FIGIRED TOLRS FIGURED TOUR' A Mathemrtical Recreation by George Jelliss This cover illustration, first published n The Games and Puzzles Journal, 1996 is a figured tour composed by the author, consisting of a reenhant knight's tour of the l2xl2 boar4 showing the first few Fibonacci numbers in a convex circuit. Each number in the Fibonacci sequence is the sum of the two preceding numbers, beginning here with I and, 2. All moves are either internal or ex0ernal to the oval" i.e. there are no knight's move lines crossing the convex polygon shown by the heavier lines. FIGLIRED TOURS Definitions Most readers will probably be familiar with the chessic terminology used in this booklet, and they can jump straight in to the main text. This page is provided mainly for newcomers to the subject. A board is a playing area divided into smaller areas called cells. Most of the work hitherto done on figo.ed tours, and ttrerefore most of the examples shown in this booklet, use the orthodox square chessboard, divided ulita 64 cells ananged in 8 ranks (by convention the lines of cells across the paper in a diagram of the board) and 8 liles (lines up and down the paper), but some o<amples on smaller and larger square boards are also included, and there is no reason why non-square rectangles or other shaped boards should not be used (as in Puzzles 3 and 8). A cell on a rectangular board is named by its coordinates (r, y) indicating ttrat it is in the rttr file from the left and thelh rank from the bouom. (In chess works the r coordinate is ofteir replaced by letters a, b, c, ... in place of numbers 1, 2, 3, ....) A piece, represented by a tokeir of distinctive shape, can be placed on any cell of the board and is able to move from that cell to certain other specified cells. An tr, slJeaper is a piece that is able to move from cell (r, y) to any of the cells (r * r, y + s) or (x + s, y * r). Many of these {r, s}-leapers have acquired individuat rutmes. On the chessboard there are five {r, s}-leapers that can reach any cell from anyother, namely: {0, 1} wazir, {1,2} knight, {2,3} zebra, {1,4} giraffe, {3,4} antelope. Other single-pattern leapers mentioned in the text are: {1, 1} fers, {0, 2} dabbaba, {0, 3} threeJeaper. Double-patternleapersmentionedare {0, 1} + {1,2} emperor, {1, 1} + {1,2} prince, and {0,5} + {3, 4} fiveleaper. These can make a move of either of the two t},?es. The standard chess rook also features, this makes any move {0, n). A link is an arc, rising into the air above the boar4 and connecting the centre of one cell to the centre of another. Seen from above this is represented in a geomenical diagram by a straight line. This line may in some cases pass through the centre-poi* of an intermediate cell, whereas in fact the link does not use that cell. To avoid this misleading effect, we sometimes draw the link as a curved line on the diagram. (A move can be represented as a directed link, i.e. with an affow on it indicating the direction of travel of the piece along it.) A path (or chain) is a geometrical figure, all of one piece, made up of links (i.e. move-arcs connecting certain cells centre to centre) in such a way that no cell is linked to more than two other cells" A path is either open which means it has two ends (cells linked only to a single other cell) or closed (no ends, every cell linkd to two other cells). A wazir-path is a path in which all the links can be traversed by a wazir, similarly we can define knight-path and so on. A reentrant path is an ope,n path along which a given piece can move and in which the move from the final cell to the initial cell is also a possible move of the given piece. By inserting this link the path can be made closed" A tour is path that uses all the squares of the board. Since a tour is a path it can be an open tour or a closed tour. A tour in which all the links can be traversod by a wazir is a wazir-tour, similarly we can define a knight-tour and so on. A journey is a sequence of moves by a given piece, not entering or leaving any cell more than once (this way of stating the definition, rather than saying that it 'passes through' or 'visits' wery cell, permits the final cell to be the snme as the initial cell). Any journey determines a unique path, along which the piece travels. The converse however is not true. An open path determines two journeys, one startrng at each end. A closed path of n cells determines 2n journrys, there being n choices of starting point and two choices of direction. These distinctions are important rrhen it comss to counting tours. A symmetric path is one that looks the same when reflected or rotated. Reflective symmetry has an axis and is called direct. Symmetry without an axis is called oblique, is rotational and has a centre. If there are two a:<es of symmefy the point where they cross is a centre of symmetry. S;rmmetry on sqlrare boards is binary, quaternary or octonary i.e. the path can be split into two, four or eight congruent parts related by rotation or reflection FIGURED TOURS lntroduction The name figured tour is appropriate for any numbered tour in which certain arithmetically related numbers are arranged in a geometrical pattern, since it combines in one concept both senses of the ambiguous term 'figure', which can mean either a numerical symbol or a geometrical shape. Tours in which all the entries participate in the arithmetical properties, for example arithmo-geometric tours in which the arithmetical properties derive from the geometrical structure such as symmetry, or magic tours which require calculation of rank and file totals, are not figured tours in this sense. The problem of constructing a numbered knight's tour showing the eight square numbers L, 4,9, 16, 25, 36, 49, 64, along one rank of the board, proposed by G.E. Carpenter and solved by S. Hertzsprung rn Brentano's Chess Monthly in 188L should probably be taken as initiating the study of figured tours, though with hindsight earlier examples can be recognised as faling in this category. For example a tour by Euler (1759) showing an arithmetical progression, tours by Jaenisch (1862) with '1., L6,I7,32,33,48, 49,64 in a circuit (resulting from the joining of four 1"6-move circuits to form a tour), and a variety of examples from India dating from 1"871. which have recently come to light. But none of these precursors had much immediate impact. The idea was taken up and developed by T. R. Dawson in the pages of the Problemist Fairy Chess Supplement (PFCS) and its continuation Fairy Chess Review (FCR), which he edited from 1930 to L95L, and the term 'figured tour' was first used by him (FCR viI94a p.96) in connection with his tours showing square numbers in paths of knight moves. A number of results by Dawson first appeared in an article in the Conptes Rendus du Premier Congres International de Rdcrdation Mathdmatique (CIRM), Brussels 1935, edited by M. Kraitchik. This booklet brings together for the first time a wide range of results of this type, including in particular most of Dawson's work, to show what has been done and to provide a stimulus for further work. Detailed sources of the tours are listed on page 20. One of the reasons this subject has perhaps not aroused wider interest is that the method of presenting the tours in FCR was merely to show the relevant numbers in bold type. By including lines joining the successive numbers, or by framing shaped groups of numbers, the results are made more attractive. An alternative presentation, which we use in a few cases, is to diagram the moves, with only the significant numbers superimposed. There is considerable scope for choice of what numbers to display in a figured tour, but the sequences of main interest are those that are formed in some regular manner and have n members within the set {1,2, ..., n2}, so that a solution may be possible on any size of square board, nxn. By far the most popular sequence of this type is that of the squares: 12, 22, 32, ..., n2 (that is 1, 4, 9, L6, 25,36, 49, & on the 8x8 board). This sequence has a property of permanence in that the first z numbers remain the same on all boards greater than m. Other sequences of this type, less used because impermanent, are arithmetical progressions (ars) with common difference (co) equal to n, for example the multiples of z (8, 16,24,32, 40, 48, 56,64 on the 8x8 board), or CD = n+L the maximum-spaced progression of n members which has 1 and n2 as end terms (1, L0, 19, 28,37, 46, 55,64 on the 8xB board). Tours showing the squares are, in general, much easier to construct than those showing arithmetical progressions. This is because the number of cells between two successive stations increases, so that having fixed the shorter routes I-4-9-16-25, with 2, 4, 6 and 8 intermediate cells, the later routes 25-36-49-64, with lO, 12 and 14 intermediate cells, are subject to less constraint.