EULER'S METHOD.
The problem respecting the placing the knight on any given square, and moving him from that square to any house on the board, has not been thought unworthy the attention of the first mathematicians. Euler, Ozanam, De Montmart, De Moivre, De Majron, and others, have all given methods by which this feat might be accomplished. It was reserved, however, for the present century to lay this down on a general plan; and the only English writer who has noticed this is Mr. George Walker, in his Treatise on Chess. The plan is this: Let the knight be placed on any square, and move him from square to square, on the principle of always playing him to that point, from which, in actual play, he would command the fewest other squares; observing, that in reckoning the squares commanded by him you must omit such as he has already covered. If, too, there are two squares, on both of which his powers would be equal, you may move him to either. Try this on the board, with some counters or wafers, placing one on every square; and, when you clearly understand it, you may astonish your friends by inviting them to station the knight on any square they like, and engaging to play him, from that square, over the remaining sixty-three in sixty-three moves. When the automaton Chessplayer was last exhibited in England, this was made part of the wonders he accomplished, though as the above plan was not then known here, he could not adopt it, but used something like the method laid down by Euler, and which we subjoin.
Our young Chess-players must remember that it does not matter on which square the knight is placed at starting; as, by acquiring the plan by heart, which is soon done, he can play him over all the squares from any given point, his last square being at the distance of a knight's move from his first. It is obvious that this route may be varied many ways, and we have often amused ourselves by trying to work it on a slate.
ANOTHER METHOD.
The problem of the knight's covering successively each square of the board, has, in all ages, attracted the attention of the first mathematicians; it is only lately, however, that this very ingenious system for performing the feat without seeing the board, has been invented by an Edinburgh gentleman. We well recollect the surprise occasioned among chess-amateurs when it was first performed; indeed it was generally considered a greater mental effort than that of playing three games of chess at the same time, without seeing the board.
The general rule for moving the Knight upon all the squares of the board, is to commence by moving him to that square which commands the fewest points of attack, and by continuing this principle he will occupy all the squares in rotation, observing, that if on any two or more squares his power would be equal, he may be placed indifferently on either of such squares. Thus we see, that there are different routes which the Knight-errant may take in his progress over all the board; still, whichever of these routes for covering the sixty-four squares may be adapted, each move forms, if we may so express ourselves, a link in an endless chain, so that whatever square we start from, by taking one known route, we are sure to arrive at a square, the last link of the chain, a Knight's move distant from the square of our departure. Consequently, if any person could commit to memory the consecutive moves of any one route over the board, he would be able to start from any one square in that route, in the same manner that any of us, if required to mention the numerals up to sixty-four, could as easily start at thirty and end at twenty-nine, as if we started at one and ended at sixty-four.
These considerations greatly reduce the apparent impossibility of performing the feat; but the reader will exclaim, "What an immense undertaking it would be, to commit to memory the moves forming a Knight's route over the sixty-four squares!" and we reply, "Certainly it would be, if we used the language of Chess to designate the squares;" and herein lies the beauty of the invention. A set of names, whose application can be understood at a glance, are invented for the squares, and the performer of the feat, having learned a route of the Knight, expressed by these invented names, thinks in the new language which he directs the moves in the terms of chess—just as many of us think in English, when we are writing or speaking French.
The diagram given above represents the chess-board; the distinction of white and black squares is not necessary for our purpose. The files, commencing from the right hand are distinguished by the consonants in alphabetical succession (C and J are, for obvious reasons, omitted.) Thus, the King's rook's file is known as B, the King's Knight's as D, the King's Bishops as F, the King's as G, the Queen's as H, the Queen's Bishop's as K, the Queen's Knight's as L, and the Queen's Rook's as M. This is all that has to be learned, in this system of Chess notation; for the lines of squares tell their own numbers—one being un, two oo, three ee, four or, six ix, seven en, eight et—being, in fact, the terminal sounds of the first eight numerals. Bun being B one, or King's Rook's square; Gix, G six, or King's sixth square. We consider it quite unnecessary to say another word in explanation of this system; its ingenious simplicity causes it to be understood and learned at a glance. All that is required now is, to select a Knight's route over all the squares of the board, and commit it to memory, not in the complicated terms of Chess, but in these simple equivalents. Suppose we start from the Queen's Knights seventh square, len, the route will be as follows:
Len het fen bet. Dix bor doo gun
Koo mun lee kun. Moo kee goo dun.
Bee div ben fet. Hen let mix lor.
Hee kiv gor hix. Liv men ket gen.
Kix giv fee hor. Gix for hiv gee.
Fiv den biv dee. Bun foo hun loo.
Mor lix met ken. Get fix det bix.
Dor boo fun hoo. Lun mee kor miv