FIGURED TOUR' a Mathemrtical Recreation

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 ochess 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:

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. With the arithmetical progressions however the number of intermediate cells is fixed throughout.

The tours are arranged in the following sequence: (1) Knight's tours showing the square numbers in various formations, sections C, D and E, (2) Knight's tours showing other numbers in formation, sections A and B, and (3) Figured tours by other pieces, section F. Tours are numbered separately within each lettered section. Some Puzzles are given as exercises. Most of these are reasonably easy, so please try them for yourself. Solutions are given at the end (page 2I). FIGURED TOURS

The Squares in Wazir Paths

The classic figured tour problem, to arrange the square numbers l, 4,9, 16, ..., 64 in a row in a knight's tour of the chessboard, was proposed by G. E. Carpenter (Brentano's Chess Monthly vol.l no.1 vL88L p.36), and solved by S. Hertzsprung (vol.l no.5 ix1881 pp.2a8-9). The Hertzsprung solutions and an alternative set by S. H. Hall are shown below. The equivalent problem on the smallest possible board, the 6x6, does not seem to have been considered until L00 years later when I found the surprising result that if the squares are to appear in order of magnitude the solution for this case is unique (C1). Other solutions are possible with the squares in different sequence on the first rank (CZ-a; note alternative routes in C3 and C4).

C1 Jelliss C2 Jelliss C3 Jelliss C4 Jelliss IZ 19 22 33 28 3r 21 6 13302334 18 11 20 27 32 29 5 2 7 113521 r0t726 3 8 15 1 4 9 L62s36

C5 Hertzsprung C6 Hertzsprung C7 Hertzsprung C8 Hertzsprung 1 4 9 L62s364961 1015 23748172635 3 r061521712 6350 6 57322952553427 8 1.524 3 50172637 1 4 9 162s364964 6053 2Lt6251 1813 3 30 5 5633285151 5 ?, 7 1035386348 u 1138 3 18473427 1 4 9 t62536496,1 58 7 2 3150532635 14 11 31 23 16 5l 18 27 5 8 t32433286350 8 59543548271419 1 4 9 162s364964 33 6 135239224762 1239 6 L962514629 5 34 7 26t5243728 8 59 20 11, 48 63 24 37 12 53 12 45 58 61 28 19 7 20115623326152 58 55 42 47 41 29 20 23 13 10 15 62 17 10 47 44 43 32 55 40 21 30 57 60 40 s7 22 43 54 5s 30 45 33 6 571031224538 60 19 t2 21 42 15 38 23 54 11 41 31 56 59 20 29 21 42 55 58 31 44 53 60 56 1L 32 13 46 39 30 2r 11 14 6L 18 39 22 43 46

C9 Flalt 1937 C10 Hall 1937 Cll Hall 1937 CIz Hal.l L937 51 21 56 41 60 45 30 43 59 41 41 54 19 30 33 52 56 33 44 21 46 39 30 19 s7 30 13 32 s9 28 6r s2 57 40 53 20 31 42 61 46 12 55 60 45 40 53 20 3r 43 22 57 32 rI 20 17 38 t4 11 58 29 54 51 34 27 22 55 32 59 52 19 44 29 7 s8 43 18 2932 51 31 58 55 31 45 40 31 18 29 3 5631 1233605362 39 58 23 12 33 28 47 62 5613 6 614639282r 2342 5 1035123718 1015 2 5550632635 14 11 38 7 18 51 3127 5 8 57r2r7223550 51592441 6 1728 13 1 4 9 t62s364964 3 8 132437 6 6348 t4 11 2 2362173827 1 4 9 t62s364964 8 17 6 1L20231637 10 15 2 5 5017 2635 1 4 9 162s364961 6053 Z 7 625tr427 5121924394421 18 1 4 9 162s364964 101524348632637 3 8 61521526 6350 18 7 404322473845

A rank or file of squares is just a particular case of a wazir path. Examples of the squares in other wazir paths are shown below: C13-15 show the three possible shapes of closed wazir paths in open knight tours. C16 shows the only symmetric open wazir path possible in a closed knight tour.

C13 Jelliss 1985 C14 Jelliss 1991 c15 Jelliss 1995 c16 Jelliss 1992 27 6 1130 43 52 39 32 62 59 42 53 38 57 44 17 21 60 296 23 62 3t 34 20 59 22 39 18 43 3CI 4L 8152851 40 31 14 53 41 52 61 58 43 46 37 56 28 57 22 6r 30 33 48 63 23 38 19 60 3 4027 11 526742 29 21 33 38 60 63 50 39 51 23 18 45 59 20 724 5 23532 5821 217 26 29 42 3l t4 50 37 51 45 sl +o@ss 24 56 27 583 47 37 21 61 10 45 28 T7 55 46 23 34 z rrl+ffizslzz 35 19 38 158 50 62 57 325 10 20 35 56 59 13 10 15 322t 8 t7 26 14 55 26 37 16 43 51 54 11 6 15 46 63 18 3 t2 61 58 47 22 30 3 12 5 28 1931 7 3918 53 12 41 41 51 10 56 63 52 35 48 t3 833 2 11 6219 18 21 60 57 11 11292033 6 27 18 54 13 10 17 52 rr 12 45 53 s0 55127 31 17 L1 FIGURED TOURS

Theorem. A 6x6 knight's tour with squares along the third rank is imnossihle. Pnoor: From b3 any tour must pass through the corner b3-al-cZ. The route must then go either d3-e1-f3, which would require two squares, on d3 and f3, differing by two, which is impossible, or e1-d3 or eL-f3, which require two squares differing by four, also impossible. It is also possible to prove that the squares cannot lie along the second rank, but my proof, which requires consideration of 24 cases, is too tedious to give here.

The Squares in Symmetric Knight's Paths

The title DAwSoNIAN roun is fittingly applied to any knight's tour in which the square numbers form a chain of knight's moves, since they were originated by T. R. Dawson in PFCS in February 1932 and he published examples in almost every issue of PFCSIFCR until June 1,948, where he wrote: "These notes complete a 16-year piece of work - nothing like sticking to it!" In this series Dawson created a complete collection of 100 tours showing the squares arranged in all possible symmetrical closed paths. There are 106 such symmetric 'octagons' in all, but three cases require a 9-rank board and three others will fit onto the 8x8 board but cannot be incorporated into a tour:

It may be noted that the second three are MoDAL TRANSFoRMS of the first three. Roughly speaking a modal transform is a sort of 45" rotation. More precisely, two diagrams are modally related if the lateral features of one correspond to diagonal features of the other, and vice-versa; thus straight paths and right angles are preserved, but acute and obtuse angles alter from lateral type (where the bisector is parallel to a board edge) to diagonal type (where the bisector is diagonal) or the other way around. This terminology of 'modal transformation' is Dawson's own: in Caissa's Wld Roses in Clusters (1937) he applied it to the transformation of chess problem themes. Dawson's 100 tours follow, arranged according to the type of symmetry in the octagon, and numbered Dl to D100. The number in brackets is the problem number in PFCSIFCR. Fuller bibliographical details are given at the end of the booklet. Just 7 octagons, D'1,-2, 5-6, 7A-72 arc modal transforms of themselves and can be described as 'self-modal'; 68 octagons, D3-4, 7-48, 73-96, form modally related pairs; the 25 remaining cases, D49-69, 97-IO0 are 'non-modal'. The tours are arranged so that the 'modal twins' are shown in adjacent diagrams. To begin with, there are five, D1-5, that have octonary symmetry (four with lateral and diagonal axes and one with skew axes. Then there are five, D6-L0, with quaternary symmetry (two with lateral, two with diagonal and one with skew axes). The rest have binary symmetry, either direct, D11-69 (19 with lateral axis, 39 with diagonal axis, and L with skew axis), or oblique, D70-100 (1"1 with the centre of symmetry at the centre of a cell, 16 centred on a corner, and 4 centred at the mid-edge of a cell). Octagons 45-52 are 'Sulian' (after as-Suli c.an900), characterised by having no cells on the axis of symmetry, while 97-100 are 'Bergholtian' (after Ernest Bergholt 1918), characterised by two moves crossing at the centre. Suli and Bergholt were the first to construct tours of these two types. The tours are generally arranged so that non-intersecting octagons are grouped together. There are 40 cases: D2-8, Il-28, 49-53, 69-71., 73-89. There are 4 octagons that contain three-move lines: D6, 49, 53, 54. There are two-move lines in 27 octagons. Another relationship that can often be noticed between pairs of octagons is that of 'folding out' or 'folding in' of a two-move segment. 6 FIGURED TOURS

Dawsonian Tours Octonary symmetry (1-5) Dl (377) D2 (608) D3 (4e2) D1 (4e3) 6235017 3815 647 r0 2e 12 >4.-{ 6 43 320 5 5263543128 622t 2 5364 115031 51 1861 2 5183914 13 56 gt3o 7 44-1. 32 6 51 2 1932296255 3 s4 63 z*\z :s rz 21 63 \#L16 37 16 7 zB t/ro 3 s4 s o\t 21 4.-7 50 53-&l 27 30 zo 6L sz-l 10 4ltgo 33 re s22%W 8 13 10 571653839233M 8 +\h-r@n s661 ss +@60 23 z+ $te 26 23 64' 9-36 41 58 45 z+ z\o 1s sz 63 ou/t 1722 t\9 1649603740 42 le s6-\ 40 7{48 2e 53 20 29 32 59 41 35 12 17 58 25-38 21 40j9 34 ro 4s192s-(\e 34 s7 s 8 se 2\s/28 37 14 30 27 22 55 10 33 42 57 26 23 oo E: G4 62 47 zt rcAt rz sf36 41 38 1843 6 5716392647 21 54 31 28 43 56 lr 34 59 18 37 22 61 48 35 50 44 tr 24 15 42 13 58 35 7 58 17 4427 16 15 38

Quaternary symmetry (6-10) Ds (32e) D6 (4so) D7 (660) D8 (r67s) 8 55 6 33 1015 4431 14 2L 54 7 L(1e 56 27 52 55 48 61 22 5t 20 37 30 45 6 1.-56 47 58 63 5 60 9.*56 43 32 11 11 s3 24 ts zo&s )-p 18 47 62 53 56-49-38 23 58 s 8 zg/+ehY4 ss 48 l\- // /\ 7 30 4s 22 13 6 8 17/28 s7 s4 sr oo hg6 rs ++ zy.t{ 7 z s1{2 se s4 /34 sel6{ 7s s6\l 61 17 42 57 1C25 12 35 52 23n2 29 4 41 10 63 46 4N0 3s l9/se 24 43 32 \r s4 t/4 {28 24e 3 40253417 +z\33 10 3 36'53 12 2r rs/s3 s8 3s 24 2e/10 47 so +t s(I 5 +oA1 s8 42 5 14 t\1 1.-62 3 50 41 36 23 26 37 34/s1 48 r'ro6142 13 2 1s/ 8 ls 1o\1 28 zz )q tr/t zz 37 so sz rle+ J7 zB/51 48 3e +o +dq zo/++ sz z se 613 34 4t r4i2s 52 39 20 23 63 z sr h'.v6 zs 27 zz 33 38 4t1fl 31 60 43 62 3 Lz 7 +hg4zz7 30 15 26 17 40 19 24 51 38

Binary direct symmetry (11-6e) De (10s3) D10 (1813) D1l (661) Drz (1674) 11 54 21 40 13 56 23 41 2 415461 1843 52 47 1714 19 8 3 125558 63 18 59 32 61 50 27 31 20 39 12 55 22 43 14 57 5558 3 425346 t9 14 207 58 3L -/\62A9 28 33 51 5r 53 r0 37 18 41 58 15 21 40 \.4 s7 60 17 48 51 ls rsPr 6-9'2 s\6 47 fze 6\3 20 3s 26 38 rg-M 9^ 36 15 42 59 o: sdtsqY Ae 32 45 20 2225 345360 5 10 1 30/57 2 11 36-55 52 19 -: \l qo 8 TWlAtzL 50 33 33 4dY3 26 3s s2 6r/42 {\f s s6zrbzs:s ,igX}4:* 1r 14 d<6-*o 27 24 24 27 36<1 30 1)51 sI 6Y3 8 3 Lo 37f8 ls 51 2 3128 5 62 1726 3g 7 12 15 22-25 31 29 47 32 29 38Y9'62 43 10 45 4<1 zz 13_16 39 24 3229 50 3 4827 6 6r 13 1037 6 35 28 23 26 28 37 48 31 11 39 50 63 42 7 .ih" g4o zl 11 L7

D13 (10s4) D14 (1814) D1s (716) D16 (1676) 7 52rr211726 1338 2 431457 8 4t 1059 1910 7 512t241526 38 23 50 57 32 59 48 63 1023 8 51 12394827 6356 1 42L358 7 40 8 ss 20 rr)6< s2 23 51 56 37 22 49-62 33 60 t\ 53 6214625503714 41 3/62\s 39/e 60 11 43 18 st z*,s L4 24 3s sz 3/sabo1 +z rt\l z2 9.J4 s/4ovs 28j9 ss Q +s '4'6t/tz zo 6 s6 s 41\7 12 sy28 3s ss 30 2L36 s3 2 61V1 qs 4s 42 s7 2e 3{ 13 s0 +o zs€+ 3 zo n +9) ss zNih\/ 6 Ia6t/zz ;4 42 L7 4-Q 58 3),&l 2e re s14q r 392s 28 zs o)x613 +4 n 3 56 19 44-l/62 33 60 s0 47 52 rT3ffi7 32 23 ;i ;T#trN;zl 26 11 8 \/7 ro 4s 18 43 2 57 31 59 30 63 53 18 35 18 33 24 29 26 259403148633833 7 281542 9 1413 18

Dr7 (1303) D18 (1301) D1e (1834) D20 (183s) 50 53 62 23 18 13 26 29 2 51 60 47 5829 6 45 11443 1859105 8 63 22 49 52 2s 28 45 42 61 48 3 51 5 4657 28 2 19 12 q5-4-J 60 11 s1 30 27 s2 se 30 ss 44 7 26 23/40 37 44 4L54 5L 43 10 6 s'/\t\z \{0 t\ ITdrzilp zrffiss36:rlo 41 16 +Loz\}+ $ ro 27 s6 41 1( 21 s6 2e 31 +p+ 20 sw6 63 22 s3f2 6L z r\o n so/n 10 15 ,\}hr\ol811 22 7\2 17 18 l-60 33 3e 6t 2r s2 s7 1{ 3s 24 5920 1 32 9 12 740 63 36,T3 42 9\ 14 17 32 rs zo 9q st13o oz z so +il30 z736QN1 13 34 3 rilszy's s MIT 40 zr:B'zsq\l rz ls 8 11 18 13-4 61 32 59 zs 3l\g4z ls sc2s 34 les8rsY13 I 396 37 21 11 20 t3-1.6 33 18 1911 5 103158 362 483L28372633 1455 FIGURED TOURS

Dawsonian Tours

D21, (2352) D22 (23s3) D23 (rer7) D24 (1e18) 1556 3 34 5 58 Lr32 1017 8 4326 153841 205346 7 221344I1 13 2 19 8 15 105154 2 534657 1033 8 59 47 8 2t5445t02311 185143505346rr ss 14 3JA{ 6 31 12 s2re6As0171243 t\42 7 20-9 48 55 52 sz (s+ 47 38) 60 7 s 48 srl8Ys 42 rs 24 ilD.t6/s6zr rz 47 +zhz3e/ r\36 61 2N13\ 30 34 59-4 +g tcgg 28 1r x d6/r\t, 21 zs zz 64{1 42{21Y8ry}6 D @t#\'@s:s 40 6r sQ\z rs 26 13 23 4o=Y9 oz ){ts zg u 20 47 56 53 22-49 62 33 oo gl-e4 3 16? 40 zs 63 34 5e 38)s 28 23 30 50 63 24 41 28 15 26 17 57 54 21 48 63 34 31 50 63 2 613257303726 60 39 62 33 58 31 14 27

Dzs (1e1e) D26 (reao) D27 (2288) D28 (228e) 15 59 18 35 62 37 32 4r 42 15 32 63 34 47 38 19 7 2 456051564758 54 5r 26 t7 21 19 58 35 17 20 15 60 33 40 63 38 31 62 43 46 39 20 35 12 4461 6 A46se30ss 27 L6_53 s0 5746 23 29 58 13 31 19 36 61 12 3r 44 1r &l/\ 33 48 13 18 37 3 8 6lf\s so s7 48 t*tbrth31 ss zr ro.sT +Mlv o 3s-el 61 30/ sYo 2'46 Lr L1 62 43/4 13 el€O 54 31 9-28\-\ 15 56 49 60 37 22 zB (oo \g3o/ts zz 17 g4z 17 42 ss slas zB 40 2e 38\3 48 61 rzsi@!V/r, \-l \l lx 7 zzYss q zd{ so 59 6 29-4\ 25-lp 55 52 20P3 ry'2s{4 4y38 3s 3 8 3\1 >6+ 1s 32 541124 5 5232815 2 27 8 sryo sP6 23 11 l{zt rs :}16 zt +a rz 41 6 l'30 43 62 17 23 8 53 t027 4651 2 7 58 3 26 9'24 51 54 22 19 24 t5 26 39 31 37 7 2 134263463141

Dze (e81) D3o (1677) D31 (216e) D32 (2468) 392 536011 65562 12 5 24 3 263918 15 35 6 5550 3 483958 42 5 21 3 2639 18 15 525940 1 5461 12 7 23 2 4L 8 r9-1rt 27 38 s4 sl t6-l- 38 s7 2 17 23 2 4t 8 L9)6 27 38 3 38 s#\ 10 63 s6 6 4;',4...l...... {f0/s L4 17 7 31/53 5fi19 4-5e 40 6_15,,4<6/tM17 50 47 4

D33 (1836) D34 (1837) D3s (2787) D36 (2788) 23264332452851 2 60 3 582362 5 30 7 11 18 37 62 13 60 5L 51 273135229564758 42332127 50 3 1629 57 2261 /\4 31 8 27 31 38 35 12 17 52 55 14 59 251283348593055 2s22 35 14 3L 48 A 52 2 se 2w3Y4 33 6 2e 19 10 63 36SLr6 53 50 2r s6 ry / 283s 26 34 3s zo l@e s8 ls ltT 51 t9 52 39/en5-10 13 21 8 33 6(25-4 29 18 20391295456063 13 10 ss rC<\rzfue 40 43 zq \/7 26 3 tlffi{ 111111 111137186158756 18 53 12 45 38 49-14 11 7 224542 1 281730 37 1015 6392243 18 38191013 8 556259 41 41 17 50 15 46 37 48 4441 6 234631 2 27 t1 7 38236219402r

D37 (2s14) D38 (2s4s) D3e (278s) D40 (2786) rr52 5 4613 18 3 62 11 18 35 46 t3 22 33 54 55 t4 57 24 29 26 13 38 4530 328 5 514122 6 17r25t /\4 63t4r7 48 45 12 17 34 55 14 23 22 59 54 15 41 37 30 27 22746311223 6 55 s3 10 4s t1q2\+q 61 2 19 r0 4746 2r)6 53 32 13 s6 23 s82sJ8 3e 42 47 41294 53 8 2r $ 48 7 32/ e-sN 1s ++ gY6/sr z41s 60 zr +o]b:6 3r 26 3s s6 7 \o $6\\{ rc4 6*\zotg 33 s4 4e

Dawsonian Tours

D41 Q7o2) D42 (2703) D43 (2701) D14 (270s) 35 2375215322362 2276237 635423r 43 32 35 28 53 30 39 14 273A 3 12t5401731 38 51 34 x21 63 11 3r 6338 /\1 2841327 34 31 27 44 31 40 15 54 51 243283t18331439 3 sl-fi/i0\rg-l 6 6r zz 26 3/40\6Ld6 s 30 43 63 42 33 36 29 52 13 38 29 26 63-4 41-,16 35 2A /\ -/ \/ t 33 8 26 4s so ss 62 38 13 +ss@r3013 nd{/q41 w]]]Q4+E37 1< zwre 5 54 4970-9 26 17 60 24 ls 60Ye 29> s8 4s 23 6T25AQ\6 17 12 45 58 5-61 9-t6 21 52 4447 8 57204r1229 s3 50 21 l(59 48 19 10 46 Y22 59 20-9 56 7 6 61 48 s*/st rz 37 55 6 454227r059 18 14 23 52 55 12 17 46 57 6t24 3 48 5 58 11 18 574659 8 49rAfi22 464356 7 58192811 51 54 13 22 47 56 rr 18 2 476021 1019 8 57 60 7 5647512350 11

D4s (2638) D46 (263e) D47 (287r) D48 (2872) 26 4r 12 3746 33 14 3r s6s3 8 A20s1 1023 t7 20 1,5 22 47 52 33 60 63 2 594641442722 11 38 zs6/n 30 17 34 7 2 ss/s^\ 22 17 so t4 23 18 63 31 61 46 5r 58 47 62) 28 21, 40 43 42 27 40 9-36 15 32 5L s1 s7 {4M\n 21 rr 1916214853M5932/\ /\ 3 t{4/to 4s 42 23 26 2ryA9 50 35 18 6 47 s8:s ldss 18 8^3v4 3s 6u3\so 4s +soNqizt24zszo3s 30 10 3 3 I\T\ \\ 2 43 8 494 23526r 46 33 42 s +e s{fu,s 2sQ8 9.-$.49 1r1 58 s 10 4N6 re 91s 30 7 46 L58\55 62 19 22 41 30 ss 34 n t!6s sz rz z\o6q\+4s 30 43 szss 6 \6/sls 11 57 48 5r6.f Zt 60 53 32 15 28 39 60 13 26 37 3726 5 10412857 2 11 8 53 50 13:16 31 34 47 6 155659546320 29 10 31 41 27 38 6r 14 6 11 40 27 56 3 1229 545r12 7 32351117

diagonal only (49-68) D4e (14s1) DsO (1se6) Dsl (lses) Dsz (2637) 15 16 2s 38 13 41 27 56 sr46 K s 186340 1751 5 4019562338 52 11 38 59 10 13 16 6r z+ ny'\s 26 s7 rz 43 2 43 50\53-61 41 6 17 6 3 185524392057 37 58 51 12 45 60 4r 14 47 r6q 3{ 41 10 ss 28 17 sz +st+/to 8 3s 62 53 L6-47 4 41 2t 37 30 10 53 64 39 50 42 62 17 34 23 48- 9\58 29 42 rr 11 3 51 49N 4 2tL6 7 46 rp/rk31 s821 tt q/$g 63 48 rs 42 17 8 35 40 l-f-t-59 54 ss 48 13zoX

Ds3 (r44e) Ds4 (14s0) Dss (1se1) Ds6 (1se3) 17 12 15 51 21 26 33 30 7 2433 4059265538 47 62 43 60 4I 14 31 34 63 48 19 32 5t 11 21 30 M ss 2227 3141 6 2556395827 44 59 46 49 32 35 40 t3 18 33 6249 20 31 52 13 ry1,322e t\ rr1.643 2ry3 24 31 31 23 8 tt/t>ye 60 37 s1 63 48 6y4\s s0 33 30 +t gd r\t so 2s zz 10 39 36 51 28 23 42 3( s7 28 6r 58 45 5 36 29 12 39 34/17 46 61 3{ 23 L2 s3 56n9t\ yA6 s -64 e{ 7 s2 4\62 3s so 16 sz 1f$38 ss 28 6 41 l-nt zob+q$d'L s sz rr _""k;$N:r1:^?i 3 8 s3 46-t

Ds7 (e10) Ds8 (181s) Dse (1306) D60 (130s) 614233515914 3 18 756355837541518 2023 1845 2 335617 46131241 8 27r43I 3431 6043 2 195845 3459 6 55 16 193851 17 44 21 64 57 46 31 34 11 40 47 41 13 30 7 28 41 62 53 32 55-4 t7 20 s 8 st selsfuo rz 14 zz rs 21 \rt7 48 ss s01s12 \< s 32 ls 303soo/<<\6s7 60 33 o+.-gsb9 s 52 3s 43 16J3 58 49/54 3s 30 3e 10 sl 48\9f,6 2e 6 53 52 zgA6-s\sB-.9 zz 63 40 13 24 ro so 37 52 49-34 254 21 56 17 I \\-\ +(ftSs silbs-frrn 28 3e 6.1€(22s 6 47 32 29 +B-1 ro zr 14 4r rs 1z\-6b*e s zB ss:A$yfus\s 3 22 51 14 37 26-:19 8 23 10 3 6227 3043 162312 6011101326 7 3851 62 s3 3o sq 2Nr zo 57 3827 5013241148 7 2831 2 4722r14245 411461 8 395227 6 376063541958232 FIGURED TOURS

Dawsonian Tours

D61 (1s2s) D62 (rs26) D63 (rs27) D64 (1s28) 2 27 4223 6 33 413r 2 5542331153115r 63 56 19 32 6t 14 47 34 17 42 53 58 19 60 23 38 6124 3 284330 7 31 41 34 1.-54\\- 15 50 15 48 18 31 62 57 48 33 60 13 52 55 18 41 24 39 34 6r 26 l-.62 41 22 5 3t 45 s6 3 32\43X4\\- J7 s2 13 55 el29 20 15 58 35 46 43 L657 54 59 20 37 22 31 40 3s 4\ 2e +0 so r\if+e{ 4s rz ss s6 s\+ zr+o 33 62 3s llpe \e 56 15 642136-9 46 19 6 s7 30 6ryEQ\2 17 53 28 I 16-21x6 39 44 rs zo so ii 36 zr 32 se so t\ufr zo 3s 10 3960528ytn522 z s sfuilq>/g 24 rr 10 s r\zg s h, 48 63 14 55 52 49 12 37 18 47 58 7 6237242720Lr 27 sz 7 \48 zz 43 40 27 11 7 \-+fq\r z 51 58 13 54 17 48 11 38 613859 8 19t02326 6 3 2651 8 111023 6 11 28 13 30 3-M 47

D6s (1816) D66 (286e) D67 (2870) D68 (381) 7 5429523156rr22 47 22 31 20 39 18 35 14 39 46 59 30 41 20 61 18 14 51 14 57 60 53 18 55 2851 6 5510233257 30 11 18 45 32 15 38 17 58 27 10 45 60 17 42 2r 45 18 15 52 17,56 59 62 5 8 533063342112 23 46 21 40 ryr6 13 34 47 38 29 26 31 14 19 62 50 13 46 43 58 61 51 19 26 12 29 z+ +{q+hzr ,/t t6 37 28 57 32 37 a6 $ zz 13 17 12 4tg[{- 20 63 \ -// ss so 43 /vs-o /63 12 33 48 zswzgf ls 6 r23s 4,\Nfi: 6 2r --ffil;1111. 28 3 set1z;fi4-:t'--/-7-- 60 7 s6 s t s+/ffia/t- r --ra-f rz 3 3 38 33\pqe 2227 30 sr 54 tzzo 5 58 11 62 53 31 4fz4 Y ro 5 14 34 11 40 Y pz 29,D1 7 4837 2 4t 18156043 2 2752571061 8 59 50555235 8 13 2 11 39 2 37 rO Zg'8' 31 28

skew Binary oblique symmetry (70-100) D6e (r1s2) D70 (3324) D7r (630s) D72 (6e32) 13 26 59 38 11 t8 57 22 29 26 31 31 t3,16 59 10 458611059562340 42 51 40 37 32 35 18 15 60 37 12 2s58 21 r0 17 32 3s zs 17 621114 7 4239 5855 39 43 50 19 I 7ff,0 t\o 2 L63r34 27 14 39/12 19-16J3 56 27 30 33/12 rs 24 s8 5 46 6057224121 sz 41 17 l\- t9 9 38 ;,6&fu+ 40 61 36 15 21 43 20-9 48 3e 61 16 1yr8 23 rz oy'\3 382s s1 zr zffzs io r/ ?6 t*oo +6uy 35 2V11 48 ,1\.8 55/41 37 6747 40 re f s7 8 47 4 35 163y20 37 26 +Qy'+424zs 8 13 oz +9s+ zt/sz d4 7 5249 3845 42/7 22 3 vA3t -/ f4 \ 3 t( zg sz rs 59 &l 57 51 15 L0 2L 26 zs 3zs)# s z 4s s4 o: +igr s+-{ zo s s6 r-48 1\7/tlsoz730 561762 5 2823127 506330334653 6 3 sos3}'6 6 ss z zr 1433 2 492831 1851 635855461162722

D73 (607) D7 4 (3388) D7s (s66) D76 (6306) 1239 8 3 2215t0r7 54 41 52 37 48 57 21 35 46 63 50 21 32 19 31 23 39 L1 43 r0 29 32 45 26 7 2 11 38/9J8 21 14 s1 2 47 -/\4249 22 31 18 12 11 .10 t5 14 27 31 3I 40 43.+4 zobr6 1r '-:?\I:-il'#-!#;: 62 416{z 2093 24 3s 13 38 16 tl /<.33 92p t€rs M37 24 t\* 3e so 4y22 17 26 se/16 Kz $ 48 +r 3017 30 8 4r/12 3716'+t/SZ ZS 18 s0 ss 2r&1 27 60 11 1633 ++ dF+ s7 ro zs 63 17 30\1 s gzs 2 hzs 4 lBstliss24f ll tn /n ir 17 &l_45 28 33/36-53 56 6328/3 18 s Ly7 10 s3 s6 z\zs tt rd v r8/y4{-t 6wr s6 sl 60 31 Drg<8 sr 26 3s 20 1{0 61 r>e 32 ls 60 s s+\6 n 26 3s r'&, 5 z0:+9 51 23 58 63 16 59 32 27 34 57 52 zs 6zrb+4i 6 13 8 55 8 59 62738 1512 6 19 2 6122 575055

D77 (602s) D78 (316s) D7e (6s81) D80 (6462) 63 60 51 40 27 38 35 32 IZ 47 58 31 14 33 26 37 760et{3s30r7r1 2851401532612 63 52 11 62 59 50 33 26 37 s7 30 13 48 27 16< 34 10 3 6v116 13 28 3r 60 46 11 28 se 32//Ls 3s7s ss 8 rrlhgzols18 '^:F'X^Wr;ffi 29 56 45 10-49 24 35n2 z sg s2J5,1 sq rs 32 27 17 42 37 30 934 y r(+s 13 sre-34 wAo z u4 60 z:t( zg 63 s8 //* 4s 24 s3 zo 26 s3 24 1T8/ 510 5 7 ot ++/z so gd zo 38 4313{ ss s0 21 16 33 'e 1358 rzi|+4r+zr\rd ll t 6 47 2 55 10 5 182320 s2 1'

Dawsonian Tours

D81 (60e4) D82 (6381) D83 (6na) D84 (646r) 38 7 54 63 22 35 52 1,1, 6346 3 6 59502730 63 38 59 56 47 36 23 26 181140 7 38133455 556237 8 53 r02r31 2 s 94{262e s8sl sB ss f4{ r{r)8 3s 41, 8 17 t2 33 54 37 14 6 39 ffi 23 36-33 t2 5l +t oz/U-le 60 31 zB 39 62/57 60-49 16 25 22 10 re 6 3e)6 35 s6 33 or so/s\+ o/s^]o-zs zo 8 t'As/ot/++zs sz s7 s4 oo.ee8 2ry5+ +s s 42 gd s),sz rs 36 | / '7t { 42 e)#\sy'zn re 40 e-Q[2ys6 rs 32 s tilpfobrd zszo ozzrfg4+o\sdft 57 48 41-4 45 16 29 26 r0 37 zo +g-la zs 24 53 53 41 26 z 4 8 194415 43 4-63 2{e ryAe 2 43 46 59 18 27 11 31, 41 18 39 I2 55 22 33 11 11 6 5t32t4421730 22 6t 2 45 &-47 30 5L 475834415301728 38 rt 42 17 31 13 54 23 52 3 12 7 1831 t413 3 44236029502748

D8s (623r) D86 (6382) D87 (6026) D88 (6027) 3423 561152 7 3863 1r t2 37 62 29 14 27 24 23 t4 45 12 27 36 53 48 17 56 3 52 5 1417 14 551235225762sr 6 38 63 40 B zry,{ 30 15 41 rr 21 37 t*\.6 3s 2 51 4655 18 15 B 43 2433 1053 8 37 eI39 rr 42 et x4=zsVg/ | \-\ zo rs 22 13 4625 28 49 Sq s7 48 P4*{. 6 ,t6 13 s1 21 36'.lJr SS/S\ ro/43/s6 2ra6 3r 10 43 rc6 38 s tA+ zg so rq 6€- tl fro sFtlz/t zzzs6ejd"4/-rS\ 4ry0-424 3s 8 ss 20 17 2/3e 6 3)54 ss so 3e/ye-{ 11 32}s 12 17 rxy,islq-+4 oa 50 r/46 s7 41 17 32 7 42 \ 20 l-s\ 61 30 63 tr6r zo :sT64 zB 3r 2629 16194845 2 4l 5948 3 52 5 341954 3 rs\zlqo s 3zses6 59 38 63 22 29 26 35 24 151827303421741 2 5158451853 633 8 $\d 1e 60 s7 62 3r 62 21 60 37 31 23 30 27

D8e (6s82) DeO (6637) Del (6e33) Dez (747s) 51 12 15 38 47 6,1 33 2 384162 7 10272459 2 3342 3948 61445I 41 3s so 11 ,/\8 63 63 8 37 40 61 58 1126 41 38 4,34 43 50 17 60 13 52 37 16 49 604 32 42 39 6 9. 2,2'5 60 23 32 3/4N 62 se sz 1s 40 43 toz$4 6zss s g\ot$W,r7 rz zt d5W 3s 46 17 sg s3 14 936/61 2a 31 6 50135V[ ne 16 19 56 2r 10 s 3w+6 s7 22 s3 42 2s22\ tz4r75679 28 s s8 19 I32 49 44 53 46 13 18 27 30 Nab\/ zo rs 18 rs s1\T 30 11 ,fr/3 31 34 55 62 345t 30 3 48 152055 6 rr2$25 8 L3512r 2623 1655 8 29 1857 52 ) 7 54 63 30 33 31 23352455147t4 2926 7 1255211911

De3 (7718) De4 (677s) Des (7174) De6 (6636) 436217 2 415237 48 583962 7 2633283r 47 6 4532535043 2 31 55 26 15 42 57 21 11 18 3 42 63 16 te

De7 (7s3s) De8 (7s36) Dee (7716) D100 (7717) 17 21 13 22 11 38 57 16 32 3 1157305946 11 2 29 6 3116631233 275429507 21114 11 2t)6 37 58 17 60 39 43 56 31 34 45 t0 29 60 7 50 3-&1 5 324562 305126 3 40 13 6A zs(z\z +r 10 4s s6 2 33 s8 3s 12 17 28 130/47 20 13 31 4r ss 28 s3 8 +y.gfi/+z /4{ \^A- t 20 lfiq/ 48 se 10 6r s3 428.5{? 48 61 28 51 8 49-4 35 10 61 44 52 31 48 7-\z\25 +dBs-64 - 5 3 26rrcz3s 8 ss11 22 L-54 5X21 63)6 13 I0 27 52p1 48 19 58 29 15 56 r9/32_.n6 n 3S 32 zg 4 4g,sz 43 62 7 4r sz rs {e\g647 62 53 21 9-36 57 60 15 1g 18 s6 t64 24 63 10 3s \ ( 27 2 3N4 rysrs1 18 21 so sqbz ru zt 26 rt 2U55),6 t3 38 59 57 14 6L 20 33 12 37 22 30 33 28 Y/so s: 6 63 51 40 L7 20 15 38 7 26 23 51 25-12 37 56 L7 14 60 17 58 13 62 2L 34 11 FIGURED TOURS 11

Additional examples by other composers are shown; these illustrate that the position of the circuit on the board can be varied, &S also can the positioning of the numbers on the circuit.

D5a G. Fuhlendorf D6a P. C. Taylor D6b P. C. Taylor D6c A. H. Haddy 48 43 46 29 50 63 56 31 18 57 8 23 1659 14 rr 41 38 29 16,19 14 27 21 23 20 t5 34 25-30 47 32 4s 28 ss 30 sr 62 7 zz 17 t{ttib;s oo 30 17 o#rb?t zo 13 1,1 17 21 zo 1e{ 'y'orfto 42 47 /\-/44 3rMSr 32 s7 s6 le 6 f 21 rs/Tz 37 3e 42 f 18 rs/s8 23 26 rs 22 7 16 3s 26Ar $ 27 36 41, 54 59 3fr1 52 s 41 3 3/6 61 26 8 3Vs0 3 2r 8 tt 2r8 76 rzss nrc/ot6+6ztzt 40Ar 26 17 22 s3/s8 33 20 ss 4 4s 62/47 38 3s 43 3s r0/s1 60 s7 3 62 s s0 2748 4s s6 TI f 22 25,14 37 rO 7 19 2 +z sz/ot z 4s 32 27 30 32/7 2 49 46 rr s1 s 4 TT 6r rzs/z {tr4z3s rz:br6e ,*t 8 s s1 1{ 1V46 2e 31 3e rQ tfroffisz 47 56 61 4J3 10159 40 5s 44 1524 t338-9 6 3 20 51 42 53.164 33 40 31, 28 6 33X4 45 48 55 62 s3 sz 11 db{' 54 43 sB 4r

Dla F. Dignal DZa H. A. Adamson DZb A. H. Haddv D3a A. H. Haddy 266324 7 2861 3019 304s 6/1{}fl1o63 56 53 6-''LQ 51 40 63 54728556613063 ,t4 57 60 23 8 27623 205560 7 7A +obo1 s3 48 7 /'55 s252 59-6(5g--64 43 50 27 4\55 50 29Al q,q*f 21, 6 2e 18 31 4rr 8 s 2 s1\23e sf7 8 s 2 4N23e 8 s\tx

D4a A. H. Haddy D7a A. H. Haddy D9a F. Dignal DLla A. H. Haddy 62 1,r239&51 16 1l L7 26 15 12 37 52 21 56 135235015463942 8 11 6 1726155829 3 zz 63 sz/qfuo 37 so 11 11 r8/25=W 55 38 53 255t44538412147 s18 r2 6r ,lqqz +s 27 16<3 48 si3 6 s7 22 53 r2 srA 49 t6^.43 10 10 t 2Ta637 3o\7 ,y:s \l /rz 4448 s3 ++$e 10 4To le 2y3e s4 3s 19 4 41 6013213936 zr s6'@'NN'ps \f 60 13 2N9 32 2s51 13 3 28 9 4049342358 11 60 5T4 I 3617 22 42 6\6 23 40 3s s6/3r \f '1\Qt s 8 17 24\T8 3s 30 +o +z/z s 8\q 62 33 6 57 t03120332635 3 20 4s s)Ae s2 1459 6 1916332655 zs /qr ++2fu 7 60 613059 8 632821 18 62 13 22 47wr 32 55 7 18155827562934 42 4s lb-rzf 61 32 63 58 7 622932193127 21 2 63 1133 48 53 50

D13a F. Dignal D15a F. Dignal D17a F. Dignal D17b F. Dignal 62 35 56 51 40 37 18 15 6322435261 6 15 54 12 15 18 2s/\ 46 35 32 27 22s\6246s728ss ss s0 63 t{ 17 14 41 38 5 17 24 1L44\e 26 47 34 63 z zt/s*27 s1 s sB 46 10 13 t6 +s 3e :l 28 31 7 s6zs :l':rx:,#;e' t\ sozrdor\t\ ffH{z,\?ei:H so +\ 4 33 z/+t 8 59 23 2f7 20 s3\0 61 48 3e 24/3 8 s3Y6 se 10 48 l\re s3 sve 24 7 37 32 re 20 \s8 3 56 38 e\22 h11 s4 2e 2049102560 9 3033 zs 3z#+4r 6 43 rz T7 3 6\fryw2s7 60 4rr\/\/zrrL4 2 473027 1045 8 23 :i":H,;s\l i; 12 g 39 '{ 43 ffi 59'e 50 55 48 19 36 43 16 13 34 3r 3128 3 46 5 221114 28 39 30 35 26 13 18 15 5 12 7 405t 566358 37 12 L7 46 35 44 15 L2

D18a F. Dignal D29a F. Dignal D57a A. H. Kniest D60a F. Dignal 3137r0r7 8 21 3019 43 46 3r &l 41 60 33 62 3914 3 58 5 123356 4510473143 I 3 18 11 16.35 26 31 18 7 22 30 s, o/^\vz 63 40 ss 2 5938133157 6 11 48sr11 \< re42 7 363tr*-g 21 s zo zs 17 41 r@ s8 61 34 ls 40 3'A\r7 8 ss 32 11 46 3s s0v>4 17 20 r\i))r r\7 1023 6 so 2s ,+4}3q ls 2 3s 60r({\+:r 10 7 s2 49-32 146 21 6 4r 56 49- 14 39 4 13 28 4l 4rR6_61\30- 9 18 53 28 31 r$froq\oos7 zz 13 szbo+q\z 3 41 :r'mt.$: ;;i; cfQtSs so zs zzrs 28 fi M 1s s6)s 60 3e 48 57 50 53 4N 12 6r 7 54rr2617241324 45 4Zn9 62 2L 21 27 52 13 30 55 26 37 62 23 58 515447586360452 1027 8 551223 1821 48 63 46 43 26 51 20 23 54 27 11 63 21 59 38 61 12 FIGURED TOURS

D68a G. P. Jelliss D73a A. H. Haddy D75a A. H. Haddy 5962524340 738 11 18 55 26 41 21 57 22 11 18 21 26 13 46 41 44 In D68a, unlike D68, 2623 6063 6 37 44 4I 51 27 I2 L7 56 21 40 37 20 27 t2 15 10 43 38 17 the axis of symmetry 61 58 398 15 10 19 4?/.538 23 58 17 I0 19 2L25 14 45 42 of the circuit is also 22 27 42 45 28 s3-r64 zo)):6 3s 28 )7 L6 s&s 48 37 an axis of the board. 29 56 2T 35 10 ,6 3 sz 3s +s s\4+ tQ+ 7 24 61 3c1z s+ Note that in D68 the 20 15 28 32 L7 46 5l ,\n 8 s z 4sgls 6 2\ss ztto/e cubes, I-8-27 -64, also 55 30 13 18 53 48 TT 31 7 4{ 34 srffi47 60 s3 8 3\60 s;trzs s8 form a circuit. More 14 L9 54 3r 12 33 52 47 30 33 6-11/46 61 50 63 30 5 52 l/lZ 59 50 63 on this in section B. The Squares in Other Formations

Open Knight Chain. E1 was the first squares-in-a-knight-chain tour made. E2 is by the 'squares and diamonds'method. E3-84 show the squares in symmetric open paths.

El Dawson 1932 E2 Dawson 1932 E3 Jelliss L992 E4 Jelliss 1995 8296 yq27 \17 27257621530236/- 6027 5617 2 15 42LI 57 2 29 8 59 1027 46 28 6\48 1ry6 s8 61 28 t 21 63 Mat s718se284r12\14 30ss81284760 11 s60yI 30 7/6r 15 rO 2s 46 49 3 26 ss/s6 zs)6 +{ zz 26 61 20 ss)6 3 10V3 3 s6 7 /s2/9 s0 4s 26 | -/l I se f 11 24 3st\ot/t 50 ss 4{tq0/r132 13 le s8 z6,zMo rg) 6 3r {4r48-2F rz 61 r*r L4 sl 40 +z 16 +s s 38 sr +bd\6 fi +s s js sAer sa,'g4, zg ss38s3 #4fu4t 24 f s8 3 34 23 51 19 42 544t 8 3720451233 sltz+ o*e 47 30 ss 8 32 Ie 34 41 36-23/62 T3 32 13 56 39 52 21 44 37 39 6 435235104718 y\{n fi 32 7 38 4s 3954172015ffi43 22 57 233 22 55 38 53 20 425340 7 16t934 11 2352354837462t 6 18 33 40 35 42 2I 14 63

Asymmetric Closed Chain. The next three tours are the only examples of this type published in PFCS/FCR; but see also the squares-and-cubes solutions 84 and 86 that follow. E5 and E6 show the same shape in different positions and differently numbered. Although the octagon of moves can be symmetric, it is not possible for the whole tour to be symmetric at the same time. In fact there is surprisingly only one symmetric tour that has the squares in a knight path. This was shown by Valeriu Onitiu, 87. ln such a tour the numbers 4 and 36 must be diametrally opposite and 1 a knight's move from 64; further the square numbers *32, i.e. 4, 17, 32, 33, 36, 41, 48,57 also form an octagon of knight moves, and since 17 is a knight move from 16 and 48 from 49 there is a considerable constraint on the possible configurations. In FCft (xi 1939 p.43) we read: "V. O. notes that he examined 144 dispositions of the squares, all that are possible for diametral symmetry, and the above is the only case leading to a tour. Moreover, every move of the tour is determined, so that the tour is uNteus in all the millions possible."

E5 Haddy 1932 E6 Dawson 1932 E7 Onitiu 1939 E8 Jelliss 1986 11 18 2t 26 13 38 13 40 33 30 37 50 21, 52 19 26 567513585 46 3r 4015 10 5 384720 3 20 27 12 15 44 41 46 37 38 63 32 29:627 22 53 536057 6 1732 43 18 11 8 39 46 r94, sr 48 17 to L9 22)sJ4 3e 42 31 3JAs6srh>q 18 8 55 2 5g/A 19 3A$ 41 11 6 fiOhT.z 2T zB z3x4 soc>36 q7 44 3s 28 17 s1V3 6r sz t{{n 44 17 42 7 rz 4*sl,r' 3zAg 52 --\ 4g | 't t eq 7 24 61 48 sry1 a3\1061 z+le r0 49,12 1-16 41 20 29 42 ss rG zt"-'l 56$thz 63 \ f \^/ 6 2g-4\-/ 55 2 35 62_49 2 11 40\45 6-9'58 55 rso\=t'uq\+ 23 4CI 27 30 1\60/l7'ffi 33 54 s3 8 3\60 sttr33 s8 47 42 rs \

In my 'squares in a cube' idea E8 the cube is geometrical instead of numerical. The square numbers are arranged at knight's moves apart, the twelve knight moves joining them delineating a cube. If the squares are taken in order of magnitude they form an asymmetric closed circuit indicating a tour of the vertices of the cube; a tour-within-a-tour! (See also Puzz,le 8 and page 22.) FIGURED TOURS T3

In the first article on tours in PFCS (vol.l no.10 iil932 p.58) Dawson attributed the problem of constructing a knight's tour with the square numbers in a knight chain to G. E. Carpenter. However, five years later (FClt vol.3 no.7 viiiL937 p.77), after H. J. R. Murray had asked him to verify the reference, he concluded that he must have modified Carpenter's proposition, so the idea is his own.

Longer Leaper Paths. We have already shown the squares in wazir and knight paths, but other leapers between squares of opposite colour can also be used, as shown in the following examples. Giraffe {1,4} paths were introduced by P. C. Taylor in a lecture to the British Chess Problem Society on 26 ii 1932, with the following three open tours E9-E11. The two closed tours E12-E13 by the same composer, published 12 years later, may have been presented at the same lecture, since Dawson was in the habit of saving up contributions for later publication. Instead of closing the tour we can close the chain, as in my Et4. In this case the tour itself must be open. 7,ebra {2,3} paths were shown by T. G. Pollard in the four examples, E15-818. An an open tour with closed zebra path is also possible, as in my EI9-820. Other tasks to be considered here are, for example, {3,4} or {2,5} or fiveleaper paths. Some of these look difficult, if not impossible. Puzzle l: Show the squares in a closed three-leaper path.

E9 Taylor 1932 E10 Taylor 1932 Ell Taylor 1932 ElZ Taylor 1914 r0 43 24 55 22 45 26 53 59 10 33 46 61 21 31 48 12 27 20 43 58 47 22 45 18 L-12 3 20 27 325 1 58 9 442s5449 46 8 4560rr32472823 D 8 rL 26 zr 41 s7 48 41 46 rDb+ 21 30 o\ A', /\ zr {\ 27 1 58 9.342s624930 281342 7 1059 4623 M17 2 4328 633 s4 "2 ,20 4\ 7 29 51 lse\8 /3s\50147 At, l\rrhn tr\/ t^\loo\/zs 3 34 23 \s6l13\18/3sP6/63F0 {l;Ar? l1;r Ai? iwl 33413167566r to\/ 6 \/oo\/rs\r 53 56 4015 6 3138631839 55 4 41 16 193651 6l 10 35 5 32391417303762 421554 5 52392037 *\/IMTilIH 1? Iffi 57 52

E13 Taylor 1944 El4 Jelliss 1985 E15 Pollard 1945 E16 Pollard 1916 6 3 41232639 1041 13 48 11 60 1546 5 58 & 47 28 41 3639 21 2r 40 51 3 60J 312r 45 22 5 24. 42 25 38 .-4 59 L6 27 z\z I +94r)nq:s 37 3e\I)6s 22 se fr ^:##fr49 14 +\zsll45 28 6 48 6Y4 1C7 40 37 ztzs 52AT ,/\ / I ._ffi?:1,\: s\llf\ 3 sdnlzo\rz 43Fo 4e 2 4s 26 ry34 :K,m=,l s8 31 ru s2 zs\d 7 ( 7 6z3t so 42 26 47 62Y_ 32 19 64t \7 \3 \z r56rs 47 20 s7 \030 616t y:s,e 16 55-36 34 U ruls o\o 3 6 ss;(x18 27 1 7 5145 48 57 10 32 se 18 4yT{s3 14 sL tlff?\tzfiztt- 8 8 \ rz sgz6, st 14 s7 6 43 29 12 55 18 31 19 48 33 60 15 50 35 54 52 37 20 33 10 35 24 3r 11 60\ s1+ 13 s8 17 sz 15 28 L3 44 t7 30 11 56

ET7 Pollard 1916 E18 Pollard 1950 E19 Jelliss 1991 E20 Jelliss 1992 14 29 264t246158 1 102912635231 6 1 1\58 51 32 63 18 49 30 48 53 14 t9 42 27 24 6l 27 42 13 62 srfti z/so 45 1,8 47 51 21 62 4r 26 30 15 28r2fas selz s lrl 52 49 20 43 28,25 60 23 6(AE, /'srzz i34534553615206r t7 46 22 63 40 27 18 45\50 61 56 3 3 r/t;6{V 47 zB 50331 39 56 59 154 15 58 3936 43 34 38/154 rr 1)hQ\412r 15 30 58 11 I 32 17 ff"ryu+46-9 38 19 54 7 26 r\22 r9A4 41 60 37 sqz 7 40 z3D)s +o 34 1''32 13 6 38 57 17 L0 33 1835 8 3720 23 20 25-40 59 38 35 12 639101326152213 311435237127 10

Satins. The eight square numbers can be taken to represent queens standing on the board in mutual unguard: one in each rank and file (which makes the pattern a 'satin'), and not more than one in any diagonal. T. R. Dawson in E21-E23 gave solutions for 3 of the 12 geometrically distinct arangements, my F;24 is another solution. AII cases have four queens on white and four on black, as is necessary for the four odd and four even squares, so probably tours for all cases can be constructed. T4 FIGURED TOURS

EZl Dawson 1935 F,22 Dawson 1935 E23 Dawson 1943 F;24 Jelliss 1995 s 18 os6 Lr 22 rs s1 31 ss@ 3s 24 rs 12 rs 26 z3@3 30 s7 18 ss 38 8 ss @o: 18 sl 30 8 s7 6 zr@ss 12 23 37 40 33 s6 13@ 23 zo 37 34 zt z+@s4 31 s8 56 15 38 7 51 Zg((Arre reg)r7 ro 3e 24 s3 14 s4 3s 38@rs zr 14 11 zz@so 3s n zg s6 47 s 40 Q6z 17 so 3L sz s8 7 20 47 s2 13 386 +t@* 32 s7 ro 17 zz 51 38 zi, zB s3 16 59@ 11 s7 6 37 zB sz zo@ Oqo 3 40 37 26 sr 12 sz ss 42 63 26 s 48@ zo Ls sz rr oo@ ++ s +t@ot ro zr 48 27 32 62 sg 32 45 48 41 6627 436231 s8@s 226 3srzu@+s 6q 8 s8 13 41 z@T 24 47 31 z 61@ zs 34Xso 60 s1 2 4s 30 27 (f)47 14 17 z 4r ro or@ +: O4zlroo+szz3s26 60 63 30 33 44@zB 3s C44 6r so 3 46 7 zB e4o 13 18 3 42 7 62 rz ss z +z z+@$ 23

ktter Shapes. As with monogram tours (knight moves delineating letters), the recognition of some letters, such as R, is dependent more on the line drawing than on the disposition of the squares used. Only some of the more geometrical examples are shown here, forming letters O, T and C. E27 is one of three by Dennison Nixon showing TRD. My E28 besides being a C is another wazir-path solution.

F,25 Hall 1935 E26 Dawson 1935 E27 Nixon L916 E28 Jelliss 1995 596 35 40 61 26 29 32 3341 7 5031 46 IT 22 10 5 2663t245284L 591052231855 20 38 41 60 7 3431 62 27 6 33245 8 23 30 47 3 62II4427121346 6 2358L75621 2 45 558 33 30 13 31 5 22r 10 6@4ory 11 60 46 19 51 42 37 1 63 2 5 48 2s 6rZ716Wq so 21 7 57 48 53 44 4I 15 63 42 13 17 24lr 30 3e 61 12 12 47 52 356 48 20 8 f+s sr 837 10 43 1 20 23 56 53 60 928 37 s7 60 2lflglr s s4 33 s2 26 15 3t 40 43 552 45 12 53 22 17 l4 4L 62 55 58 39 26 17 14 18 23 58 55 20 35 38 31 35 62 13 28 33 38 51 30 41 IT 51 17 16 13 52 2T 51 57 40 6L 18 15 38 27 59 56 t9 22 37 32 53 34 L1 27 34 63 5A 29 32 39 Other Numbers in Formation

Arithmetical Progressions. In his L759 paper on knight tours the mathematician l-eonhard Euler gave (among much else) the eight symmetric knight's tours of the 5x5 board. AL shows the first of these tours. The numbers along one diagonal in this tour happen to form an arithmetical progtession with common difference (co) of 6. This is the only one of the eight tours to have this property, which makes it a figured tour. A2 and A3 show similar symmetric open tours for n = 7 and 9 (cn 8 and L0).

A1 Euler 1759 A2 JeLliss 1991 ,A'3 Jelliss 1991 A4 Dawson 1935 2318 5 r02s 23 38 17 40 35 32 49 77 16 27 16 79 72 49 60 81 3A 17 t2 15 31 53 48 51 6 rr241914 16 7 2237484134 11 L7 78 47 28 59 80 71 62 11 11 31 36 47 50 33 51 172213 4 9 212439 6 333631 45 76 15 26 73 48 61 50 39 18 29 16 13 32 35 52 49 L2 7 2 $2A 8 45202530 5 12 18 13 74 29 58 51 40 63 70 1 1019283746556/. 1 1621 8 3 19 11 17 44 rr 26 29 7541255241305738 7 2027 2 9 56633845 16 9 2 132843 4 r21942312453 8 6961 5 8 232611415760 1 18 15 10 3 1227 43322L31 9 5667 6 37 2421 6 3 6259 4239 2A112235435 4 6568 7 1 25221310 6158 1 223310 3 665536 5

For n even the numbers in the progression have to be arranged along a rank or file instead of diagonally, since they are alternately odd and even, and so on cells of opposite colour. No solution by the knight is however possible in the 6x6 case, even if we drop the requirement that the numbers be in order of magnitude. The 8x8 case was solved by T. R. Dawson in 1935 (Aa). The 9-move segments of this tour are alternately in the lower and upper ranks of the board. Puzzle 2: Construct a 10x10 example (cD 11). Arithmetical progressions can be found in any tour that contains a pattern of moves repeated at regular intervals, since the numbers in the successive positions in the pattern will all differ by the number of moves connecting it to the previous or next repetition. FIGURED TOURS 15

We now consider other arithmetical progressions in order of increasing common difference. The even and the odd numbers of course form two separate arithmetical progressions of Co 2, and they occupy squares of opposite colour. Nevertheless, a figured tour is possible even in this simple case. A5 shows the first nine even numbers forming a 3x3 magic square.

A5 Dawson 1932 .4.6 Indian 1871 A7 Jelliss 1992 A8 Indian L87I 41 38 43 48 53 36 21 26 54 49 58 63 28 33 44 39 63123 8 2714 1916 38 43 18 2r 1,6 45 52 49 44 47 40 37 20 25 50 35 57 62 55 50 45 10 27 32 4p@wffi4 L9 22 37 46 51 48 13 46 39 42 45 54 49 52 27 22 48 53 64 59 31 29 38 43 ssl62 rr 28 7 20 ls 18 I 42 39 20\/ 17 36 15 50 53 46 I 21 L9 31 5r 61 s6 sr 46 4t 36 3t 26 ro\s-lo-r $zgEisl 23 34 29 40 ?s 12 49 14 530 23 28 52 47 60 35 30 25 42 37 s7 s4 e(e s{z+ t7 42 28 1r Z+ E{50 61 s4 11 f \--= 1.--y- \ '/\ '10 613 10 7 I 58 33 1 4 7 10113 16 re 22 47 s2 3e 60 43 30P3 33 2 5'26 7 57 60 [s0ys_ss,t9&4WL/36 1 156 29 60 8 11 2 SlZ ZtL4r7 1 27 61 31 58 55 62 I I 61 L 32 57 3 6 e rzlrs 18 23 20 48 51 46 59 40 35 41 3r 1 32 3 6 63 8 5956

Euler's L759 paper also contains the three elementary 3x4 knight's tours. Two of these show arithmetical progressions with cu 3. These have been incorporated into 8x8 tours by Indian composers, as in 46. This also includes a progression with cD 5. A7 with cn 4 is a symmetric knight's tour with the progression 1,5,9,...,61 in a symmehic circuit formed of two dabbaba paths joined by fers moves (a symmetric circuit of 4n moves by a single-pattern leaper on squares of one colour is impossible). A8 has multiples of 5 in a cross. Puzzle 3: Omit four cells and form a 60-move closed tour with 90" rotary symmetry having 5, '1,5, 25, 35, 45,55 on the odd diagonal and 10, 20, 30, 40, 50, 60 on the even diagonal (not in sequence since x must be opposite x + 30).

A9 Dawson 1935 A10 Dawson 1935 A11 Dawson+Hall 1938 AI2 Hall 1938 27 6 4724 3 183120 24591661261130 9 26 5 34372455 8 39 r2r73155 1019 41 37 481526 5 3021 2 17 47 14252229 8 1 12 333625 6 43382354 3354Ir 184536 92r 7 28234625 4 t932 58 23 60 45 62 27 10 31 4 27 44ps se zl+o s 16 t3 46 38 43 41 4918 29 22132 L6 1 43 4lFlzf-|'3zr3 z 4s 32 s7l|28 63 42lsz zz 47 32 53 578 9 53 143 50 15162 37 34 20 s71|42 63 141 3 36 33 s8 3 ozl+g tt ztllo 41 52 15 48 22 39 42 5|61 s7 36lse 14 6r 4r szl+s so rs I o 15 4 31 46 29 64 61 18 t3 52 31 62 1 50 41 58 256 55 10 53 40 63 12 35 38 50195439M173437 259481550112A17 25160 29427 40 23 52 4t 56 11 58 39 60 13 534051185538 5 16 173016019165112 6130 3 0 5921 526

The four examples A9-AI2 show nine numbers in arithmetical progression with ct 7, arranged in 3x3 formation. A12 employs the ruse of starting the numbering from 0 so that the progression consists of the multiples of 7; in this the numbering could be reversed (63 -- 1, 0 + 64) so the progression is L...57. The full progression of L0 numbers (1, 8,'/.5,22,29,36,43,50,57, &) cannot be shown in triangle formation since this needs 4 odd, 6 even, or 4 even, 6 odd, not 5 of each. Now four examples with co 8 (see also A2). Al4 was composed without knowledge of the earlier A13. The 8 octuples appear in order of magnitude along a diagonal. In A15 the octuples are still along the diagonal, not in sequence, but now the tour is symmetric. 4,L6, octuples in octagon, is new.

A13 Indian 1871 AI4 Dawson 1935 A15 Jelliss 1978 4'16 Jelliss 1995 53 20 51 36 55 60 45 &+ 1350292553447M 45 20 43 60 47 58 37 40 41 58 19 43 56 17 34 37 54 19 46 63 56 61 28 3 L24930 1 5635 261461938414857 20 1 4257 514 215235505948 1 41 51 14 31 51 33 48 63 46 2144 1 4259563936 1 38 33 r8 47 40 31 62 57 127521140153657 62 31823&t3I 5449 17223932495843 2 15 10 5 3253 5841 62 17 22 63 32 55 50 35 30 39 1 23r 10 7 24 5 14413027 26 7 245911392037 4 7 2427 10331253 22 3 3661 931 23L6 9 12 2 28 3 12 23 16 9 6 2rt86I42 2516 9 6 51 142934 49 385 27 8 11 6 15 4 132629 8 25221760433819 8 5 26I528Lr5213 4 235037 6 295233 r6 FIGURED TOURS

AI7 Bennett 1952 A18 Bennett 1949 A19 Bennett 1949 420 Bennett 1950 18 15 48 27 62 25 50 39 6413 2 5134 ss6057 351637 4 3958 4960 18 61 40 57 36 63 42 55 47&t71449386L24 3 5063 1245 583354 2 5 3415 48614051 395817624156456/_ 16 19 28 63 26 51 40 37 1,1 1 44 35 52 61 56 59 173633857505962 60 19 26 37 16 35 54 43 2946135241 6 2360 49 4 25 62 11, 46 53 32 6 1 28331447 524r 1 38592025M3346 205342312259365 24 15 48 43 36 31 r0 29 29187 5627426346 6 27 2 1s342r5053 453021123532 7 58 5 18212647283740 8 rr201332452453 3 12 5 24 9 5247 32 5411 2 4356 9 4 33 202316 7 423930 9 1930 9 2255264364 287L4113049225r 1 445510 3 3457 8 17 6 192227 8 41 38 r0 2r t2 31 44 23 54 25 13 429 8 23 103148

A2I Indian 1871 A22 Indian 1871 A23 Indian I87I A21 Jelliss L992 21 38 57 L0 19 46 55 12 55 41 59 50 21 46 37 62 16 37 46 43 48 27 52 55 36 53 60 19 42 55 58 25 58 7 2047 5611 1315 5851 564536612047 4512 1538 5154 59 26 6r t8 37 54 59 24 43 56 39223728 9 12t351 43544t604922 6338 3617 44472849 56 53 52 35 20 4t 38 57 26 23 6 ss-tT4t/\/\/ 2s 44 L7 zB s7 52 35,4064&rs 4r 14 5,smBo 25 58 17 6233 4621 40 7 44 23 40 27 36 43 t6 53 53 42 27 82 23 18 39 18 3s A0 2e 6 57 34 22 27 /\II 14 M &6r syLlffi-r

The tours Al7-A.20 by E. W. Bennett show arithmetical progressions with co L0 (see also A3). ln M1-M4 the numbers are arranged at the spokes of a knight wheel around the initial cell.ln M4 the powers of.2 (2, 4, 8, '16, 32, 64) are placed on the spokes around 1 (which is 2 to the power 0); this of course is not an arithmetical but a geometrical progession. Puzzle 4: To place the even numbers 2, 4, 6, 8, I0, 12, 14, L6 on the eight spokes emanating from L, in a knight's tour of a 9x9 board, with the odd numbers 3, 5, 7,9, l'l-, 13, 15, 17 n a surrounding circuit. Plzzle 5: Form a closed knight's tour that can be numbered so that any eight successive odd or even numbers occupy the spokes of a knight wheel. Punle 6: You may have noticed how often pairs of repeated digit numbers, like 22 and 33, or pairs that are reversals of each other, like 23 and 32, occur on adjacent squares in tours. Do they really occur more often than one would expect from chance - or is it just that we notice such coincidences? The task is to show as many such coincidences as possible in the central 4x4 area.

Squares and Cubes. The diagrams in this section show the squares L, 4,9,16,25,36, 49, 64 in one knight chain and the cubes 1,8,27,64 in another, an idea suggested by H. A. Adamson (PFCS vol.l no.ll iv 1932 p.64). Since 1 and 64 belong to both sequences, the two paths together form a 10-move circuit. In the first example published, 81, both chains are open but asymmetric. It is also possible to solve the task with both paths symmetric, as I have shown in the new results Bl2-83. The other examples are of closed tours in which, 1 being a knight move from 64, the path of squares and the path of cubes are also separately circuits. 84 and 85 are closed path examples by Dawson (D68 shown previously is another example by him). In the Frans Hansson examples 86-88 the cubes circle the centre point of the board.

81 Dawson 1932 B2 Jelliss 1992 83 Jelliss 1992 84 Dawson 1935 23 62 19 46 17 12 37 18 51 35 14 23 52 37 48 29 3L 34 57 40 59 Z0 47 42 46 13 48 59 44 39 32 61 20 1s 22 63 3&7 14 rr 13 24 53 3631 28 51 38 56 39 32 19 46 41 60 2r 55 58 45 11 33 60 43 38 6r 24 3s rvbDile 38 34 ss z/tslr1 30 17 3330355817 24348 12 47 56 49Ji0 35 62 31 34 zr +q zi46g to L rz ztsz2z 4fue 50 38 55 18 22 6l s7 12 ls I1.3fu45 -\\- s4 s 3$r+ :z 3 60 33 4o-{ida s+ s\i3'-8,61.6 lTh +s 29 12 3746-| \\-\ 3lrD8ag | qq 6 rL 50A4e 41.30 63 32 43 -5 26 s3,:6 sl 11 t zo/t 42 61 40 51 L5 4..24 64,5/62 23 s3 24 \ \-\'- I 411iq\3)z 59 z +lf:o 5, -8' 55 28 1g 57 B'-g/ z0 59 14 63 11 28 13 52-9-'61 7 50 10 fl\z/sr 8>1s zo 29 42 31 58 \.,50 29 52 7 s 10 le s8 43 62 4L 60 1453 10 5 26512163 3 52 9 L82I2823 26 RGURED TOURS L7

85 Dawson L942 86 Hansson 1938 87 Hansson 1938 88 Hansson 1938 541950 3 48233137 17 10 19 28-25 32 51 30 5 21 9-54 56 21 1g IO 59 42 13 46 28 -/\ t\ 7 6t 3l 51 2 53 18 33J6 47 24 20 z9rfrr s0Ye 26 33 ro s3/ 6 2T16 re 28 s7 1I 14 II 60 43 30 47 62 zo 55-.4 4g6,g\ss :s 9.'r8\ 21 2+2U6." 31 52 37 {\ zs'6.| ' '.s5 zz 17 zo s8 A 18 4s 1347 32 2e 17 4o\s +o 8."4b72.ss 34 37 11)08,.15 r6zMz z\s 52 46 27 58 29 15140\5t\;, 8'17,',14 63 4g sdl}ol\q n/zots 43 b {p'.po 3}64 s3 s8 3 ,{'rl{ ss;30 41 46 63 8-6r.59 4116 45 12 14 7 +Nt4s ss 38 63 12 s\+ 3ry4 4s 60 3r ,l*ig&*%ii1;\/ 60 57 6'il 10 43 14 zg 3 42 5 4661405754 3s z \g4 33 62 47 42 s6 3 22 37 sNy24 3s 7 6259421528II41 6 45 Z 415647 6239 50 3 34634843326r 21 38 55 2 23 36 53 50

Triangular Numbers. The sum of the first ru numbers, I+2+3+4+..'+n = n(n+1Y2, is the nth triangular number. On the 8x8 board the sequence of triangular numbers has L0 members: 1, 3, 6, 10, t5, 2'1, 28, 36, 45, 55; and L0 is itself a triangular number. Dawson's tours 89-14 show these triangular numbers in triangular formation. However, if the number of triangular numbers is triangular this does not necessarily mean they can be arranged in a triangle in a tour, since the balance of even to odd may not be correct. There are 1,5 triangular numbers not geater than I21,, of which 8 are odd and 7 even, but a triangle requires 9 of one and 6 of the other. Puzzle 7: Form an LLxl1 knight's tour with triangular numbers in a rectangle 3x5.

89 Dawson 1935 810 Dawson 1935 811 Dawson 1935 RIz Dawson 1935 23263344 9 165712 43 18 31 60 33 16;s 58 312633101954438 4 1 56 9 51513447 t\ 32 29 24 27 56 43-10 47 30 61 44 17 s6/se/31 47 34 rr 3v2sv4 e s6 s3 57 8 3.1235 485350 /\ l\ 2s 22 31 34 4s6f$ s8 19 42 63 32 45 36 57 51 273245185550 7 42 2 5 105552134633 T\ f\ 30 3s zB4 60 ss 48 11 62 zs ror+{ s{st +t 35 12 3s/24 2e 46\37 52 s7 7 -/ t\ 58 1r\ 3645204914 zL6le 36 +gAo ss s1 9 20/t 28 15 2637 52 6328473651 6 4120 61 43 6 -21-28-15 32 lg I t\ ts\l 16 61 6t sl rz 3s 2 8 23f10 sr 11 19 rgr3 61 23 38 19\s8 s 59 40 61 44 37 18 27 24 \t ,5 t\ 63 20 3.s0f7 14 s s2 2L-10-3-6 27 1225 38 I- 62-15 -19 -3 -60-21 40 42 63 38 29 22 25 16 3r 2 17 6115 1 51 38 13 4 7 22rr24 3950 13 L1r7 2 61 2239 1 59 39 60 11 62 17 30 23 26

B13 Dawson 1935 814 Dawson 1935 B15 Dawson+Hall 1936 816 Dawson 1936 8 532447 2 51 3849 3942 9 1455 16 53 62 s0 15 62 27 60,\ 17 42 39 50 39 14 33 52 6L 12 63 2326 7 5237 4 11 40 +z/ob 63 17 63 48) 8 tl 56 28 49 16 41 38 59 18 ls 36 sl 38 L3'6}sr OO s4 s 46 zs.uf"o 3e 41 38 13 10 15 54 6L 52 M sr ze e{ q8 40 19 // le +cls 31// L7'32 59 62t11 27 zz ss{6 4s 40 s 61 12 7 U/Sg S\ 33 t8 57 2564291237113158 35 16 37 48 29 24 3L 54 -/ ll l. // nds-zsa{s6 63 31 4r 3724 1 6 45 58 51 32 52L324 7 30472015 4 4L 182358471025 /r // // 29 18 11 L4 35 44 57 60 2 s/26 23 3f 29 48 L9 1 8 3 3611345732 1203287305546 \t \/ 16 13 20 3t 62 59 42 33 25 36-3-28AL 46 31 50 4 53\./ 1023 6 55 162r 12 5\/ 2219415726 9 19 30 17 L2 13 32 61 58 427223530192047 9 2 5 5435223356 21 2 43 6 27 I 1556

Odd Primes. The numbers most resistant to being arranged in patterns are possibly the primes. There is only one even prirne, 2, and so in knight's tours it is inevitably distinguished from all the other primes by being on the opposite colour of square. In B15-81"6 the odd primes form a rectangle. Other Figurate Numbers. Figurate numbers are any that can be represented by arrays of counters in regular geometrical formations. BL7 shows 'double triangular numbers', i.e. of the form n(n+\). Bl"8 shows 'pentagonal numbers', of the f.orm n(3n-L\2, i.e. n2 + (n-I)n12, that is a square plus a triangle. B19 has 'cubes' in the comers, while B20 has 'octagonal numbers', which are of the form n(3n-2). This is a tour of squares and diamonds type. 18 FIGURED TOURS

Bl7 Dawson 1932 818 Dawson I91I B19 Jelliss 1985 820 Jelliss 1986 45 54 41,32 43 64 7 10 16 63 t1 55 44 47 20 49 8 11, 16 13 44 57 40 27 8 5 48574623602L 40334155 6 9 4 1 13 56 17 60 19 50 43 46 47t17 1039264356 4956 7 4 59204524 5346314263 2 11 8 62 15 64 57 51 45 48 2r 6 9 1245 5841 2825 6 9 58472613226r 34395247s6 5 623 57 12 61 18 59 22 53 42 1548 5 3829215542 55 50 3 10 19 6225 44 17 48 35 30 61 L2 57 21 36 33 58 1 5229 6 23 5037 16 3 5459223r 32 rr 54 63 42 27 38 17 38 29 16 51 20 23 60 13 11 2 3532 s 264128 17 4 49312330 6360 51 2 311135 184128 49 18 27 36 15 58 25 22 3437 4 9 303924 7 3651 2 t962533221 t2 33 61 53 30 39 16 37 28 37 50 19 26 2r 14 59 3 1031 3825 8 27 40 1 18355233206r6/} 1 52133415362940

Our frontispiece is a L2xl2 tour showing the Fibonacci numbers in a convex polygon. These numbers are formed by starting with L and 2 and adding the two preceding numbers to get the next. The sequence runs: L,2,3,5,8, L3,21,34,55,89, L44.T\e odd-odd links 3-5, l3-2I,55-89 in the polygon cannot of course be knight moves; my solution uses {L,1} fers moves for these links. Puzzle 8: Construct a L6-cell knight's tour showing the first four squares L, 4, 9, 16, forming a square. The board must be connected but cannot be a simple 4x4 square. This is a two-dimensional analogue of my cube tour E8. The last page of the booklet shows the four-dimensional analogue!

~~Figured Tours by Other Pieces~~

Wazir. In the simplest possible tour by the simplest possible piece, that of the wazir on the 2x2 board (F1) it so happens that the first two square numbers, L and 4, are in the same row. This may seem a trivial observation, but when we try to extend the feat to larger boards it begins to appear much more interesting. To have all the square numbers in a row cannot be accomplished in wazir tours on boards of sizes 3x3, 4x4 or 5x5, but when we consider the 6x6 board it is possible once again (F2) and moreover the solution is closed and unique!

F1 F2 F3 F3a 14 18 19 20 2t 22 23 68 69 70 71 72 73 74 75 76 77 23 t7 8 7 6 521 67 16 45 41 43 42 1L 10 39 78 16 I 36 1 42s 66 17 18 19 20 2L 22 23 38 79 15 10 35 2 326 654817 8 7 6 5 243780 t4 tr 31 31 30 27 &t49r69 C 1 4 2s3681 13 12 33 32 29 28 6350151099 2 3263582 62 5I 14 Ir 98 29 28 27 34 83 61 52 13 12 97 30 31 32 33 84 60 53 51 55 96 93 92 89 88 85 59 58 57 56 95 94 91 90 87 86

The next possible case is the 10x10, as in F3, though here the path can be varied slightly as in the diagram on the right. The problem is solvable on any square boards of side 4k-2. The next case is I4xL4. Pazzle 9. Construct a I4xI4 example. Here are some simple examples of the squares in other formations: F4-F5 show the squares in a square or rectangle; F6-F7 show the squares in a satin (one entry in each rank and file). The first satin is also a knight path. The idea of a wazir tour with squares in a knight path F8-F9 complements that of a knight tour with squares in a wazir path, as in the Carpenter task. F8 includes F6 in one corner, FIGURED TOURS T9

'We and the gnomon can be transposed (25 e1, 36 f3). then turn to some examples with arithmetical progressions (F10-F14). In F13 the wazir accomplishes a feat that the normally more manoeuvrable knight cannot.

F4 F5 F6 n F8 F9 123 4 123 4 2 1 1011 2 1 1011202r 14 1518r9202r t4t3 6 5 876 s tfyyn 'ri*: z/e-.g rzrszz 13 16 17z4z3zz 15t27 8 9 r2t3 16 44 t\, g.*ti'rls 14 {'4 r\* ft23 rz1\10\: s2627 161110 9 10 11 t4 15 5 6 1516 10hio rs s 6 rsloqz+ t, 8 ) :\s zs 32 31 30 2e z92s tY/ u 33 36 2e 33 34 35 3627 26 3 ',4 5 32 31 30 F10 Fl1 Ftz Fl3 319 r3r43 4 3 q@222r 4 5 18 193233 F14 3736 31 30 5 4 3 2 5 8 r2r5 2 5 2 5 242320 3 6 t7203134 38353229 6 7 2 1 6 7 LL161 6 C6O18@ 2 7 1621 30 35 3934332827/ \ 8/ 1 8 10 8 1716 1 8 1s222936 4041 422s26 910 109 7 9 /\l 11 rz@t4 rs 10 9 14232827 49 44 43 21 17 16 11 11 12 13 24 25 26 48 45 22 23 18 15 12 47 16 21 20 19 11 13

Emperor. W. E. l.ester showed that a knight + wazir (emperor) can perform the squares-in-a-row task without crossing its own path, F15-F18. He noted that the first of these is easily generalised to any larger board; on the 9x9 the number 21 will be on a9. Note that F18 incorporates a 4x4 solution.

~~F15 Lester 1938 F16 Lester 1938 FI7 Irster 1938 F18 Lester 1938~~

~~Prince: Irster also showed that a knight + fers (prince) can do a tour with the squares in sequence along a diagonal. A non-intersecting solution is impossible. F19, incorporates a 5x5 solution.~~

F19 Irster 1938 F20 Dawson 1933 Rook: The example F20 may be compared 28313445484341M with A5. [t shows the first nine odd 32 27 29 35 46 40 49 42 numbers forming a magic square in a 30 33 26 47 44 36 63 39 partial rook tour. The problem was 6 1517t9 50 6L 37 110 9 8 ',l given with a white R at e6 and white 3 7 5 1618 62 38 5l pawns at c5, d7, f4 with stipulation: 142 921 52 55 60 Pawns not moving, rook plays fewest 8 4 r2r0 22 59 53 57 moves, numbering squares I,2,3,... 1 r3232r 11 56 58 54 to form a 3x3 magic square. 20 FIGURED TOURS

~~Sources of Figured Tours~~

Months are shown by lower case roman numerals i - xii. Entries are in the form: 1(10) ii/32 p53 #329 (DS), which means: volume 1, issue 10, February 1932, page 53, problem number 329, @5 in our catalogue).

Euler (1759), Memoires de I'Academie Royale des Sciences et Belles Lettres, 15 pp.310-337, Berlin.: A1. Harikrishna MS 1871, quoted in S. R. lyer Indian Chess 1982#35,41,61-63, 67 (A22,21,23,8,6, 1,3). Brentano's Chess Monthly: G.E. Carpenter 1(1) v 1881 p36; S. Hertzsprung 1(5) ix 1881 pp248-9 (C5-8). Problemist Fairy Chess Supplement (PFCS) vol.l: l(10) iil32 p53 #329 (D5), p58 text @l); 1(11) ivl32 pp64-5 #376-81 (DZa, 1,5a,6a, B1, D68); l(I2) vrl32 p73 #448-51. @6b-c, 6,E2); l(I3) viiil3Z p81 #490-3 @3a, 4a, 3, 4); l(14) x/32 pp89-90 #5&-7 @75a,85, D75, E6); 1(15) xitl32 p97 #605-8 p73a, 2b, 73, 2); 1(16) iV33 p105 #658-61 (D7a, 11a,7,11); 1(17) ivl33 prI3 #716-9 (D15, E9-11). We Riennaise: l9lxl32 (N\ Evening Standard, lnndon: 1932 @17); 1933 (F20). PFCS vol.2:2(1) viii/33 rt #907-lO (D1a, 15a, 57a,57); 2(2) xl33 p14 #980-1 p29a,29); 2(3) xitl33 p24 #lD5t-4 @9a, l3a, 9, 13); 2(4) iil34 p35 #1132-5 @17a-b, 18a, 60a); 2(5) ivl3a pp55-6 #1303-6 (D17-L8, 60, 59); 2(7) viir?a p72 #1449-52 P53-4, 49, 69); 2(8) xl34 p83 #1525-8 (D61-a); 2(9) xitl3a pp9r-2 #1593-6 (D56, 55,51, 50); 2(10) iil35 pl04 #1,674-7 @12,8, 16, 30); 2(ll) ivl35 pllT #1705-6 @25-6), pl21 #1813-16 (D1.0, 1,4,58,65); 2(12)vil35 pl'?S #1834-7 (Dl9-20,33-34);2(I3)viiil35 pl35 #1917-20 Q23-6); 2(17) iv/36 p182 #2178-9 (A11, 815 with 816 in text); 2(18) vrl36 pl87 #2288-9 @27-8). Comptes Rendus du Premier Congres International de Rdcriation Mathdmatique (CIRM) Brussels, 1,935, edited by M. Kraitchik: Essay by Dawson on figured tours (A4, 14, 9-lO,84, 9-14, E2l-22). Fairy Chess Review (FCR) vol.3:3(1) viii/36 p3 #2352-3 p2l-2); 3(2) x/36 pl8 #2468-9 @32, 31); 3(3) xil36 p29 #2544-5 @37-8); 3(4) il37 p4r #2637-9 (D52,45-6);3(5) ivl37 p54 f2702-5 @al-a); 3(6) vrl37 p65 #2185-8 (D39-40, 35-6); 3(7) viirl37 p77 #2869-72 (D66-7, a7-8); 3(8) V37 pp86-7 #2933-5 (C9 version by G. Fuhlendorf, C9-10); 3(9) xir/37 p99 #2930-l (F15-16), #3037-8 (C11-12); 3(10) iil38 p110 #3108-9 (86, F19), #3035-6 @17-18); 3(11) ivl38 p120 #3180 @f; 3(12) vi/38 p130 #3251 @8); 3(13) viii/38 pl4o #3249 (Ar2), pl4l #3324 @70); 3(14) x/38 p150 #3388 (D74); 3(15) xii/38 p159 #3465 (D78). FCR vol.4: aQ) xirl39 p.43 #4135 text; 4(6) vt/40 p93 #4t35 @7). Brilish Chess Magazine: t/4I (BI8); rl Z @5); i/43 (E23). FCRvol.S: 5(I2) vil4a p96#6024-7 (E12,D77,87-88); 5(13) viiilaa p101 #6093-4 (E13, D81); 5(15) xiVzA pIrT #6230-1 (D83, 85); 5(16) iil4s #6305-6 (D71.,76);5(r7) ivla5 pr32 #6381-2 P82,86); s(18) vil45 pr4o #646t-2 (D84, 80). FCR vol.6: q4 xl45 p8 #6581-2 (D79, 89), p9 #6460 in text @15); 6(3) xiila5 pI7 #6636-7 (D96, 90); 6(a) irl46 p26 #6635 (816);6(5) ivla6 p33 #666I-4 @27,two similar tours by D. Nixon showing R and D, D94); 6(7) viii/a6 p5o #6849 in text @17), #6932-3 (D72,9r);6(14) x/47 pll0 #7474-5 (D95, 92); 6(15) xil47 prIS #7535-6 @97-8); 6(18) vil48 pl38 #7716-8 (D99, 100, 93). FCRvol.T: 7(6)vrl49 p.46 #8081 in text (A18); 7(12) vil50 pl10 #8534 in text (A19);7$Q xl50 pl22 #8537 in text @18); 7(15) xii/50 p131 #8723 (A20). FCR vol.8: 8(6) x/52 p 5 #9285 (Al7) Chessics: 1(5) 1978 p8 (A15, also version of C12 by A. S. M. Dickins); 2(21) 1985 p56 (F1-3, P9);2(22) 1985 p61 (819, Cl, 13, E14); 2(25) 1986 p106 @20, D68a, E8). The Games and Punles Journal: 1(8+9) 1988 p143 (F3a); 1(11) 1989 p178 Ga-D; 1(12) 1989 plga (F6-7); 2(r3) p20r @21). Mathemntical Spectum: '25(l) viiil92 pp16-19 (A2-3, C14, E1,9, F8, 13, P6-7). Figured ?ozrs (new): A7, t6,24,82-3, C2-4,15,83-4,20,24,28,F9-12,14, P1-5, 8. FIGURED TOURS 2r Puzzle Solutions

These tours are all of my own composition. In tours 2 and 7 underlining adds 100. Pl (1992): Open knight tour with squares in a closed 3-leaper path. Y2 (L994): 10x10 knight tour with arithmetical progression of co LL in a row. The Ll-move segments of the tour are alternately in the lower and upper ranks. The tour has maximum symmetry about the vertical axis, only the moves 45-46,50-51 and 55-56 having no reflection. PTi (1994): Tour of 8x8 less 4 cells, with 90' rotary symmetry, and ap of cn 5 on main diagonals.

~~122958 5 M1542 7 ss/4-ft3-eGt6\ 3q11 257 1443841 \ 3\6031 10272217 461 32)r 62 2r s6 47 40 2s\ 26 37 18 6{sz ss 48 23 I s{r so 63 za 3s 24 3e sL'ryyryj}36 f~~

P4: Knight tour 9x9 with L-L7 forming a star, even numbers on the knight wheel centred on L. P5 (199a): Closed tour with any eight successive odd or even numbers in a knight wheel. P6 (1991): Repeated (LL,22, 33, 44) and transposed digits (12121., 23132, 34143, 45154) in centre.

27506376948617123 38 59 40 57 36 9 26 29 61 lL26 49/62V5 24-7 60 41 56 37 L0 3r 28 358 s1 2{n6zs\4/273 60 39 58 25 30 27 78 6t)2 3s 4a 37 6{e 46 55 42 724 rsQzs38\31 zr@s 46 6I 20 1 66 7e) 4 4r:dr:q +45 SA 49 52 17 1 176 n3a/g){ 33)QVo7r 621350fie 15 219 80 1s-32 sry} 6e 1r3 41 514863143 18 516 31 54 81 68 t7 56 13 70 t9

W (1994): 11x11 knight tour with triangular numbers in a rectangle. Within the rectangle the triangular numbers, in order of magnitude, give a rook tour of alternating 2 and 1 unit moves. P8 (1996): knight tour of a 16-cel board with the square numbers in a square" P9 (1985): L4xl4 wazir tour with squares in a row.

7 LLM 43 J6 5 129 34 57 42 1J 12U, 8 35 56 13 16 13 10 Ls41l| 3 58 33 18 41 o+ Ogl. 36 17 t1 03 os le 461 9L 32 59 40 83 067e , { 27 20 4102 47 38 65 60 31 84398293, 76 19 26 oL48 99 88', 23 68 7I 988550811 52 25 62 4900e786. 95 72 69 FIGIJREDTOIIRS Postscript

My tour E8 (p.12) first published in Chessics in 1986, has the square numbers so arranged that the twelve lmight moves joining them delineate a picture of a cube. The heavier lines, joining the square numbers in order of magnitude, form a tour of the vertices of the cube. In two dimensions it is impossible to show the first four squares l, 4,9, 16, in a square of knight moves on tlre 4x4 board because no complete knight's tours are possible on that board. However, it can be done if we use an irregularly shaped board (of 16 cells, cormected dge to edge) as shown in P8. It me recently ocnrred to to see if this task could be exrtended to show a hypercube - the fourdimensional analogue of the cube - which has 16 vertices and therefore requires a tour of a l6x 16 board to show it. The result is shown below. Readers may like to try to improve the pattern of the connecting moves which is not particularly elegant.

~~Jelliss 1996 - the squares forming a hy'percube~~

~~I hope that this booklet will inspire readers to construct new examples of figured tours. There are many other possibilities yet to be investigated.~~

~~G, P. Jelliss September 1997~~