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262
THE INDIAN ANTIQUARY
[DowBER, 1924
THE KNIGHT'S TOUR AT CHESS.
By L. R. RAMACHANDRA TYER, M.A. TAERE was a discussion on the Knight's Tour at Chess, ante, vol. LII, pp. 351-364, in the course of which plates were given of an "Indian Knight's Tour by Figuren " and of a "Correct Knight's Tour," also by figures. It was pointed out that the problem to be solved is to move the knight over every square on the chess-board in 64 consecutive moves ; that is to say, if the top left-hand square is counted as No. 1, No. 64 must come within a knight's move of square No. 1. The "Correct Knight's Tour" solved the problem, but the "Indian Knight's Tour" did not, because, although it filled up every square in consecutive moves up to 64, the last "move ” was not within a knight's move of square No. 1. Therefore there were only 63 moves, as No. 1 is a station and not a move.
But the interest in that "Indian Knight's Tour" lies in the fact that the first half of the board is filled by 32 " moves," in such a way that by merely repeating the moves" in the first half in the same order in the second half the whole board becomes filled. The tour failed, however, to solve the problem, because neither did move 32 fall within a knight's move of square No. 1, nor did move 64 fall within a knight's move of square No. 33.
The "Correct Knight's Tour" is indeed correct, in so far as move 64 is within a knight's move of square No. 1, but it does not arrange that when half the board is filled, the rest of it can be filled up automatically, as in the case of the "Indian Knight's Tour."
It is, however, possible so to arrange the 10ves that when half the board is filled up, the remainder can be filled up automatically, and yet to bring move 84 within a knight's move of square No. 1.
To make my meaning clear I here repeat the plate of the "Correct Knight's Tour" and the "Indian Knight's Tour by Figures :" see Plates I and II attached. I also add thereto Plate III which I have called a "Symmetrical Knight's Tour."
It will be observed from Plate III that the moves 1 to 33 are 80 arranged that No. 33 falls in the bottom right-hand corner of the board ; i.e., exactly at the opposite corner dia. gonally of No. 1, which is at the top left-hand corner. So the second half of the moves oan be worked backwards to No. 1 in exactly the opposite direction to the first half working forwards from No. 1, and yet No. 64 falls within a knight's move of No. 1. It will therefore be seen that Plate III exhibits not only a "correct " knight's tour, but also a more perfoot tour than that previously given, as the last half of the board can be filled up automatically. The Symmetrical Knight's Tour combines in fact the advantages of the Correot Knight's Tour (Plate I) and the Indian Knight's Tour by Figures (Plate II).
The point can be made yet clearer by observing Figures A and B of Plate IV, which show the distribution of the first 33 moves in Plates I and II respectively, move 33 belonging to the second half of the board. It will be seen that in the first case the moves are distributed 16 in each half of the board divided vertically: in the latter case they are 16 in each half of tho board divided horizontally.
Plates V and VI represent the moves of the "Symme trical Knight's Tour." In Plate V, Fig. A, moves 1 to 32 are distributed, 15 in the upper half of the board and 17 in the lower half the board being divided horizontally. In Plate V, Fig. B, moves 33 to 64, these facts are reverged, and the distribution is 17 in the upper half and 15 in the lower. Similarly in Plate VI. Fig. A, moves 1 to 32 are distributed, 21 in the left half of the board and 11 in the right half : the board being divided vertically: and in Plate VI, Fig. B, the reverse ooours, the distribution being 11 moves in the left half and 21 in the right ball of the board,
