Show THE KNIGHTS TOUR AMUSEMENT FOR THE LOVERS OF CHESS enormous number of combinations poss ble in this diverting exercise A mental recreation that has few equals what is termed the knight s tour in chess and which consists in amov ing the knight in such a manner as to cover every square on the chessboard in sixty tour moves counting the starting point is a much more inter esting problem than Is generally sup posed and as a game or menial recie atlon it has few equals it has been asserted that some of the chess masters have memorized the knight s tour so as to be able to start from any square on the board and fin ish on any other square but analysis proves that this is impossible it is possible though to start from any black square on the board and finish on any white square or to start fromey white square and finish on a square and this would give the enormous number of 1 distinct combinations without including any of the variations of which it can be demonstrated that there are tour or more in aery combination let any one try to work out just one tion of the knights tour without re as to beginning or ending and he will have some appreciation of the nature of such a mental feat it may be mentioned here that the four minor variations increase the total 16 times to form an idea of this let us ex amine some of the diagrams which sho v a reg ilar pattern in the center these are formed by moving the knight in the same relative direction when entering and leaving each of the tour middle squares now the knight can move in eight different di erections ions from any of those squares and as there are twenty eight corn diagram no 16 endless tour bi nations of two in eight numbers so there are lust that number of varia alons possible in regard to the knight a movement to and from them each of the tours is capable of not less than four minor variations which do not entail any alteration in the cen aral arrangement of the diagram taking for instance diagram no 16 in which had the knight been moved from square 15 to square 32 leaving square 29 open it could later have been moved from square 47 to 29 without disturbing the middle of the diagram similar changes can be made in the same diagram by trans posing at squares ie 48 17 19 33 35 etc and the embodiment of these four changes alone collectively and sep aracely increases the original 28 designs to having the same distine alve central pattern tor there are first the original squares 28 4 times 28 with one of each change in 28 squares 6 times 28 with 2 of 4 changes in 28 squares 4 times 28 with 3 or 4 changes in 28 squares all of 4 changes in 28 squares 28 total or 16 times 28 although this article is not intend ed to give an exact mathematical dem on of the subject we may elaborate yet a little more just to show that only a very small proper tion of the combinations possible have been stated tor though the moves of the knight to and from the four mid die squares of the board will at three different points conflict with each other there still remain five flicking moves to or from each square yielding ten different combinations tor each of the tour squares and a total of original combinations for the four salares combined and each one of these subject to the same minor variations as the 28 already described which would give a total of 10 dis tours 9 plus dependent on irregular moves to and from these central squares alone |