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Show WrKNIGILT'S,rTOIJR AMUSEMENT FOR THE LOVERS OF CHESS. Enormous Number of Combinations Possible In This Dlvertlnp, Exercise' A Mental Recreation That Has Few Equal. What Is termed tho "knight's tour" In chcsB, nnd which consists in moving mov-ing the knight In such a manner as to covor every square on the chessboard In sixty-four moves, counting tho. starting point, Is n much moro Interesting Inter-esting problem than Is generally supposed, sup-posed, and as a gamo or mental recreation recre-ation it has few equals. It has been asserted that somo of tho chess mastors havo memorized tho knight's tour, sd as to be ablo to start from any square on the board and finish fin-ish on any other Bquaro, but nnnlysU proves that this Is Impossible. It Is posslblo, though, to start from any black squaro on tho board and finish on any white square. Or to start from nny whlto squaro nnd finish on a, black square, and this would glvo tho enormous number of 1,024 distinct combinations without Including any of tho variations, of which it can bo demonstrated that thoro aro four or moro In every combination. Let nny ono try to work out Just ono solution solu-tion of tho knight's tour without restriction re-striction ns to beginning or ending, and ho will havo somo appreciation of tho naturo of such a mental feat. It may bo mentioned hero that the four minor variations incrcaso tho total 1G time. To form an Idea of this, lot us examine ex-amine somo of tho diagrams which show a regular pattorn In tho ccntor. Theso aro formed by moving tho knight In tho samo rclatlvo direction when entering and leaving each of tho four mlddlo squares. Now, tho knight can move In eight dlfferont directions di-rections from any of thoso squares, and as thoro aro twenty-eight corn-Diagram corn-Diagram No. 16 (Endless Tour), blnatlons of two in eight numbers, so thoro aro just that number of variations varia-tions posslblo In regard to tho knight's movement to and from them. Each ' of tho tours Is capable of not less than four minor variations, which do not ontall any alteration In tho central cen-tral arrangement of tho diagram. Taking, for instance diagram No. 1G, in which had "tho knight bcon movod from squaro 1G to squaro 32, leaving squaro 29 open, it could later havo bcon moved from squaro 47 to 29 without disturbing tho middle of tho diagram. Similar changes can bo made in tho samo diagram by transposing trans-posing nt squares 4G, 48, 17, 19, 33, 35, otc, and tho embodiment of theso four changes alono, collectively and separately, sep-arately, Increases the original 28 designs de-signs to 448, having tho same distinctive distinc-tive central pattorn; for thoro aro, first", tho original squares, 28; 4 times 28, with ono of each chango in 28 squares, 112; G times 28, with 2 of 4 changes in 28 squares, 1G8; 4 times 28, with 3 or 4 changes in 28 squares, 112; all of 4 changes In 28 squares, 28; total. 448, or 1G times 28. Although this artlclo is not intended intend-ed to glvo an oxact mathematical demonstration dem-onstration of tho subject, wo may olaborato yet a llttlo moro Just to show that only very small proportion propor-tion of tho combinations posslblo have been stated, for, though tho moves of tho knight to and from tho four middle mid-dle squares of tho board will at three dlfferont points conflict with each othor, thoro still remain flvo non-conflicting moves to or from each square, yielding ten dlfferont combinations for each of tho four squares and a total of GOO original combinations for tho four squares combined, nnd each one of theso subject to tho samo minor variations aa tho 28 already described, which would glvo a total of 10,048 distinct dis-tinct tours (9.G00 plus 448) dopendent on irregular moves to and from theso central squares alone. |