ON CHESS
XXII - The Knight's Move - I
While studying the various powers of the pieces at chess, we cannot fail to be struck with the remarkable move of the knight: we have made it probable that the move of this piece originated in a compound of the shortest moves of the bishop and rook; but in modern chess this piece is the only one which is allowed to move over the heads of other pieces. The peculiar power which this privilege gives to the knight in actual play, it is not our purpose here to discuss: another interesting question will occupy our attention. A little consideration show that the king, provided no other piece were on the board, could pass in succession to every one of the sixty-four squares, either with or without going twice over the same square; the queen could do the same, and so likewise could the rook. But the pawn, as it can only move straight forwards (except in capturing, and even then it moves obliquely forwards), cannot traverse the sixty-four squares; nor can the bishop do so, for one consequence of his diagonal move is to confine him to squares of one colour: consequently, he can traverse only thirty-two squares. The knight is yet remaining, and a question arises, - Can the knight traverse the sixty-four squares without stepping on any square twice? The solution of this question is one of the most remarkable circumstances in the history of chess; for as it was soon found that the problem could not be solved by mere inspection, the difficulty attending it drew the attention of ingenious persons towards the subject. Difficulties act upon scientific and ingenious minds rather as incentives than as discouragements; and this problem of the knight's move attracted the notice of firstrate mathematicians, who might not otherwise, perhaps, have paid any attention to chess and its associations. Among the distinguished men who have endeavoured to solve this problem are Euler, Bernouilli, Mairan, Demoivre, Montmort, Willis, and Dr. Roget; and we propose in the present chapter shortly to consider the results at which they arrived.
Most of the solutions of the problem (for we may here state at once that it can be solved,) have been arrived at by repeated trials, without proceeding in accordance with any particular law; and, we doubt not, that most of our readers could, with a little patience and ingenuity, carry the knight over the sixty-four squares, after many trials. But the object of such a man as Euler, whose profound mathematical talents led him to seek for principles in every department of study, was to elicit some general law by which the required object might be attained. He was successful in tracing the outline of a rule or law by which this might be accomplished; but the practical application of it was so difficult, that we doubt whether any one but himself has ever adopted it. The thorough mastery of the subject can only be attained when we are able to solve the problem in all its varieties, that is, to begin the circuit of the knight at any given square, and to end at any other given square.
In order to trace the power of the knight step by step, an anonymous writer, about twenty years ago, gave representations of imaginary chess-boards, rectangular, but containing a smaller number of squares than a real board; and he was able to demonstrate, that if the board contained 12, 20, 21, 24, 25, 28, 30, 32, 35, 36, 40, 42, 48, 49, or 56 squares, the knight could be carried over the whole of them, without going twice on the same square. These moves of the knight may be represented either by numerals, or by lines drawn on a diagram: the latter 13 the more perspicuous and pleasing of the two and we will here give representations of the modes of proceeding in a few of these cases. Let us suppose there are three boards, containing respectively 5x5, 6x6, and 7x7 squares, the knight can be carried over them in the following manner: -
The angles represent the various positions of tie knight; and the lines, his paths from one square to another.  Beginning with fig. 1 (a), we see that if the tour commences at the left hand bottom corner, all the twenty-five squares in succession can be traversed without any one being covered twice; and the route terminates at the central square. In fig. 1 (b) the tour commences at the right-hand bottom corner square, and, after extending over the thirty-six squares in succession, ends at the square next above the initial square. In fig. 1 (c) the route is over all the forty-nine squares, and the terminal square is it a considerable distance from the initial one.
These examples show that the knight may make the tour of a chess-board containing a smaller number of squares than the regular board; and there is little doubt that it might also be done on a board of more than sixty- four squares* (* Ciccolini has solved the problem of the knight's move over a board of one hundred squares, as well as over a circular board of sixty-four square). These imaginary boards have helped to devise systems whereby the problem can be solved on a real board.
We will now give three diagrams, representing three modes of solving the problem on a regular chess-board; and the reader would gain a clear idea of the subject by performing the same operation with a knight: he will do well to mark each square with a counter, as the knight steps on it, in order not to go twice on the same square. In the first diagram we shall commence at one corner and terminate at another: in the second, we shall cover all the thirty-two squares of one half of the board, before proceeding to the other half: in the third we shall give a re-entering route, that is, one in which the last square is situated at exactly a knight's move from the first square, so that the tour may be re-entered on, and performed in precisely the same way any number of times.
In fig. 2 (a,) the regular board of sixty-four squares is traversed by the knight, beginning at one corner, and ending at another; this, it will be seen, forms a figure having some degree of symmetry, but less so than one or two which we shall hereafter give.
In fig. 2 (b,) the squares are separated into two portions, one of which is traversed before the knight crosses over to the other.
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Fig. 2 (c,) possesses this distinguishing property, that we can commence the tour on any square: in drawing the diagram, we commenced at the right-hand bottom corner, and ended at the knight's third square; but any other initial square might have been selected, because the route is an interminable one, re-entering into itself.
Many other ingenious modes have been devised, some of which we shall notice hereafter; but no satisfactory attempt to give a general solution to the problem had been made public, until the month of April, 1840, when Dr. Roget communicated a short but admirable paper to the Philosophical Magazine, unfolding a method by which the problem could be solved in any form, that is, by beginning at any given square, and terminating at any other given square of the opposite colour*. We will now attempt to explain this ingenious method. (*Since the knight at each move, goes to a square of a different colour from that which he before occupied, all the odd squares are of the same colour as the initial square, and all the even squares must be of the opposite colour; consequently the sixty-fourth square, which is the terminal one, must always be of the opposite colour to the initial one.)
In the first place, the reader must conceive the board to be divided into four quarters, of sixteen squares each, by two lines passing through the middle at right angles to each other, and parallel to the edges of the board. Then selecting any quarter, we shall find that the sixteen squares may be divided into four systems, each of which consists of four regular knight's moves. These systems are shaped, two as perfect squares, and two similar to the rhombus, lozenge, or diamon'd (in future we shall use the last of the three names).  Thus in fig. 3 the sixteen squares, constituting one quarter, are divided into four systems, represented by these four kinds of lines,  : forming two squares and two diamonds; and it will be seen that the four sides forming each of these figures, are regular knight's moves.
In the next place, it will be found, that, after passing over the four squares of one system in one quarter of the board, we can pass to the same system in an adjoining quarter; and, after traversing that system, can pass on to another quarter, and so on; thus, in sixteen moves, we can traverse the sixteen squares forming one system of the whole board. We will demonstrate this as to two of the systems, and the reader will then readily admit its truth as to the other two. In fig. 4 (a,) we traverse all the sixteen squares of the system -------; and in fig. 4 (b,) all those of the system ....... The diamonds in the former case, and the squares in the latter, appear to be incomplete, because only three out of the four sides are represented; but this necessarily results from the conditions of the problem, for we must not go twice on the same square, which we inevitably should do if we drew the four sides of each figure: the knight, however, steps on the squares representing the angles of each figure, and this is sufficient to make our description correct.
Now the question which arises, is this: - Can the knight after having traversed the sixteen squares of one system, pass on to another system? He can do so under certain conditions: he can pass from a square to a diamond system, or from a diamond to a square system; but not from a diamond to a diamond, or from a square to a square. Moreover, the sixteenth, or last square of each system ought to be as near the centre of the board as possible, since, if it be at or near a corner, the passage to another system may be difficult, or even impossible. If we examine fig. 4 (a,) we shall see that, beginning at the corner square, the terminal one of that system is such as to allow the knight to step on to either of the square systems, there being a choice of four moves, of which two belong to each of the square systems: similarly, from the terminal square in fig. 4 (b,) we can select four squares to move to, of which two belong to each of the diamond systems.
If the necessary precautions be attended to, it will now be evident that the problem may be solved by the method under consideration. Let the initial square, for example, be in one corner: it will then belong to a diamond system. After traversing the sixteen squares of that system, the knight passes to a square system, which is succeeded by the other diamond, and this by the other square, when the tour terminates. A little practice will give the necessary facility, provided the player attends to these two points: - 1st, to complete the sixteen squares of one system before he passes to another: 2nd, to terminate each system rather towards the centre o the board than towards one corner. Generally speaking, he may pass round either to the right or to the left ad libitum, and may choose any one of the sixty-four squares he pleases, as the initial square.
In our next article we will apply this method to several remarkable forms of the problem under consideration.
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Chess Archeology 
