your

status and quality of life. However, studies have shown that there are other benefits to

playing chess, although the correlation between these may not always be as obvious.

One study by Stuart Margulies involving 9 mid-elementary schools showed that chess

improves reading skills9. A study in China in the late 1970s concluded that chess

improves math and science skills as well, and another study in Canada involving 500 fifth

graders in the 1980s corroborated this finding. Various other studies and surveys have

shown increases in creativity, memory, and self-esteem due to chess as well.10

While

these studies may have been based off playing legitimate games of chess, there is no

reason to believe that the same wont hold true for puzzle variations of the game.

Intuitively, it certainly seems that many of the same qualities that develop in a good

chess player such as focus, patience, creativity, and anticipation of future

consequences will also develop in a good chess puzzler.

Beyond the perhaps more tangible benefits mentioned above, chess puzzles should

be valued in their own right. Often preceded by the words the art of, chess is a

beautiful game. The elegance of a good puzzle is something to be appreciated, and

chess puzzles posses this quality. They manifest elegance in simplicity, in directness.

There is no trick, no attempt to fool or lead anyone astray with distractions or red

9Byrne, Beverly10Ferguson, Robert C.

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herrings; the puzzle is straightforward and success is based purely upon will and focus

and determination. The emphasis on strategy and economy (how each piece has a

completely unique and necessary function without which the puzzle is impossible) only

adds to the elegance of the solution. Furthermore, because chess puzzles are purely

intellectual, (and not based upon unraveling complex and abstract word problems for

instance,) the feeling of accomplishment you feel when you finally find the solution is

magnificent. Thus, whether you look at it from an academic viewpoint, a practical

viewpoint, or a personal one, the result is that, in the end, chess puzzles are simply

worth all the time and effort they require.

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Works Cited

Byrne, Beverly. "Scientific Proof: Chess Improves Reading Scores." Chess Coach

Newsletter6 (Spring 1993): Mephisto and Fidelity. Web. 15 May 2013.

.

Fairy Chess Piece. Wikipedia: The Free Encyclopedia. Wikipedia Foundation, Inc. 29

April 2013. Web. 22 May 2013.

Ferguson, Robert C. "Benefits of Chess."Benefits of Chess. Web. 14 May 2013.

.

Godden, Kurt. "A Tour of the Knight's Tour." Chess.com. 3 Mar. 2008. Web. 15 May

2013. .

"The History of Chess." ThinkQuest. Oracle Foundation, 22 Sept. 2010. Web. 14 May

2013. .

Jelliss, George. "Introducing Knight's Tours."Introducing Knight's Tours. 2004. Web. 15

May 2013. .

McKay, Brendan D. Knights Tours of an 8 x 8 Chessboard.Australian National

University. 1997. PDF file.

Watkins, John J.Across the Board: The Mathematics of Chessboard Problems. Princeton:

Princeton UP, 2004. Print.

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Image Bibliography

Figure 1: Bain, John. "Chess Puzzles!" ChessPuzzles.com. Web. 22 May 2013.

.

Figure 2: Weisstein, Eric W. "Knight Graph." MathWorld. Wolfram. Web. 21 May 2013.

.

Figure 3: Keen, Mark R. "The Knight's Tour." Mathematics Dissertation 'The Knight's

Tour'. 2000. Web. 21 May 2013. .

Figure 4: Knights Graph. Wikipedia: The Free Encyclopedia. Wikipedia Foundation, Inc.

22 March 2013. Web. 21 May 2013.

Figure 5: "Puzzling Graphs: Problem Modeling with Graphs." Good Math Bad Math.

Scientopia, 10 Sept. 2007. Web. 21 May 2013.

.