ON CHESS
XXIII - The Knight's Move - II
When the method of solving the problem of the knight's move, as stated in our last article, is thoroughly understood, the young chess student may pass on to that which constitutes the peculiar feature of Dr. Roget's method, and which confers on it that generality and comprehensiveness never before attained: viz., the power of ending as well as of beginning on any given square, provided, of course, that the two squares be of opposite colours. When the two given squares are named, the player must attentively notice to what systems they belong: whether both are in diamond systems or in squares, or one in a diamond and the other in a square : also, if both are diamonds, whether the two form parts of the same diamond system or not. The determination of these points will decide the mode of procedure. If the two squares belong to the same system, we must depart from one of the instructions given in the last article: we must not complete that system before passing to another, because one square belonging to it is to be the very last of the 64. We must therefore pass on to another system before completing the first one, and it is optional to leave as many as we please, to assist in forming links to conduct to the terminal square. Dr. Roget recommends that one or two squares of the system should be left to the last, but we incline to the opinion that it will be better to leave a greater number, - that is, after covering two or three squares of that system to which the initial square belongs, pass on to the other three systems successively, complete the 48 squares of which they consist, and then cover the remaining 13 or 14 squares of the first system. We will illustrate this by a problem. Required: to commence at the king's rook's square, and to terminate at the king's bishop's 6th square. These two squares belong to the same diamond system; consequently we must pass on to another system before completing this one. In the diagram (fig. 5) we begin at the rook's square, and cover only two squares of the diamond system to which it belongs: we then pass on to a square system, the 16 squares of which we complete; after this we traverse the 16 squares of the other diamond system, and then the 16 of the other square; finally, we cover the remaining 14 squares of the first diamond system, and end at the required position.
If the initial and terminal squares are respectively in the two diamond or the two square systems, another modification is required, arising from the circumstance that the knight cannot pass from one diamond system to the other, nor from one square system to the other. Let the initial square be in one diamond system, and the terminal square in the other. Complete the first diamond system; then one of the square systems; then traverse a portion of the second diamond system, omitting that square which is to be the terminal square, as well as some others; after this, cover the second square system; and lastly, traverse the remainder of the second diamond system, ending on the required one. By transposing the words "square" and "diamond" in this description, it will lie available for that variety of the problem which begins in one square system and ends in the other.
If the initial square be in a diamond system and the terminal in a square one, or vice versa, the solution is easier than in either of the cases before supposed because all the four systems can be completely traversed in succession, by bearing in mind that the second system traversed must not be that which contains the terminal square.
Wo have endeavoured to impress on the mind of the reader, that attention to the respective systems in which the initial and terminal squares are contained, is the point of most importance in giving a general solution to the varieties of this problem. When this is once attended to, minor difficulties are more readily surmounted. Among these are, the quarter of the board on which the terminal square is situated. Not only must the tour of the knight, in a given problem, end in a particular system, but also in a particular quarter of the board; and as the tour may generally be made from left to right or from right to left at pleasure, we must choose that direction which, while it obeys the conditions of the problem as to systems, shall terminate in that quarter which contains the terminal square. We may illustrate this by referring again to fig. 5. The terminal square is in the right hand upper quarter. After covering 2 squares of the first diamond system, and then traversing the 48 squares which constitute the other three systems, we find the knight in the left hand upper quarter, only two squares distant from the terminal square; and as we have still 14 moves to make, we manage to go into all the other three quarters of the board before arriving at that one which contains the terminal square. In every instance, if the rules which we have given are attended to, and any difficulty arises towards the end of the tour, a reconsideration of a few of the last moves will enable the player to surmount the difficulty. The moves which it is in the power of the knight to make at any given moment, varying from one to eight in number, give such interminable variety to the modes of solution, that the judgment of the player must be exercised as to the choice of the mode of proceeding in each particular instance; but it is only in the last few moves that this judgment is particularly called for, provided the prescribed rules are attended to. Of the number of ways in which the problem can be solved no estimate has yet, as far as we are aware, been made; nor do we know of any means but actual trial by which it could be determined, since the regular arithmetical law of permutation will not here apply. If the squares of the board were numbered from 1 to 64, and these numbers were noted down in the order in which the knight moved, we have very little doubt that this order might be varied in more than a million different ways; there are 64x32 = 2048 modes of varying the initial and terminal squares alone; and in each mode the intermediate moves are susceptible of variation at almost every step of the process.
Such is the result to which Dr. Roget's extremely ingenious investigation enables us to arrive. Until his method appeared, no one, we believe, was able to insure a solution to the problem when both the initial and terminal squares were prescribed; except in the limited instance of a re-entering route, where the terminal square is a knight's move distant from the initial one. By the method which we have just endeavoured to explain, the problem can be solved whether the terminal square be far removed from the initial one, or contiguous to it; the only condition being, that the squares must be of different colours.
To shew the interesting variety of which this problem is susceptible, we will here give three additional representations, each of which possesses some peculiar property capable of being committed to memory: they are partly original, and partly altered from methods already known; and the whole of them differ from Dr. Roget's mode of solution. Fig. 6 is produced by attending carefully to this one simple rule: - Keep as far from the centre of the board as passible. In obedience to this direction, the tour of course commences in one corner, no matter which, and every successive move is determined according to the distances, from the centre of the board, to those squares open to the knight; the greatest distance being always chosen. It might appear from this rule, that the terminal square ought to be still nearer to the centre of the board than it is seen to be; but it will be found that in the course of the preceding moves, the four central squares have necessarily become occupied;
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since it happens in some cases that there is only one square left open to the knight, and that one may probably be near the centre of the board. No difficulty will occur, provided we adhere strictly to the one rule laid down.
Fig. 7 is produced by adhering to the following rule: Play the knight to that square where he has least power. Supposing the board to be unoccupied except by the knight, the reader can easily satisfy himself, that the knight can command 2, 3, 4, 6, or 8 squares, according to his position: if in one corner, he commands only 2 squares; if he be on the knight's square, he commands 3 squares; if on the bishop's square, 4 squares ; and as he approaches the centre, the squares commanded are 4, 6, or 8 in number. Now the rule requires, that in every instance the square chosen for the knight's leap be that which, of all those remaining open to the knight, will give him least power. If at any move there are two open squares of equal power in this respect, either one may be chosen. In many points this solution resembles the last, since, generally speaking, the knight has "least power" when "farthest in the centre;" but a comparison of the two figures produced will show that the routes are by no means identical.
Fig. 8 (a) is possessed of a most remarkable numerical property, and belongs to a class of problems which would be found fertile in interesting combinations. In order to exhibit this property, we have in a separate diagram or table, fig. 8 (b) numbered the squares in the order in which the knight stepped on them. The tour commences on one of the central squares, which we have marked 1, and terminates on the king's bishop's third, which is therefore marked 64. Now it will be found, that if we select two squares on opposite sides of the centre, and equidistant from it, the difference of the two numbers occupying those squares will be always equal to 32. Thus, the opposite corner squares are 16 and 48, 27 and 59; and 48-16 = 59-27 = 32: the four central squares are 1 and 33, 14 and 46; and 33-1 = 46-14=32. In the same way we may select any two squares, provided the centre of the board is precisely between them, and equidistant from them, and we shall find that the smaller number subtracted from the greater will invariably leave 32.
There are other remarkable circumstances connected with this last solution. The route is a re-entering or interminable one, and the figure produced, as seen in fig. 8 (a) is one of the most symmetrical which we have yet given. The route being interminable, may be commenced on any square, and as the initial square must always be marked 1, the distribution of the numbers over the board would vary with the varying of the initial square, every square being affected alike. Now it will be found, that at whatever square the route conmences the same numerical law will hold good; there will in fact be 128 modes of varying the order of the numbers, in all of which the same figure will be produced, and the same remarkable law will be observed; because any square out of the 64 may be selected as the initial square", and from each we may begin the route either to the left or the right.
Another example of the maintenance of a particular law throughout the numbers obtained, we will here give in order to shew the reader how varied may be tho results to which he can arrive by a little ingenuity. Fig. 9 (a) is a very pleasing and symmetrical figure, produced by a route of Which the numbers afn entered in fig. 9 (A). If these numbers are examined, it will be found, that the difference of two numbers situated on opposite sides of the centre, and equidistant from it is 16, - half the amount of constant difference; in the last case. This route is not a re-entering one, and we do not think it could be made so, with a constant difference of l6.
The reader will now have had sufficient proof of the diversified solutions of which the knight's problem is susceptible. We have never heard of a chess kaleidoscope, but the instructions we have given will enable him to form one out of the numerous other modes of solution which may be left to his ingenuity to produce. Nor will the study of this subject be without its use to the chess-player; since it not only teaches the art of manoeuvring this "beautiful piece, but brings the fact into forcible notice, that the knight has less power of moving, and therefore becomes less valuable, when he approaches the corners and sides of the board.
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Chess Archeology 
