© 2001 by George Jelliss, 1 January 2001.
Results for Camel on 5×n boards, Giraffe 7×8 tour, Antelope 7×22 tour and further Zebra tours 5×16 added June/July 2014.
Page rewritten and most diagrams redrawn with larger cell size November 2022, to correspond to the 2019 PDF #10 version of this topic
and December 2022 to include results by Awani Kumar and Andrew Usher, notes on theory from Donald Knuth, T. H. Willcocks
and Nikolai Beluhov, and notes on nomenclature from Charles Gilman.
On this page we gather results concerning tours by the 'Big Beasts' which are the single-pattern {r,s} leapers whose moves are in other directions than those of the regular chess pieces, that is not along rook, bishop or nightrider lines. The best known Big Beasts are the Camel {1,3}, Zebra {2.3}, Giraffe {1,4} and Antelope {3,4}. We arrange these according to their longest coordinate, which is why Zebra precedes Giraffe. It is not just the greater length of the moves of a Big Beast that is significant but their direction of move. For example a Lancer {2,4} does not count as a Big Beast since it moves along knight lines and its touring properties are essentially the same as a knight, just restricted to alternate cells of one colour. Arithmetically this means the coordinates of the moves of a big beast must have no common factor, i.e. they are co-prime. Thus we also leave out {2,6}, {3,6} and {4,6}.
Some general results for {r,s} leapers were published by Donald Knuth in his (1994) paper on Leaper Graphs, which was motivated in part by my articles with T. H. Willcocks on the "Five Free Leapers" in Chessics #2 (July 1976) and #6 (August 1978), and by my article on "Generalized Knights and Hamiltonian Tours" written in 1992 but not published until three years later in the Journal of Recreational Mathematics (Vol.27 #3, 1995).
We give simplified statements of his theorems here (some of which duplicate results in our Theory pages):
Theorem 1. An {r,s} leaper (with 1≤r≤s) can reach any cell on an m×n board (with 2≤m≤n) if and only if
three conditions hold: (i) (r+s) and (r−s) have no common divisor, (ii) n≤2s, (iii) m≤(r+s).
Theorem 2. If an {r, r+1} leaper (with r>2) has a closed tour on a board m×n (with m≤n)
and if (2r + 1)<m then m≥(4r + 2).
Theorem 3. If an {r, r+1} leaper (with r>3) has a closed tour on a board (2n + 1)×n
then n≥(r^2 + 5r + 2) if r is odd and n≥(r^2 + 6r + 4) if r is even.
Theorem 4: An {r,r+1} leaper can make a closed tour on a square board of side 4r + 2.
Theorem 5. A {1,2k} leaper can make a closed tour on a square board of side 4k + 2.
Theorems 6 & 7. A {1,2k} leaper can make a closed tour on a rectangular board (4k + 1)×(4k + 2), but not on any smaller board.
Theorem 8. An {r,s} leaper cannot make a closed tour on a rectangular board m×n (with m<n) when 2s≤m and n<2(r+s).
T. H. Willcocks conjectured (Chessics 1976) that any {r,s} free-leaper can make a closed tour on a square chessboard of side 2(r+s). A proof of this was recently given by Nikolai Beluhov (arXiv:1708.05810 "A Proof of Willcocks's Conjecture" 2017). Knuth's Theorems 4 and 5 are special cases of this.
The shortest skew {r,s} leaper after the {1,2} knight is the {1,3} Camel whose move length is Ö10 = 3.16 approximately.
W. W. Rouse Ball Mathematical Recreations and Essays (1939 p.185) states that Euler applied his construction method "to find a reentrant route by which a piece that moved two cells forward like a castle and then one cell like a bishop would occupy in succession all the black cells on the board", in other words a camel tour, but he gives no reference. The earliest tour using camel moves that I have found is a magic four-camel tour by Charles Bouvier 1887, where the ends of the camel paths are joined by {0,6} and {0,7} rook moves [see the Augmented Leapers page]. The first purely camel tour I know of is that by T. R. Dawson in 'Caissa's Playthings' in Cheltenham Examiner 1913, where he also introduced the name camel. Frans Hansson (Fairy Chess Review Oct/Dec 1932 prob.535) and Maurice Kraitchik (Le Problème du Cavalier 1927) also provided examples.
The {1,3} leaper has many attributes similar to the knight except that it is confined to cells of one colour. In particular the angles between its moves are the same as for the knight and it has the same facility for forming a wider variety of circuits than longer leapers [see the Theory of Moves page for more on this].
By a camel tour on any board we mean a tour of all the cells of one colour when chequered.
No camel tours are possible on boards 2×n, 3×n, since these boards are too narrow to admit any 'vertical' moves.
The camel moves alternately to cells on the odd and even ranks of any board. On a chequered board with an even number of ranks the patterns formed by the light and by the dark cells are the same (after reflection top to bottom), so if one colour is tourable so is the other. On the 4×3 board, taking the top left cell to be light, and seeking a tour on the light cells, there are 2 light cells on each even rank, total 4, and 1 on each odd rank, total 2. The difference of these is 4 - 2 = 2. On a longer 4×odd board, each added pair of files adds 2 light cells to the even ranks and 2 to the odd ranks, so the difference remains at 2. The difference for a closed tour must be 0 and for an open tour 1. So no tour is possible. [This result was stated more briefly in previous versions, and I am not sure this more explicit version is any clearer!]
On the 4×4 board the six possible camel moves on squares of one colour form a closed circuit, omitting the two central cells of the same colour.
On the 4×6 board there is one closed camel tour given by T. R. Dawson (L`Echiquier Aug 1928 problem 285). This has Bergholtian symmetry (i.e. rotative, crossing at the centre) and by deletion of one move produces 7 reentrant tours (2 of which are symmetric) but in addition there are 4 non-reentrant open tours (1 symmetric), these four are shown below. The closed tour is the simplest n = 1 case of a general method noted by Dawson for a {1, 2n + 1} mover on cells of one colour on a board (2n + 2)×(4n + 2). See also the {1,5} and {1,7} cases below. This is the smallest board on which the camel has a tour of all the cells of one colour.
The 4×8 board has no closed camel tours since the moves at a3, g3 and at b2, h2 combine to form two 4-move circuits. However, 3 open tours (2 symmetric) exist, having one end-point in each circuit.
The 4×10 board admits no camel tours since the moves at b2, h2 and c3, i3 form two 4-move circuits (P and S), preventing a closed tour, while the moves at a1, a3 and j2, j4 form two paths (Q and R) with no connection between them, and to form an open tour connections must be in the sequence PQRS or PRQS, the ends being in P and S.
The 4×12 board admits 2 closed tours as shown (1 symmetric); open tours not studied.
The 4×14 board admits only one closed tour (asymmetric), but 18 symmetric open tours (4 a1–n4, 5 a3–n2, 2 b2–m3, 1 c1–l4, 2 c3–l2, 1 e1–j4, 1 e3–j2, 2 g3–h2); of these, 13 have the middle move vertical and 5 have it horizontal; there are also asymmetric open tours, not counted.
The 4×16 board has no closed tour, but there are symmetric open tours. One example shown.
The 4×18 board has some closed tours, symmetric (one shown) and asymmetric.
On 5xn boards no camel tours are possible except for 5×5 and 5×7. This is because each camel move is from odd rank to even rank or vice versa. So the number of cells of one colour on these ranks must be equal or differ by 1. On the 5×5 board equality occurs for the minority (non-corner) colour and on the 5×7 board the difference is 1, allowing open tours. For the 5×6 board the cells of one colour consist of 9 on the odd ranks and 6 on the even ranks, a difference of 3. Adding two extra files adds 3 to the odd ranks and 2 to the even ranks, so the difference is increased. So no tours are possible on boards 5×n where n is greater than 7.
On the 5×5 board there is a unique closed camel tour of the cells of the minority colour. This has biaxial symmetry. On the majority colour however there is no move at the centre cell and the moves on the eight outer cells form two circuits of 4 and one of 8 that snag at the four inner cells.
On the 5×7 board there are 13 symmetric open tours. They are oriented here with the central pair of moves all in the same direction. On this board there are 8 cells on the even ranks and 9 on the odd ranks, so the ends must be on the odd-rank cells.
Four with ends in the middle rank:
Nine with ends in the outer ranks:
On the 6×6 board there are 9 geometrically distinct camel tours of which 4 are symmetric (2 rotative, Eulerian type,
and 2 reflective, with diagonal axis of symmetry) 5 asymmetric.
Three 7×7 camel tours were given by M. Kraitchik Le Problème du Cavalier (1927).
On the 8×8 board there are 4 symmetric tours, all in Bergholtian symmetry. These were given by F. Hansson [Problemist Fairy Chess Supplement April and June 1933 problem 715]. Hansson also considered tours on an 8×8 cylinder board [see the Bent Boards section]. Dawson's 1913 tour mentioned in the introduction is asymmetric.
Tours on the cells of one colour in a rectangular board can be transformed into tours by related pieces on shaped boards. For example, moves of a bishop on the chessboard are equivalent to moves of a rook on a board of a serrated shape, representing the cells of one colour, rotated by 45 degrees. The same is true of other half-free movers, such as the camel, whose moves are equivalent to knight moves on this board, as pointed out by T. R. Dawson (Problemist Fairy Chess Supplement vol.1 #18 Jun 1933 p.125) and rediscovered by others from time to time since.
Conversely knight tours on square boards are equivalent to camel moves on serrated boards. Kraitchik (1927 p.70) constructed this remarkable double camel pseudotour on an 84-cell serrated board consisting of a 48-cell tour on the majority colour (except for the centre cell) and a 36-cell tour of the minority colour, both in 90° rotational symmetry. The equivalent knight tours are shown alongside on the right.
A zebra {2,3} move (length Ö13, approximately 3.61) is from corner cell to corner cell on a 3×4 board, so on a 3-rank board it has no moves at all on the middle rank, and on a 4-rank board it has only one horizontal move from the cells a2, a3, b2, b3 etc, and so no tours are possible. Tours only become feasible on 5-ranks.
On the 5×5 board the zebra has moves that form two octonary circuits, one of 8 moves through the corner cells and one of 16 moves, leaving only the centre cell unused.
On the 5×6: board the zebra is a free leaper on the, and on any rectangle of larger dimensions, i.e. it can reach any cell from any other, however it cannot tour that board. In fact we can prove:
THEOREM: The zebra cannot tour any board with sides 6, 7 or 8. Proof: (a) No closed tours are possible since the path of the zebra through a2 and b1 is determinate and passes also through d4, but its paths through y1, y2, z1, z2 (where y is the penultimate and z the last file) are also determinate and at least one of these on a 6, 7 or 8 file board also uses d4, so we would have three moves impinging on d4, which is impossible in a tour. (b) In an open tour we have end cells, so we could avoid such a triple point by deleting one of the three moves and taking the cell at its other end as an end point. However, we only have two end-points, and we always have either more than two triple points or two points where at least four moves converge, or one point where at least five moves converge, so two deletions are insufficient. These interference points I call snags.
On the 5×9: board, since each zebra move is from one colour to another no closed tours are possible on odd×odd boards, and an open tour, if possible, must start and end on cells the same colour as the corner cells (which we take to be white), so the moves through the black cells b3, h3 form a 4-move short circuit, and no open tour is possible.
On the 5×10 board the moves through b3, h3, c3, i3 form two 4-move circuits, so no closed tour is possible; and an open tour with one end in each circuit is impossible, since the moves through j2, j4, i1, i5 meet at f3, preventing moves c1-f3 or c5-f3, so that the moves through a2, a3, a4, b2, b4, c1, c5 form a 16-move circuit.
On the 5×11 board the moves forced at the black squares a2, a4, b1, b5 and k2, k4, j1, j5 fix the path through d2, d4, h2, h4 also; but now moves through f1, f5 must go to c3, i3, forming a 4-move short circuit.
On the 5×12 board the moves through a2, a4, b1, b5 use the cells d2 d4. The moves through l1, l5 use the cell i3. Moves through l3, k2, k4 form a six-move path h2 to h4, and the moves through c3 connect f1 to f5, but now f1, f5 can only connect to h4, h2 forming a ten-move circuit, since the other cells they can reach d2, d4 and i3 are already used. (A further argument is needed to eliminate an open tour.)
On the 5×13 board the moves through the minority colour squares a2, b1, a4, b5 and m2, l1, m4, l5 fix the moves through d2, d4, j2, j4; but now the moves through g2, g4 must go to e1, e5, i1, i5, where they join with the moves from b3, l3 to form an 8-move short circuit.
On the 5×14 board the moves through a3, b2, b4 use d1, d5, so the moves at g3, m3 form a 4-move circuit.
On the 5×15 board no closed tour is possible since it requires an even number of cells.
Thus the smallest rectangular zebra closed tour requires the surprisingly large 80-cell 5×16 board. This tour and two similar I found in 1992 and were published in the Journal of Recreational Mathematics (vol.27, No.3 p.191-200, 1995) in my article on 'Generalised Knights and Hamiltonian Tours'.
I give here numerical forms of the three symmetric closed tours possible on this board. These were Figures 6 a, b, c in the 1995 article. Other numerical ways of presenting these tours are possible, for example by renumbering the other ends of the central cross 41 to 40 as 1 to 80, but there are only three geometrically distinct cases.
36 57 52 63 04 73 12 31 42 71 06 79 16 21 26 47 59 66 33 54 61 38 01 68 09 40 49 14 23 44 19 28 64 03 74 35 56 51 70 05 76 11 30 25 46 07 78 17 53 62 37 58 67 32 41 72 13 80 43 20 27 48 15 22 34 55 60 65 02 75 10 39 50 69 08 77 18 29 24 45
36 55 60 65 04 75 10 31 42 69 08 79 18 29 24 47 53 62 33 58 67 38 01 72 13 40 49 20 27 44 15 22 64 03 74 35 56 51 70 05 76 11 30 25 46 07 78 17 59 66 37 54 61 32 41 68 09 80 43 14 23 48 19 28 34 57 52 63 02 73 12 39 50 71 06 77 16 21 26 45
66 47 53 57 72 05 62 79 42 21 74 11 26 37 32 13 45 54 69 50 59 64 01 18 77 40 03 28 35 16 23 30 56 71 06 67 48 43 20 73 08 61 38 33 14 75 10 25 51 58 65 46 53 78 41 04 63 80 17 22 31 12 27 36 68 49 44 55 70 07 60 39 02 19 76 09 24 29 34 15
More recent work on 5-rank boards has been done by Awani Kumar, published in the Swiss chess magazine idee & form. Issue 153 January 2022 and Issue 156 October 2022. His first article examines the longest paths possible on the 5×6 to 5×15 boards and includes two open tours 5×16 with partial magic properties and open tours 5×17, 5×18, 5×19 and a closed tour (not symmetric) on the 5×20 board. In the second article he notes that there are over 60 tours having three magic lines on the 5×16 board, and gives three examples. He also shows solutions 5×17, 5×18, 5×19, and a semimagic tour 5×20. He shows further examples 5×21, 5×22, 5×23 and 5×24, including semimagic on the even boards. He also combines pairs of the partially magic 5×16 and 5×18 tours to form semimagic tours 5×32 and 5×36. We show some of these tours here arranged according to the board sizes.
A. Kumar 5×16 symmetric open tour, diametrically opposite pairs of numbers adding to 81,
with three magic lines (sum 648 = 8 × 81), and two non-magic lines adding to 648 ± 8.
45 64 59 54 73 06 79 12 39 30 77 10 25 14 19 34 (640) 66 57 48 61 52 43 70 03 50 41 32 23 16 37 28 21 (648) 55 72 05 46 63 68 01 74 07 80 13 18 35 76 09 26 (648) 60 53 44 65 58 49 40 31 78 11 38 29 20 33 24 15 (648) 47 62 67 56 71 04 51 42 69 02 75 08 27 22 17 36 (656)
A. Kumar 5×16 asymmetric open tour with three magic lines (= 648) and two non-magic lines adding to 648 ± 4.
This combines with a copy of itself to form a semimagic tour 5×32: see below.
19 08 03 14 35 52 27 42 37 54 29 48 67 78 73 62 (648) 10 01 22 05 12 17 40 57 24 45 64 69 76 59 80 71 (652) 15 34 51 20 07 38 55 32 49 26 43 74 61 30 47 66 (648) 04 13 18 09 02 23 36 53 28 41 58 79 72 63 68 77 (644) 21 06 11 16 33 50 25 44 39 56 31 46 65 70 75 60 (648)
A. Kumar 5×16 asymmetric open tour with properties similar to the previous tour.
19 08 03 14 33 52 27 44 39 54 29 46 67 78 73 62 (648) 10 01 22 05 12 17 36 57 24 41 64 69 76 59 80 71 (644) 15 34 51 20 07 38 55 32 49 26 43 74 61 30 47 66 (648) 04 13 18 09 02 23 40 53 28 45 58 79 72 63 68 77 (652) 21 06 11 16 35 50 25 42 37 56 31 48 65 70 75 60 (648)
A. Kumar 5×20 asymmetric closed tour
Numerical form (100 shown as 00):
95 74 69 64 83 38 53 90 01 32 85 36 43 58 05 28 09 14 19 48 76 67 92 71 62 97 80 41 60 99 34 55 88 25 50 07 16 45 12 21 65 82 39 94 73 78 31 84 37 52 57 02 29 86 23 18 47 04 27 10 70 63 96 75 68 91 00 33 54 89 42 59 06 35 44 13 20 49 08 15 93 72 77 66 81 40 61 98 79 30 87 24 51 56 03 26 11 22 17 46
A Kumar 5×20 symmetric open tour
Numerical form which is semimagic: ranks all add to 1010 (100 shown as 00)
93 80 75 70 39 66 47 88 37 58 45 16 05 52 99 60 29 18 23 10 82 73 90 77 68 95 86 03 50 97 14 55 34 63 12 27 20 07 32 25 71 40 01 92 79 84 57 42 65 48 53 36 59 44 17 22 09 00 61 30 76 69 94 81 74 89 38 67 46 87 04 51 98 15 06 33 24 11 28 19 91 78 83 72 41 02 49 96 85 56 43 64 13 54 35 62 31 26 21 08
A Kumar 5×32 semimagic tour. Add 100 to underlined.
19 08 03 14 35 52 27 42 37 54 29 48 67 78 73 62 01 86 91 96 13 30 05 24 19 36 11 26 45 50 55 40 (2576) 10 01 22 05 12 17 40 57 24 45 64 69 76 59 80 71 84 93 98 89 82 03 16 33 08 21 38 59 52 43 48 57 (2576) 15 34 51 20 07 38 55 32 49 26 43 74 61 30 47 66 95 14 31 00 87 18 35 12 29 06 23 54 41 10 27 46 (2576) 04 13 18 09 02 23 36 53 28 41 58 79 72 63 68 77 90 81 02 85 92 97 20 37 04 25 44 49 56 39 60 51 (2576) 21 06 11 16 33 50 25 44 39 56 31 46 65 70 75 60 99 88 83 94 15 32 07 22 17 34 09 28 47 58 53 42 (2576)
The second 5×16 tour above has the property that if it is inverted along its middle row and then placed end-to-end with the original tour, then these two tours can be combined to form a semi-magic tour (SMT) having magic constant 2576 on a 5×32 board, as shown here. This was the first semi-magic tour of a leaper, other than knight, that was discovered on a board of any size (the 5×20 example was later).
This tour can also be linked with itself to get semi-magic tours on a 5×64 board. We have just to place two tours end-to-end and link cell 160 with cell 1. In this way, semi-magic tours on board sizes 5×32, 5×64, 5×96 … ad infinitum can be easily obtained.
On the 8×8 board the snag theorem above shows that no zebra tour, open or closed, is possible on the normal chessboard. The longest zebra paths achieved are 54 moves (55 cells) open, 52 closed symmetric rotary (Jelliss, undated) and recently 54 closed symmetric axial this is by Vaclav Kotesovec (Dual-free Leaper and Hopper Tours, Prague 2009, p.41. In Czech but with notes in English).
On the 10×10 board. A zebra closed tour on this board was found by A. H. Frost as long ago as 1886 (in M.Frolow, Les Carrés Magiques, Paris 1886, Plate VII). This subject was rediscovered by Kraitchik (1927 p.70-73) who gave two examples, open and closed, and details of how they were constructed by Euler's method. H. H. Cross gave an open tour in Fairy Chess Review February 1941 (problem 4709) - not shown here, which led T. H. Willcocks to compose a closed tour about the same time but which was not published until 1978 in Chessics. This led to a birotary example by W. H. Cozens in Chessics 1978 and one by myself, as shown here.
Tom Marlow (March 1998) applied the computer program he had used to count 10×10 knight tours with birotary symmetry to the zebra problem, with minor adaptations, and found there were only 6 geometrically distinct zebra tours of this type in all; these results were published in The Games and Puzzles Journal (vol.2 #16 1999 p.290). The other four are shown here.
To conclude this Zebra section: this page on Ed Pegg's MathPuzzle site provides proofs that there is no zebra open tour on boards 8×8, 9×9, 11×11, 12×12 (and so no closed tours 8×8 or 12×12).
Andrew Usher [2022] has constructed this 10×12 open Zebra tour, to show that a zebra tour is possible on this rectangular board.
Here is a numbered version of the tour (Andrew's original numbering was 00 to 119), but I prefer the more conventional 1 to 120. The middle move is 60-61. Underline adds 100. This is not magic, the rank and file sums vary.
37 90 85 54 05 68 05 28 83 52 47 74 92 03 26 87 14 01 72 19 50 07 70 45 55 04 67 36 89 84 53 48 75 06 29 82 86 13 38 91 04 27 06 69 46 73 20 51 25 88 93 02 03 18 15 30 71 44 49 08 66 35 56 13 12 33 00 61 20 81 76 07 39 02 15 58 11 64 09 22 79 18 31 98 94 11 24 17 60 41 96 77 16 09 62 43 57 12 65 34 01 14 19 32 99 08 21 80 16 59 40 95 10 23 10 63 42 97 78 17
A giraffe {1,4} move (length Ö17 approximately 4.12) needs a 2×5 board, so the smallest board on which it can move on every cell is 2×8.
On the 5×5 board it has 16 moves that form a circuit that tours all the edge cells.
The 5×8 is the smallest rectangle on which the giraffe is a free leaper, and three distinct tours are possible, all given by T. R. Dawson (L'Echiquier Dec 1930). These are formed from two half-board tours A and B, joined AA, AB, BB with common central link e1-d5.
T. R. Dawson (L`Echiquier, Jul 1928 problem 278) gave a general method for a {1,2n}-mover touring a board (2n+1)×(4n). For n = 1 we get the corner-to-corner 3×4 knight's tour, and for n = 2 the similar 5×8 giraffe tour. The next case is a 7×12 tour by {1,6}-mover. Kraitchik noted that numbers in the same rank are congruent mod 2n + 1.
Awani Kumar has published a study of partial tours by Giraffe on boards 5×n in idee & form #155 July 2022.
On the 5×16 board he gives a symmetric tour.
Like the similar zebra tours it has three magic lines adding up to 648 and two non-magic lines adding to 648 ± 20.
We show it in numerical form.
22 27 20 67 18 05 50 45 40 35 72 77 62 13 60 55 (668) 01 06 49 44 23 28 71 66 17 12 51 56 39 34 73 78 (648) 24 29 70 65 02 07 48 43 38 33 74 79 16 11 52 57 (648) 03 08 47 42 25 30 69 64 15 10 53 58 37 32 75 80 (648) 26 21 68 19 04 09 46 41 36 31 76 63 14 61 54 59 (628)
The table below, from the Kumar article, summarises the longest giraffe paths on 5×n boards for n = 5 to n = 20. It is surprising that none of the boards from 5×9 to 5×15 and then from 5×17 to 5×20 has a giraffe tour visiting all the cells.
Board Size (5xn) 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 Number of cells 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Longest path 16 22 31 40 41 44 46 49 58 64 72 80 82 86 92 94 Number of non-visited cells 09 08 04 00 04 06 09 11 07 06 03 00 03 04 03 06
On the 6×6 board the giraffe cannot move on the centre cells and its moves from the corners meet to form 4-move circuits, but on the other cells it has a 28-move qaternary tour with diagonal axes.
On the 7×7 board the giraffe has no move at the centre cell, but forms octonary circuits of 8, 16 and 24 moves which together cover the other 48 cells.
On the 7×8: board Andrew Usher [2022] notes that there are two half-tours. They are shown in diagram form below as A and B. These can be combined by a link e2-d6 to form three open tours, two symmetric AA, BB and one asymmetric AB. Only the first symmetric tour was previously shown here.
Andrew Usher [2022] also notes that on the 7×9 board a maximal tour of 62 cells is possible, omitting 1 cell, as shown in this example where the unused cell is in a corner.
The 8×8 board can be covered with 16 four-move giraffe circuits, analogous to the knight's squares and diamonds, but the question of whether a giraffe tour on the 8×8 board is possible puzzled investigators for many years. T. R. Dawson wrote in Cheltenham Examiner 22 May 1913: "I see no reason why a giraffe tour of the board cannot be made, analogous to the knight's, but have not succeeded in working it out at the present." Kraitchik in L'Echiquier (1927 p.257) says: "Mr T. R. Dawson believes that a tour of the 8×8 board by a (1,4) leaper is impossible". In Chessics 1976 and 1980 I gave proofs that a Giraffe closed or open tour is impossible, as shown below.
In L'Echiquier (May 1928 problem 267) S. Vatriquant gave a 63-cell tour (omitting h8) intended to be a giraffe tour, but move 55–56 is a camel move! Dawson corrected this error (L'Echiquier Dec 1930 p.1085) with a 63-cell open tour omitting h8 (or c3 by diverting the route g4-h8-d7). Fifty years later I published a 62-cell closed tour (omitting c3 and f3).
THEOREM: A closed giraffe tour of the 8×8 board is impossible. Proof: Letter the cells of the board ABCD as shown. Every A communicates by giraffe moves only with Bs, so in a closed tour every A must be preceded and followed by a B, but the number of Bs is the same as the number of As. This equality is possible only if the As and Bs form a closed circuit. But this would exclude the Cs and Ds. (Jelliss Chessics #2 1976).
THEOREM: An open giraffe tour of the 8×8 board is impossible. Proof: The previous argument to prove the impossibility of a closed tour shows that an open tour must be of the form ABABAB. ... AB–CD ... CDCDCD. Now combine the lettering with the numbering shown in the other diagram above (which chequers each quarter board, corners being the same colour 1). This has the property that cells A1 communicate only with cells B1. These cells are equal in number (8), so the A end of the tour must start on an A1. Similarly it must end on a D1. So the whole tour becomes A1B1...A1B1 – A2B2...A2B2 – C2D2...C2D2 – C1D1...C1D1. But the flaw in this scheme is that there is no move of the type B2 – C2 needed to link the two halves of the tour! (Jelliss Chessics #9 1980).
Quarter Tours 8×8. In October 2013, as reported in my online Jeepyjay Diary, I revisited the above proofs. This led me to find two 16-move closed tours of the A1-B1 cells and two 16-move closed tours of the A2-B2 cells, forming four closed quarter tours. In each case one tour is symmetric and the other asymmetric.
At the same time I combined two copies of a 32-cell path to form a Magic Two-Giraffe tour, where the link is a {0,7} rook move, but have since discovered that Prof C. E. Reuss constructed a Magic Four-Giraffe tour with {0,4} and {07} links as long ago as 1887 [see Augmented Leapers page].
Half Tours 8×8. There are also 16-cell open paths of course. Two such paths can be joined to form a 32-cell open path
as in this 32-cell giraffe open path given by E. Huber-Stockar (Fairy Chess Review 1933 problem 3938):
On the 9×9 board, an open giraffe tour was given by T. R. Dawson (in The Problemist Jul 1926 problem 41) and the same tour is in Kraitchik (1927 p.73).
On the 9×10 board E. Huber-Stockar (FCR Feb/Apr 1945 problem 6304) gave this unique axial symmetric giraffe tour
On the 10×10 board. This case was solved with an asymmetric closed giraffe tour by A. H. Frost as long ago as 1886 (M. Frolow, Les Carrés Magiques, Paris 1886, Plate VII). Quaternary examples have been shown by Kraitchik (1927 p.73) and by W. H. Cozens (Chessics 1978) and by T. W. Marlow (1998). Marlow found 50 solutions, but looking back in my records he does not seem to have sent me a printout of these.
A Giraffe symmetric open tour of a 128 cell board, equivalent to a {3,5} tour on a board 16×16.
The antelope or {3,4}-mover has move length Ö(9+16) = Ö25 = 5 units. The {0,5} move is also of length 5 units and a piece combining the two moves {0,5} and {3,4} is called a 'Fiveleaper'. This combined piece (invented as a fairy chess piece by George Leathem in Chess Amateur in the 1920s) is important in its own right. For more on this see the separate Fiveleaper page. The separate name 'antelope' for the {3,4} mover was suggested by A. I. Houston (Chessics #2 1976 p.2).
The antelope needs at least a 4×5 board to move. On a 6-rank board the antelope has at most one move from the end four cells of central ranks, so no tours are possible.
On the 7×7 board the antelope has no move at the centre cell, but forms octonary circuits of 8, 16 and 24 moves which together cover the other 48 cells (similar to the Giraffe).
On the 7×8: board it is a free leaper but cannot tour it; W. Langstaff (FCR problem 5798, Dec 1943 p.69 and Feb 1944 p.78) gave a 47-move path.
The antelope requires a 7×22 board (154 cells) for a closed tour (and probably for an open tour also). Here is the only symmetric closed tour I can find. It is of Eulerian type, not passing through the centre point. Diametrally opposite numbers differ by 77. It is numbered from the Bergholtian origin for comparison with the zebra and giraffe tours on a minimal rectangle (5×16 in those cases).
Numerical form:
45 146 97 90 107 6 59 134 53 114 153 88 39 128 55 138 81 32 25 18 71 124 144 99 112 151 92 105 8 43 140 1 64 119 86 41 126 79 34 23 66 27 16 73 101 110 3 48 149 94 103 142 77 62 131 50 117 10 75 36 21 68 121 84 29 14 108 5 60 133 46 147 96 89 38 129 58 135 52 115 12 19 70 123 56 137 82 31 91 106 7 44 145 98 113 152 87 40 127 54 139 154 65 26 17 72 125 80 33 24 150 93 104 143 100 111 2 49 118 9 42 141 78 63 120 85 28 15 74 35 22 67 47 148 95 102 109 4 61 132 51 116 11 76 37 130 57 136 83 30 13 20 69 122
The above tour was published in my article on 'Generalised Knights and Hamiltonian Tours' in the Journal of Recreational Mathematics vol.27, No.3 p.191-200 (1995).
On the 8×8 board: F. Douglas (Problemist Fairy Chess Supplement, Problem 403, April 1932) noted that the {3,4} "may describe two doubly symmetrical closed tours of 44 moves". Half-tours are shown, the other halves reflecting these in the a1-h8 diagonal. These have quaternary symmetry with diagonal axes. W. Langstaff (FCR, Problem 5589, 1943) gave a 50-move open path; and later (Problem 5732, October 1943 p.61 and December 1943 p.70) a 52-move solution, but these were surpassed by some 54-move examples by A. H. Haddy and T. H. Willcocks (FCR, Problem 5817, February 1944 p.74 and April 1944 p.85; Problems 6095-6, August 1944 p.101) Willcocks later gave a closed solution.
On boards of 8 to 13 ranks: As for the zebra, we can prove, by considering the formation of 'snags' that no antelope tours are possible on boards 8×n to 12×n. On the 13×13 board its moves through a3, b2, b3, c3, and cognate cells form three short circuits, so no tours are possible.
On the 14×14 Board T. H. Willcocks gave a quatersymmetric {3,4} tour in Chessics 1978.
For reasons of clarity, the hand-drawn diagrams used there only showed one quarter of the path. Now that drawings can be made with computer precision we are able to show the full tour with reasonable clarity.
Naming of Leapers. The {1,6} leaper has recently acquired the name of Flamingo among problemists, for instance in the Glossary of Fairy Chess Definitions published by the BCPS in 2018. The leapers with coordinate 5 are not mentioned there. However, Charles Gilman at the chessvariants website offers some partially systematic names. We list them here and give explanatory notes below. Zemel {1,5}, Satyr {2,5}, Gimel {3,5}, Rector {4,5}, Flamingo {1.6}, Parson {5,6}, Namel {1,7}, Stork {2,7}, Samel {3.7}, Ox {4,7}, Famel {5,7}, Curate {6,7}. Bittern {1,8}, Huckster {3,8}, Agronome {5,8}, Deacon {7.8}, Remel {1,9}, Albatross {2,9}, Outsetter {4,9}, Somel {5,9}, Bimel {7,9}, Verger {8,9}, Macaw {1,10}, Lyrebird {3,10}, Runaway {7,10}, Zany {9,10}, Pamel {1,11}, Quail {2,11}, Xomel {3,11}, Mystic {4,11}, Humel {5,11}, Scryer {6,11}, Lamel {7,11}, Flybynight {8,11}, Mamel {9,11}, Eccentric {10,11}, and others.
The "-mel" suffix from Camel indicates those with odd coordinates, the Ca showing its 45 degree rotation relationship with the Cavalier (i.e. Knight). Similarly the Zemel is related to the Zebra, the Gimel to the Giraffe, the Namel to the Antelope, and so on. The ecclesiastical names Rector etc are on the principle that they are leapers whose coordinates are nearly the same, so are akin to Bishops. The bird names are on the similar principle that they are pieces moving almost like Rooks!
In the list below I have adopted some of these names, but prefer Gazelle for the {2,5} since I have used it before, and doubt whether many of the other names will get accepted as standard.
The {4,7} and {1,8} movers are both of length Ö65, and similarly the {6,7} and {2,9} movers are both Ö85 leapers. See the Double-Pattern Leapers page for tours using these double-pattern fixed-distance leapers.
This example on a 6×10 board is the n = 2 case of the general method noted by T. R. Dawson for a {1, 2n + 1} mover on cells of one colour on a board (2n + 2)×(4n + 2), similar to the closed tour by camel on 4×6 board shown earlier.
7-Rank Boards. F. Douglas PFCS problem 1415 (1934): "The root-29 leaper, moving a1-c6 and the like may play a 42-move closed tour on the 7×8 board. FD gave all possible closed paths of the leaper up to 42 moves. The longest tour on the 7×7 board is the 32-move one.
On the 14×14 board T. H. Willcocks gave a quatersymmetric tour in Chessics 1978. For reasons of clarity, the hand-drawn diagram used there only showed one quarter of the path.
This tour (Jelliss 2018) by a {3,5} leaper (Ö34 mover) on the cells of one colour on a 16×16 board is related to an equivalent Giraffe tour on a serrated board shown above.
A {4,5} leaper tour on a board 18×18 is given as an example of an {r, r+1} mover on a square board of side (4r + 2) in Donald Knuth's paper on Leaper Graphs. We show here the first half of the tour with cells numbered 1 to 163. The second half of the tour rotates this path 180 degrees. [Note that the numbering in the original paper is from 0 to 400, since the numbers there are shown in base 9. As usual we prefer the traditional numeration.] It will be seen that a large part of the pattern is taken up by zigzags of moves within each quarter. Add 100 to underlined numbers. It appears that a tour with quaternary symmetry (by 90 degree rotation) is impossible in this case.
The name Flamingo for this piece seems to have been recently introduced. This tour on the 7×12 board is the n = 3 case of Dawson's general method for a {1, 2n} mover touring a board (2n + 1)×(4n). Compare 3×4 knight and 5×8 giraffe tours.
Board 8×14: T. R. Dawson. General method for a {1, 2n + 1} mover touring cells of one colour on a board (2n + 2)×(4n + 2). This is the n = 3 case. See {1,3} and {1,5} cases above.
The name Namel is derived from Antelope-Camel in the Gilman system as noted above.
A {1, 10} leaper tour on a board 22×22 is given as an example of a {1, 2k} mover on a square board of side (4k + 2) in Donald Knuth's paper on Leaper Graphs to illustrate his Theorem 5. We reproduce his numbering here since the tour cannot be shown clearly in a small graphical diagram. To give a standard numbering 1-484 the 0 at top left can be replaced by 484 (= 22×22) since this is a closed tour.
0 143 386 139 384 133 378 127 372 457 360 269 172 257 18 251 12 245 6 243 2 145 361 270 171 258 17 252 11 246 5 242 1 144 387 140 383 134 377 128 371 458 359 268 54 467 388 465 390 463 392 461 394 459 396 185 170 187 168 189 166 191 164 241 162 333 397 274 399 196 401 194 403 192 405 238 55 332 81 320 83 318 85 316 87 314 355 184 158 331 80 307 94 309 92 311 90 313 354 275 446 231 412 233 410 235 408 237 56 471 449 276 445 230 415 228 417 226 419 224 157 472 79 306 97 304 99 302 101 300 353 180 156 327 78 293 108 295 106 297 104 299 450 179 444 217 426 219 424 221 422 223 60 473 451 280 443 216 429 214 431 212 433 210 61 326 77 292 111 290 113 288 115 286 349 178 152 35 76 37 74 39 72 41 118 285 348 281 442 203 440 205 438 207 436 209 62 477 347 174 345 26 343 28 341 30 339 32 151 478 65 48 67 46 69 44 119 284 455 282 150 483 142 385 138 379 132 373 126 367 456 173 262 19 256 13 250 7 244 3 146 479 271 362 263 22 259 16 253 10 247 4 335 52 141 382 135 376 129 370 123 366 267 358 468 53 466 389 464 391 462 393 460 395 272 357 186 169 188 167 190 165 240 163 334 161 273 398 197 400 195 402 193 404 239 406 469 160 321 82 319 84 317 86 315 88 183 356 330 159 322 95 308 93 310 91 312 89 182 447 198 413 232 411 234 409 236 407 470 57 277 448 199 414 229 416 227 418 225 420 329 58 323 96 305 98 303 100 301 102 181 352 328 155 324 109 294 107 296 105 298 103 278 351 200 427 218 425 220 423 222 421 474 59 279 452 201 428 215 430 213 432 211 434 475 154 325 110 291 112 289 114 287 116 177 350 34 153 36 75 38 73 40 71 42 117 176 453 202 441 204 439 206 437 208 435 476 63 175 346 25 344 27 342 29 340 31 338 33 64 49 66 47 68 45 70 43 120 283 454 482 149 50 137 380 131 374 125 368 121 364 265 24 261 20 255 14 249 8 337 480 147 363 264 23 260 21 254 15 248 9 336 481 148 51 136 381 130 375 124 369 122 365 266