ȘTOUR HLPNULL HLPqGTOUR SUBiDODEC HLPƯKTOUR HLP+'3X HLP>4X HLPS5X5 HLPg 6X6 HLPlDODEC TOUnѳDODEC AQMqDODEC COMDGRID HQP3GRID HQP.4GRID HQP5GRID HQP p6GRID HQPBIBL HQP 3X5 COM<3X6 COM3X7 COM+3X8 COMd3X9 COM w 4X3 COM@4X4 COMJ4X5 COMG4X6 COM4X7 COM  mH4X8 COM *5X5 COM 76X6 COM) @3X5 TOU50f3X6 TOU73X7 TOU:3X8 TOU=o3X9 TOU@$4X3 TOUD8l4X4 TOUF4X5 TOUH/[4X6 TOUK+4X7 TOUO,N4X8 TOUS.5X5 TOUX]6X6 TOU\Q3X5 AQMb33X6 AQMt!3X7 AQM3X8 AQM33X9 AQMhg4X3 AQM@4X4 AQM[ 4X5 AQM24X6 AQMa4X7 AQM.p4X8 AQMM#CL5X5 AQMp -6X6 AQM'/TOUR1 HQP QTOUR2 HQP] introduction guide to usage dodecahedron - Hamilton's original problem knight's tours - on chessboards of various sizes bibliography :: (TOUR)TOUR1.HQP :: (TOUR)TOUR2.HQP :: (TOUR)DODEC.HLP :: (TOUR)KTOUR.HLP :: (TOUR)BIBL.HQP :[end] [Harold V. McIntosh, 15 July 1985] : This is a null .HLP file A:LU.COM -O TOUR.LBR 63 -A TOUR.HLP NULL.HLP TOUR.SUB DODEC.HLP KTOUR.HLP 3X.HLP 4X.HLP 5X5.HLP 6X6.HLP DODEC.TOU DODEC.AQM DODEC.COM DGRID.HQP 3GRID.HQP 4GRID.HQP 5GRID.HQP 6GRID.HQP BIBL.HQP 3X5.COM 3X6.COM 3X7.COM 3X8.COM 3X9.COM 4X3.COM 4X4.COM 4X5.COM 4X6.COM 4X7.COM 4X8.COM 5X5.COM 6X6.COM 3X5.TOU 3X6.TOU 3X7.TOU 3X8.TOU 3X9.TOU 4X3.TOU 4X4.TOU 4X5.TOU 4X6.TOU 4X7.TOU 4X8.TOU 5X5.TOU 6X6.TOU 3X5.AQM 3X6.AQM 3X7.AQM 3X8.AQM 3X9.AQM 4X3.AQM 4X4.AQM 4X5.AQM 4X6.AQM 4X7.AQM 4X8.AQM 5X5.AQM 6X6.AQM TOUR1.HQP TOUR2.HQP -L -X INTRODUCTION DODEC.TOU - for a list of connections in the network DODEC.ASM - for the assembly language program DODEC.COM - to generate all of Hamilton's tours DGRID - for a map of the network :Introduction. The Irish mathematician and Physicist, Sir William Rowland Hamilton, proposed a puzzle based on the regular dodecahedron. His idea was to take twenty cities spaced around the globe, corresponding to the twenty vertices of a dodecahedron. Supposing that there were three routes leading to of from each city to one of its neighbors, the idea was to work out all the possible world tours. The three routes corresponded to the three edges meeting at each vertex of the dodecahedron. Because of his example, problems requiring the determination of all the routes through a network or maze, which visit each node once and only once, are referred to as Hamiltonian circuit problems. Although the original puzzle contemplated closed circuits, it is just as interesting to accept open tours, provided only that all the nodes have been visited. The puzzle may have been somewhat artificial, but the dodecahedron is the only regular solid yielding a substantial challenge. The tetrahedron, cube, and octahedron are rather simple, with only short and easily discovered circuits. The icosahedron produces a much larger number of solutions, so that much work is required to find and classify them all. - Because of the symmetry of the dodecahedron, only one starting point need be considered, and only one of the three edges leading out from that node need be tried. Because of reflective symmetry, only the right hand fork needs to be taken after arriving at the second node. To study the solutions first print DGRID, then execute DODEC.COM. Use ^S to stop the display and copy one of the solutions on the map that you have printed. How many round trips are there? How many tours are there altogether? How many tours coincide except for side trips which can be taken in one direction or the other? Can you make a tour from the North Pole to the South pole? :: (TOUR) DODEC.TOU :: (TOUR) DODEC.AQM :: P=(TOUR)DODEC.COM (TOUR)NULL.HLP :: (TOUR) DGRID.HQP :[end] [Harold V. McIntosh, 15 July 1985] introduction 3XN tours 4XN tours 5X5 tours 6X6 tours :Introduction. A maze problem of long standing, although it may not be thought of as such, is to enumerate all of the knight's tours on a chessboard. The rather capricious move of the knight - two squares in one direction, then one at right angles, irrespective of whether the intervening squares are occupied or not - makes its movements somewhat more obscure than those of the other pieces. The first question to be solved is, whether there are any knight's tours at all? That is, we wonder whether it is indeed possible for the knight to jump around the board, touching each square once and only once. Maybe, even, he can return to his starting square, so that we can try for round trips as well as simple tours. Chess players have probably found a tour at one time or another; the next question is one of counting - how many tours are there altogether? Either to help in solving the first two problems, or as a subsequent generalization, all kinds of different boards may be examined. Small ones are that much easier to analyze, but they also emphasize details of the rather particular nature of the knight's move. - Books on mathematical recreations generaly devote a section to knight's tours. Due to their popular nature, they often content themselves with citing a few known results, rather than exhibiting a detailed mathe- matical treatment of the subject. Many famous mathematicians have been attracted to the problem; two centuries ago Leonhard Euler studed the matter in some detail. Knight's moves on narrow strips are sufficiently restricted that some definite mathematical results can be obtained. We may think of 2xN, 3xN, or 4xN strips, all of which are essentially one dimensional. Square boards of increasing dimension, such as 5x5, 6x6, 7x7, and finally 8x8 can be studied. Whether square or rectangular, once the board has more than 30 squares or so, the number of tours is extraordinarily large, and theorems about random numbers begin to become useful. This HELP file contains programs for two strips and two square boards, from which quite a bit of information may be deduced. :: (TOUR)3X.HLP :: (TOUR)4X.HLP :: (TOUR)5X5.HLP :: (TOUR)6X6.HLP :[end] [Harold V. McIntosh, 15 July 1985] introduction 3x4.TOU 3x4.ASM 3x4.COM 3X5.TOU 3X5.ASM 3X5.COM 3X6.TOU 3X6.ASM 3X6.COM 3X7.TOU 3X7.ASM 3X7.COM 3X8.TOU 3X8.ASM 3X8.COM 3X9.TOU 3X9.ASM 3X9.COM 3GRID :Introduction. Insight into a maze problem frequently arises from grouping the nodes into subsets in some clever fashion. The requirement is that any transitions between subsets be consistent with transitions between the nodes themselves. For the knight's tour, the classification of squares of the chessboard into red and black is fundamental, because the knight always alternates between them. For a board with an odd number of squares - as when the edges are of odd length - one color must predominate. Thus all tours must begin and end on this color, which precludes any round trip on an odd order board. Sometimes a finer partition still can be found, as on a 3x7 board. +-------+-------+-------+-------+-------+-------+-------+ | E | D | C | B | C | D | E | +-------+-------+-------+-------+-------+-------+-------+ | D | A | D | E | D | A | D | +-------+-------+-------+-------+-------+-------+-------+ | E | D | C | B | C | D | E | +-------+-------+-------+-------+-------+-------+-------+ - The transitions between the lettered subsets are A <---> B <---> C <---> D <---> E 2 2 4 8 5 where the sizes of the sets are indicated just below their letters. If all the transitions between A and B are not used up before (or after) all the others, it will not be possible to complete them later. Also, unless the tour actually begins on A, it will not be possible to pick up both A's later. If we get a 3x5 board by dropping the first and last column, the same argument applies, but since A must lie on a minority color, the tour just can't be possible. Similar reasoning eliminates tours on a 3x6 board. These sets were made up by an astute choice of initial squares, thence adding new points according to the least number of moves by which they can first be reached. :: (TOUR)4X3.TOU :: (TOUR)4X3.AQM :: P=(TOUR)4X3.COM (TOUR)NULL.HLP :: (TOUR)3X5.TOU :: (TOUR)3X5.AQM :: P=(TOUR)3X5.COM (TOUR)NULL.HLP :: (TOUR)3X6.TOU :: (TOUR)3X6.AQM :: P=(TOUR)3X6.COM (TOUR)NULL.HLP :: (TOUR)3X7.TOU :: (TOUR)3X7.AQM :: P=(TOUR)3X7.COM (TOUR)NULL.HLP :: (TOUR)3X8.TOU :: (TOUR)3X8.AQM :: P=(TOUR)3X8.COM (TOUR)NULL.HLP :: (TOUR)3X9.TOU :: (TOUR)3X9.AQM :: P=(TOUR)3X9.COM (TOUR)NULL.HLP :: (TOUR)3GRID.HQP :[end] [Harold V. McIntosh, 15 July 1985] introduction 4X3.TOU 4X3.ASM 4X3.COM 4X5.TOU 4X5.ASM 4X5.COM 4X6.TOU 4X6.ASM 4X6.COM 4X7.TOU 4X7.ASM 4X7.COM 4X8.TOU 4X8.ASM - won't fit 4X8.COM 4GRID :Introduction. We have a collection of knight's tours for a 4xN strip, with N running between 3 and 8. According to Kraitchik, the key to understanding the 4xN strip is to group the squares into four classes - outer red, outer black, inner red, and inner black. A knight ALWAYS alternates between red and black squares. For the 4xN strip, marked as shown, +-------+-------+-------+-------+-------+-------+-------+-------+ | A | B | A | B | A | B | A | B | +-------+-------+-------+-------+-------+-------+-------+-------+ | a | b | a | b | a | b | a | b | +-------+-------+-------+-------+-------+-------+-------+-------+ | b | a | b | a | b | a | b | a | +-------+-------+-------+-------+-------+-------+-------+-------+ | B | A | B | A | B | A | B | A | +-------+-------+-------+-------+-------+-------+-------+-------+ a more specialized transition diagram holds: A <---> a <---> b <---> B - The rule is that ALL the A's must be visited before ANY B's (or conversely). Moreover, only A's and B's can be terminal points, otherwise the gap between a's and b's could never be bridged. Since there are equal numbers of A's, a's, B's and b's, any premature jumps between a's and b's will use up one of the a's that has to be paired with an A. Kraitchik also reports a symmetry, beyond the obvious rotational and reflective symmetry of a rectangle - small and capital letters may be interchanged. That is, the top two rows may be exchanged, simultaneously the bottom two rows, and you will get another tour. The number of tours for various cases should be: N = 3 4 5 6 7 8 tours = 8 0 82 744 6378 31088 :: (TOUR)4X3.TOU :: (TOUR)4X3.AQM :: P=(TOUR)4X3.COM (TOUR)NULL.HLP :: (TOUR)4X5.TOU :: (TOUR)4X5.AQM :: P=(TOUR)4X5.COM (TOUR)NULL.HLP :: (TOUR)4X6.TOU :: (TOUR)4X6.AQM :: P=(TOUR)4X6.COM (TOUR)NULL.HLP :: (TOUR)4X7.TOU :: (TOUR)4X7.AQM :: P=(TOUR)4X7.COM (TOUR)NULL.HLP :: (TOUR)4X8.TOU : (TOUR)4X8.AQM - too big to fit in memory :: P=(TOUR)4X8.COM (TOUR)NULL.HLP :: (TOUR)4GRID.HQP :[end] [Harold V. McIntosh, 15 July 1985] introduction 5x5.TOU 5x5.ASM 5x5.COM 5GRID :Introduction. According to Kraitchik, there are 1,728 tours on a 5x5 chessboard, of which 8 are symmetric. We only have to calculate tours starting in one octant because all the others are related to these by either rotation or reflection. However, because those arising from a single octant may still be symmetric, or may begin and end in the same octant, counting should be carried out with some discretion. :: (TOUR)5X5.TOU :: (TOUR)5X5.AQM :: P=(TOUR)5X5.COM (TOUR)NULL.HLP :: (TOUR)5GRID.HQP :[end] [Harold V. McIntosh, 15 July 1985] introduction 6X6.TOU 6X6.ASM - file too big to display 6X6.COM 6GRID :introduction :: (TOUR)6X6.TOU : (TOUR)6X6.AQM - File too big to display. :: P=(TOUR)6X6.COM (TOUR)NULL.HLP :: (TOUR)6GRID.HQP :[end] [Harold V. 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