Fiveleaper Tours

by T. W. Marlow and G. P. Jelliss, May 2002. Further historical examples added July 2014.
Further revisions March 2018 including the work of J. D. Beasley.

Sections on this page: — Introduction — Fiveleaper Tours on Small Boards — Double 8×8 Tours — Other Fiveleaper Tours 8×8
— Magic Fiveleaper Tours 8×8: Notes — The 42 Closed Magic Tours — The 16 Open Magic Tours — Fiveleaper Pseudotours 8×8

é Introduction

Just as the knight makes moves of length root-5 that have coordinates {1,2}, a fiveleaper is a type of generalised knight that makes moves of length 5 units, with coordinates either {0,5} or {3,4}. I'm not sure when the fiveleaper was first introduced as a fairy chess piece, but T. R. Dawson gave an analysis of multipattern fixed-distance leapers, of which the fiveleaper is the simplest example, in Chess Amateur August 1925. For more details see the section on Compound Leapers on the Theory of Moves page. The leaper having only the {3,4} move is known as an Antelope and some results using it, including a tour on the 14×14 board, are given on the Longer Leapers page. Because of difficulty in showing fiveleaper tours clearly in graphical form, they are given here as numerical arrays.

The catalogue of magic fiveleaper tours by T. W. Marlow, completed in 1990, is published in full here for the first time.
This work was previously reported briefly with a few examples in The Probemist and Variant Chess in 1991.

The further examples by O. E. Vinje and E. Huber-Stockar, and an early one of my own were added in 2014.
The latest revision (2018) makes some corrections and adds details of the work by John Beasley (2009, 2010).

é Fiveleaper Tours on Small Boards

T. H. Willcocks, Chessics (#24 Winter 1985 p.93); Open 5-leaper tour on 7×7 less centre cell.

04 23 46 27 42 11 02

25 40 29 36 19 44 09

16 13 34 07 38 17 14

21 32 01 05 22 31

48 43 10 03 24 47 28

37 18 45 26 41 12 35

06 39 30 15 20 33 08

T. W. Marlow Chessics (#24 Winter 1985 p.93); Closed 5-leaper tour on 6×9 board.

54 21 14 31 18 11 28 15 52

23 50 07 46 33 04 49 08 45

36 43 26 39 02 35 42 25 38

19 12 29 16 53 20 13 30 17

32 05 48 09 22 51 06 47 10

01 34 41 24 37 44 27 40 03

This is symmetrical about the horizontal axis, as shown by the pairs of numbers adding to 55, so that the tour is semimagic, adding to 165 in the columns.
Marlow indicated that this board can be shown to be the smallest rectangular area, even for an open tour.
On rectangles with fewer cells some on the middle lines will have only one move or none available,
as in the above 7×7 example where there is no exit from the central cell.

é Double 8×8 Tours

In Variant Chess (v.1 #6 Apr-Jun 1991 p.75) I made the following observation: “Since the fiveleaper has four moves at every square of the 8×8 board it follows that in every closed tour the unused moves are also two at every square, and therefore form either a tour (is this possible?) or a pseudotour (i.e. a set of closed circuits). To use network-theory terminology, this would be a pair of Hamiltonian tours that together form an Eulerian tour. A trivial example of this is provided by the moves of a wazir on a 2×2 torus.” The term ‘Eulerian tour’ is used here in the sense of a path that uses every branch of a network once.

It is only in the course of revising this page that I found that this problem was in fact solved long ago by Maurice Kraitchik in what is probably the first fiveleaper tour of the 8×8 board constructed. It appears in his little book Le Probleme du Cavalier (1927 p.74). He mentions there that the fiveleaper has four moves available at every cell, but omits to point out that the unused moves in this case also form a tour. I show the tour as given and alongside it the complementary tour.

M. Kraitchik, Le Probleme du Cavalier 1927.

5-leaper double tour, part 1

40 33 16 59 62 13 34 47

45 26 51 06 29 44 23 52

64 11 56 19 38 03 08 57

31 14 35 48 41 32 15 36

28 61 22 53 46 27 60 21

39 04 17 58 63 12 05 18

42 25 50 07 30 43 24 49

01 10 55 20 37 02 09 54

5-leaper double tour, part 2

06 63 58 15 24 05 08 59

41 38 29 26 61 48 39 36

18 51 10 43 34 19 50 53

23 12 45 02 55 32 21 16

14 47 04 07 64 57 30 25

27 62 49 40 37 28 09 60

42 33 20 17 52 11 44 35

01 56 31 22 13 46 03 54

The first tour uses 36 lateral and 28 skew moves, while the complementary tour uses 12 lateral and 52 skew moves.
There are 48 lateral moves and 80 skew moves on the board, to be shared between the pair of tours.

My question of whether such a double tour is possible was in fact answered in the affirmative by Tom Marlow in a letter to me of 17 November 1991, but due to an oversight his result was not published until ten years later, in the last issue of The Games and Puzzles Journal (v.2 #18 Mar 2001 p.347). The following is Marlow's solution; in his own words:

“The 5-leaper has exactly four moves available on every square of the 8×8 board. In all there are 128 leaps, each being possible from either end. The two closed tours below make use between them of all these leaps. The method of construction was to build a tour starting at a1 and at each leap to mark as unavailable the corresponding leap after 180 degree rotation; e.g. the opening a1-a6 barred h8-h3 and h3-h8. When the tour was complete the same route, rotated 180 degrees, could be travelled using the barred leaps. That tour was then renumbered to start at a1.”

T. W. Marlow, Games and Puzzles Journal 2001.

5-leaper double tour, part 1

20 47 62 55 06 21 46 63

31 42 57 50 11 34 29 44

02 59 16 09 52 03 36 17

13 22 39 26 19 14 23 38

54 05 28 45 64 61 56 07

51 48 35 30 43 58 49 10

32 41 24 37 12 33 40 25

01 60 15 08 53 04 27 18

5-leaper double tour, part 2

46 37 60 11 56 49 04 63

39 24 31 52 27 40 23 32

54 15 06 21 34 29 16 13

57 08 03 64 19 36 59 10

26 41 50 45 38 25 42 51

47 28 61 12 55 48 05 62

20 35 30 53 14 07 22 33

01 18 43 58 09 02 17 44

In Marlow's method of construction the complementary tour is in fact the same geometrical tour rotated by a half-turn and renumbered from a1.
For each pair of diametrally related moves one is included and the other excluded, so any such tour must have 24 lateral and 40 skew moves.

Marlow's work was followed up by John D. Beasley in articles in Variant Chess: ‘Complementary five-leaper tours’ (#62 Oct 2009 p.131) and ‘Complementary five-leaper (and other) tours with rotational symmetry’ (#64 Aug 2010 p.232-233). He describes the Marlow tour as ‘rotationally antisymmetric’. Instead he looks for tours unaltered by a half-turn (i.e. rotationally symmetric), but in which a quarter turn gives the complementary tour, and shows the following example. (Note however that the first tour has been cyclically renumbered to start at a1, instead of a8, to conform with the other examples above.)

J. D. Beasley, Variant Chess 2010.

5-leaper double tour, part 1

28 31 34 13 08 27 30 33

11 36 57 16 47 10 37 56

54 19 50 23 44 53 20 49

07 26 03 64 61 06 41 14

46 09 38 29 32 35 58 39

17 52 21 12 55 18 51 22

24 05 42 15 48 25 04 43

01 62 59 40 45 02 63 60

5-leaper double tour, part 2

06 29 22 51 12 59 16 33

03 10 57 14 31 24 41 36

64 47 26 43 08 55 62 39

45 20 17 34 05 28 21 18

50 53 60 37 02 49 52 13

07 30 23 40 11 58 15 32

04 09 56 63 46 25 42 35

01 48 27 44 19 54 61 38

By applying a computer programme John found 224 tours of this new complementary type, showing the above example in print. A text file listing all 224 tours is available on the JSB website: http://www.jsbeasley.co.uk/puzzles/fiveleapertours.txt.

An explanatory note from the website: “In this file, each of the 224 geometrically distinct tours appears eight times (it can be flipped about the leading diagonal, it can be numbered in either direction, it can be rotated by 90 degrees to give the complementary tour, and each of these transformations can be applied independently). Thus tour 1g is tour 1a flipped about the leading diagonal, tours 1c and 1f are tours 1a and 1g numbered in the other direction, and tours 1e, 1d, 1h, and 1b are tours 1a, 1g, 1c, and 1f respectively rotated through 90 degrees and renumbered to put 1 at the top left corner.”

é Other Fiveleaper Tours 8×8

Using the above-mentioned computer programme John Beasley found that there are 125217 rotationally symmetric fiveleaper tours, of which 373 are also laterally symmetric (in other words they have biaxial symmetry), though he points out that this work has not been independently verified.

S. H. Hall had proposed the fiveleaper tour problem in Fairy Chess Review (Dec 1938 problem 3463) and the following solution by O. E. Vinje was published the next year. It has axial symmetry (not asymmetry as was stated here previously).

O. E. Vinje, Fairy Chess Review (Nov-Dec 1939). Fiveleaper closed axi-symmetric tour 8×8, with 20 straight and 44 skew leaps.

19 35 35 60 05 30 11 46

44 22 22 17 48 43 62 21

57 32 09 14 51 56 33 08

06 29 12 39 26 53 36 59

49 42 63 20 45 02 23 16

18 55 34 61 04 31 10 47

27 52 37 58 07 28 13 38

40 01 24 15 50 41 64 25

This was originally numbered 0 to 63 beginning at a1. However I have renumbered it from b1 to g1 (the ends of one of the two symmetric cross-axis rook moves) all pairs of numbers on either side of the vertical axis add to 65, which means it is semimagic (adding to 260 in every rank). If numbered 1 to 64 from a1, as in the other examples above, 25 pairs of numbers add to 51 and 7 pairs to 115 and the tour is not semimagic in this numbering.

E. Huber-Stockar, Fairy Chess Review Aug 1942. Fiveleaper closed symmetric tour 8×8 with 24 straight and 40 skew moves.

57 34 63 20 13 50 35 64

22 15 60 43 06 29 48 55

39 10 27 46 53 40 09 26

12 51 36 01 24 17 62 37

05 30 49 56 33 04 19 44

58 41 08 21 14 59 42 07

23 16 61 38 11 28 47 54

32 03 18 45 52 31 02 25

The solution to this was given in coordinate form: d5, g1, b1, ..., b4, f1, a1, and so on to e4 carrying on in diametral symmetry. This tour seems to have been published twice in FCR as I have a note of another version numbered 0, 1, ... 63 beginning a1-f1.

G. P. Jelliss, constructed 22 January 1973, unpublished. Fiveleaper closed tour, with 37 straight and 27 skew leaps.

10 21 40 29 52 09 20 63

61 50 13 06 43 32 49 12

04 57 24 37 16 03 56 27

45 34 19 64 59 46 35 18

42 31 54 11 62 41 30 53

07 22 39 28 51 08 23 38

60 47 14 05 44 33 48 15

01 58 25 36 17 02 55 26

This was one of my earliest compositions. It appears to be an attempt to show the maximum of rook moves.

G. P. Jelliss, Chessics vol.1 #5 1978 p.8. Fiveleaper centro-symmetric closed tour, with 32 straight and 32 skew leaps.

60 31 34 17 08 57 30 33

19 36 53 14 47 22 37 52

10 45 26 41 50 11 44 27

07 56 03 64 61 06 55 16

48 23 38 29 32 35 24 39

59 12 43 18 09 58 13 42

20 05 54 15 46 21 04 51

01 62 25 40 49 02 63 28

G. P. Jelliss, unpublished, constructed 22 March 2018. Fiveleaper closed tour with biaxial symmetry (12 rook and 52 skew leaps).

03 48 29 22 11 04 49 30

64 27 20 41 56 13 06 33

25 18 51 58 39 46 15 08

10 53 60 31 02 37 44 23

55 12 05 34 63 28 21 42

40 47 14 07 26 19 50 57

01 38 45 24 09 52 59 32

62 17 36 43 54 61 16 35

The tours in this section are not of the double type. The unused moves form pseudotours.

é Magic Fiveleaper Tours 8×8

Notes

The following text and results are by Tom Marlow, September 1990.

The diagrams show 58 five-leaper magic tours. All are magic in the sense that all ranks and files sum to the magic constant of 260. Numbers #5 and #47 [red diagrams] are fully magic because additionally their diagonals have the same sum. (These two were published in The Probemist March 1991, and the other results, with two examples, were reported in Variant Chess, issue 6, April-June 1991, p.75.)

The method of construction is to find sequences of 32 five-leaper steps that, when reflected in the horizontal axis, cover the remaining 32 squares of the board. Then if the 32nd square is on the second or seventh rank its reflection is five squares away and the second half of the tour proceeds in reverse order to the first half. [The cells containing the numbers 1, 16, 17, 32, 33, 48, 49, 64 are highlighted.] The result is that all vertical columns sum to 260 because each consists of four reflecting pairs such as (64,1) or (60,5) which each sum to 65. It remains to find cases where the horizontal rows also total 260.

42 of the tours begin on the second rank so are closed, i.e. the end is one five-leaper move from the start. Consequently they can be renumbered from 1 to 64 in the sequence 32, 31, ..., 1, 64, ..., 33 and in most cases remain magic. 34 cases, marked («) [blue diagrams] are unchanged by this transformation because the 32 square sequence is symmetric about the vertical axis. Of the rest, 7 marked (*) remain magic. The exception is number #20. Number #5 which is fully magic is in the (*) category and remains fully magic under the transformation.

Furthermore, tours #12 and #15 [mauve diagrams] can be renumbered from 1 in the sequence 49, ..., 64, 1, ..., 48 when they remain magic and have the («) category.

é The 42 Closed Magic Tours

#1 *
11	52	21	42	07	60	29	38
64	19	16	09	40	33	62	17
55	14	31	02	47	50	35	26
24	59	04	37	12	45	22	57
41	06	61	28	53	20	43	08
10	51	34	63	18	15	30	39
01	46	49	56	25	32	03	48
54	13	44	23	58	05	36	27

#2 «
35	48	25	12	21	08	49	62
64	27	10	51	46	23	06	33
29	18	41	60	37	56	15	04
20	43	58	31	02	39	54	13
45	22	07	34	63	26	11	52
36	47	24	05	28	09	50	61
01	38	55	14	19	42	59	32
30	17	40	53	44	57	16	03

#3 *
19	42	61	14	35	20	07	62
64	25	16	37	12	33	56	17
27	22	05	10	39	44	59	54
52	31	08	47	02	41	50	29
13	34	57	18	63	24	15	36
38	43	60	55	26	21	06	11
01	40	49	28	53	32	09	48
46	23	04	51	30	45	58	03

#4 *
13	62	35	20	09	24	61	36
64	33	22	15	58	39	18	11
51	06	27	46	55	42	05	28
08	25	48	53	02	31	44	49
57	40	17	12	63	34	21	16
14	59	38	19	10	23	60	37
01	32	43	50	07	26	47	54
52	03	30	45	56	41	04	29

#5 * diagonals magic
03	38	59	22	11	30	37	60
64	25	08	45	52	33	24	09
19	48	15	42	55	18	49	14
12	31	36	61	02	39	58	21
53	34	29	04	63	26	07	44
46	17	50	23	10	47	16	51
01	40	57	20	13	32	41	56
62	27	06	43	54	35	28	05

#6 *
03	38	51	58	23	30	11	46
64	25	16	05	44	33	56	17
61	28	13	08	41	36	53	20
22	31	10	47	02	39	50	59
43	34	55	18	63	26	15	06
04	37	52	57	24	29	12	45
01	40	49	60	21	32	09	48
62	27	14	07	42	35	54	19

#7 «
51	48	29	08	25	04	49	46
64	27	34	23	10	63	06	33
13	54	21	58	39	12	43	20
56	03	60	15	18	37	30	41
09	62	05	50	47	28	35	24
52	11	44	07	26	53	22	45
01	38	31	42	55	02	59	32
14	17	36	57	40	61	16	19

#8 «
57	38	29	14	19	04	59	40
64	17	34	21	12	63	16	33
09	54	23	50	47	10	43	24
52	03	60	07	26	37	30	45
13	62	05	58	39	28	35	20
56	11	42	15	18	55	22	41
01	48	31	44	53	02	49	32
08	27	36	51	46	61	06	25

#9 «
19	48	13	36	61	20	49	14
64	11	34	27	06	63	22	33
09	58	25	42	55	08	39	24
60	03	44	15	18	53	30	37
05	62	21	50	47	12	35	28
56	07	40	23	10	57	26	41
01	54	31	38	59	02	43	32
46	17	52	29	04	45	16	51

#10 «
23	40	11	52	45	22	57	10
64	27	34	15	18	63	06	33
29	48	03	60	37	30	49	04
46	21	58	09	24	39	12	51
19	44	07	56	41	26	53	14
36	17	62	05	28	35	16	61
01	38	31	50	47	02	59	32
42	25	54	13	20	43	08	55

#11 «
24	17	46	35	62	51	16	09
11	64	21	60	37	12	33	22
40	27	58	31	02	39	06	57
29	52	15	42	55	18	45	04
36	13	50	23	10	47	20	61
25	38	07	34	63	26	59	08
54	01	44	05	28	53	32	43
41	48	19	30	03	14	49	56

#12 «
04	43	24	47	50	09	54	29
27	64	07	52	45	26	33	06
62	21	40	31	02	57	12	35
19	16	55	60	37	42	17	14
46	49	10	05	28	23	48	51
03	44	25	34	63	08	53	30
38	01	58	13	20	39	32	59
61	22	41	18	15	56	11	36

#13 «
04	49	14	53	44	19	48	29
27	64	07	42	55	26	33	06
62	09	40	31	02	57	24	35
11	20	47	60	37	50	13	22
54	45	18	05	28	15	52	43
03	56	25	34	63	08	41	30
38	01	58	23	10	39	32	59
61	16	51	12	21	46	17	36

#14 «
52	17	46	25	08	51	16	45
11	64	23	38	59	10	33	22
62	05	36	31	02	61	28	35
07	56	15	44	53	18	41	26
58	09	50	21	12	47	24	39
03	60	29	34	63	04	37	30
54	01	42	27	06	55	32	43
13	48	19	40	57	14	49	20

#15 «
08	43	24	47	50	09	54	25
27	64	05	52	45	28	33	06
62	21	36	31	02	61	12	35
19	16	55	58	39	42	17	14
46	49	10	07	26	23	48	51
03	44	29	34	63	04	53	30
38	01	60	13	20	37	32	59
57	22	41	18	15	56	11	40

#16 «
40	27	62	21	12	35	06	57
19	64	15	42	55	18	33	14
24	37	04	45	52	29	60	09
11	48	31	58	39	02	49	22
54	17	34	07	26	63	16	43
41	28	61	20	13	36	05	56
46	01	50	23	10	47	32	51
25	38	03	44	53	30	59	08

#17 «
46	17	62	05	28	35	16	51
11	64	23	58	39	10	33	22
20	25	56	53	44	41	08	13
27	36	31	50	47	02	61	06
38	29	34	15	18	63	04	59
45	40	09	12	21	24	57	52
54	01	42	07	26	55	32	43
19	48	03	60	37	30	49	14

#18 «
36	27	62	21	12	35	06	61
19	64	15	42	55	18	33	14
24	39	08	45	52	25	58	09
11	48	31	60	37	02	49	22
54	17	34	05	28	63	16	43
41	26	57	20	13	40	07	56
46	01	50	23	10	47	32	51
29	38	03	44	53	30	59	04

#19 «
44	17	52	07	26	45	16	53
23	64	09	36	61	24	33	10
28	47	14	59	38	19	50	05
03	40	31	54	43	02	57	30
62	25	34	11	22	63	08	35
37	18	51	06	27	46	15	60
42	01	56	29	04	41	32	55
21	48	13	58	39	20	49	12

#20
36	03	42	53	46	35	04	41
27	64	55	14	17	44	33	06
08	49	22	39	58	09	50	25
47	20	31	60	37	02	11	52
18	45	34	05	28	63	54	13
57	16	43	26	07	56	15	40
38	01	10	51	48	21	32	59
29	62	23	12	19	30	61	24

#21 «
60	03	46	09	24	51	30	37
11	64	21	58	39	12	33	22
06	49	18	55	42	15	48	27
25	52	31	36	61	02	45	08
40	13	34	29	04	63	20	57
59	16	47	10	23	50	17	38
54	01	44	07	26	53	32	43
05	62	19	56	41	14	35	28

#22 «
60	03	38	25	08	59	30	37
13	64	19	42	55	14	33	20
22	17	50	53	44	47	16	11
09	58	31	36	61	02	39	24
56	07	34	29	04	63	26	41
43	48	15	12	21	18	49	54
52	01	46	23	10	51	32	45
05	62	27	40	57	06	35	28

#23 «
60	03	38	11	22	59	30	37
09	64	25	52	45	08	33	24
14	49	18	55	42	15	48	19
21	58	31	36	61	02	39	12
44	07	34	29	04	63	26	53
51	16	47	10	23	50	17	46
56	01	40	13	20	57	32	41
05	62	27	54	43	06	35	28

#24 «
46	03	50	11	22	47	30	51
05	64	27	56	41	06	33	28
20	25	58	53	44	39	08	13
23	48	31	36	61	02	49	10
42	17	34	29	04	63	16	55
45	40	07	12	21	26	57	52
60	01	38	09	24	59	32	37
19	62	15	54	43	18	35	14

#25 «
54	03	44	07	26	53	30	43
05	64	19	56	41	14	33	28
10	17	50	59	38	47	16	23
25	52	31	36	61	02	45	08
40	13	34	29	04	63	20	57
55	48	15	06	27	18	49	42
60	01	46	09	24	51	32	37
11	62	21	58	39	12	35	22

#26 *
46	03	30	07	50	43	54	27
25	64	33	52	05	24	09	48
20	61	36	57	16	21	12	37
59	42	55	18	39	02	31	14
06	23	10	47	26	63	34	51
45	04	29	08	49	44	53	28
40	01	32	13	60	41	56	17
19	62	35	58	15	22	11	38

#27 «
53	04	59	08	25	38	29	44
14	17	64	55	42	33	16	19
11	62	31	50	47	02	35	22
24	39	28	45	52	05	58	09
41	26	37	20	13	60	07	56
54	03	34	15	18	63	30	43
51	48	01	10	23	32	49	46
12	61	06	57	40	27	36	21

#28 «
09	46	25	38	59	08	51	24
44	17	64	29	04	33	16	53
55	62	31	22	11	02	35	42
60	07	50	13	20	47	26	37
05	58	15	52	45	18	39	28
10	03	34	43	54	63	30	23
21	48	01	36	61	32	49	12
56	19	40	27	06	57	14	41

#29 «
11	58	23	62	35	10	39	22
60	17	64	13	20	33	16	37
53	08	41	04	29	56	25	44
46	31	50	27	06	47	02	51
19	34	15	38	59	18	63	14
12	57	24	61	36	09	40	21
05	48	01	52	45	32	49	28
54	07	42	03	30	55	26	43

#30 «
23	46	11	62	35	22	51	10
40	17	64	05	28	33	16	57
59	20	53	26	07	44	13	38
36	31	50	09	24	47	02	61
29	34	15	56	41	18	63	04
06	45	12	39	58	21	52	27
25	48	01	60	37	32	49	08
42	19	54	03	30	43	14	55

#31 «
23	46	11	62	35	22	51	10
08	17	64	37	60	33	16	25
27	20	53	58	39	44	13	06
04	31	50	41	56	47	02	29
61	34	15	24	09	18	63	36
38	45	12	07	26	21	52	59
57	48	01	28	05	32	49	40
42	19	54	03	30	43	14	55

#32 «
29	46	03	40	57	30	51	04
38	17	64	23	10	33	16	59
43	20	53	26	07	44	13	54
56	31	50	05	28	47	02	41
09	34	15	60	37	18	63	24
22	45	12	39	58	21	52	11
27	48	01	42	55	32	49	06
36	19	62	25	08	35	14	61

#33 *
13	58	03	30	47	12	59	38
50	09	64	33	20	49	10	25
23	44	61	36	41	22	05	28
46	17	54	39	14	57	02	31
19	48	11	26	51	08	63	34
42	21	04	29	24	43	60	37
15	56	01	32	45	16	55	40
52	07	62	35	18	53	06	27

#34 «
29	48	05	58	39	28	49	04
34	19	64	09	24	33	14	63
21	42	11	52	45	22	55	12
40	27	50	03	30	47	06	57
25	38	15	62	35	18	59	08
44	23	54	13	20	43	10	53
31	46	01	56	41	32	51	02
36	17	60	07	26	37	16	61

#35 «
11	56	27	36	61	06	41	22
34	17	64	13	20	33	16	63
53	08	39	30	03	58	25	44
46	05	50	23	10	47	28	51
19	60	15	42	55	18	37	14
12	57	26	35	62	07	40	21
31	48	01	52	45	32	49	02
54	09	38	29	04	59	24	43

#36 «
07	56	27	36	61	06	41	26
34	17	64	13	20	33	16	63
53	10	43	30	03	54	23	44
46	05	50	25	08	47	28	51
19	60	15	40	57	18	37	14
12	55	22	35	62	11	42	21
31	48	01	52	45	32	49	02
58	09	38	29	04	59	24	39

#37 «
06	19	54	35	62	43	14	27
25	48	09	64	33	24	49	08
60	45	12	29	04	21	52	37
31	42	15	58	39	18	55	02
34	23	50	07	26	47	10	63
05	20	53	36	61	44	13	28
40	17	56	01	32	41	16	57
59	46	11	30	03	22	51	38

#38 «
20	39	28	35	62	05	58	13
11	48	23	64	33	10	49	22
46	25	56	15	18	41	08	51
31	04	59	44	53	38	29	02
34	61	06	21	12	27	36	63
19	40	09	50	47	24	57	14
54	17	42	01	32	55	16	43
45	26	37	30	03	60	07	52

#39 «
10	49	60	03	30	37	48	23
21	58	13	64	33	20	39	12
56	15	46	25	08	51	18	41
31	36	27	54	43	06	61	02
34	29	38	11	22	59	04	63
09	50	19	40	57	14	47	24
44	07	52	01	32	45	26	53
55	16	05	62	35	28	17	42

#40 «
38	19	54	03	30	43	14	59
25	48	09	64	33	24	49	08
28	45	12	61	36	21	52	05
31	42	15	58	39	18	55	02
34	23	50	07	26	47	10	63
37	20	53	04	29	44	13	60
40	17	56	01	32	41	16	57
27	46	11	62	35	22	51	06

#41 «
58	05	40	13	20	57	28	39
11	48	23	64	33	10	49	22
02	61	30	53	44	03	36	31
19	56	15	38	59	18	41	14
46	09	50	27	06	47	24	51
63	04	35	12	21	62	29	34
54	17	42	01	32	55	16	43
07	60	25	52	45	08	37	26

#42 «
62	05	40	13	20	57	28	35
11	48	23	64	33	10	49	22
02	59	26	53	44	07	38	31
19	56	15	36	61	18	41	14
46	09	50	29	04	47	24	51
63	06	39	12	21	58	27	34
54	17	42	01	32	55	16	43
03	60	25	52	45	08	37	30

é The 16 Open Magic Tours

#43
02	39	58	51	30	23	10	47
45	28	05	32	49	44	53	04
64	25	08	13	36	41	56	17
15	22	11	62	19	38	59	34
50	43	54	03	46	27	06	31
01	40	57	52	29	24	09	48
20	37	60	33	16	21	12	61
63	26	07	14	35	42	55	18

#44
14	53	58	19	38	15	04	59
41	30	17	02	55	40	43	32
64	11	08	23	34	49	62	09
37	26	21	60	13	36	47	20
28	39	44	05	52	29	18	45
01	54	57	42	31	16	03	56
24	35	48	63	10	25	22	33
51	12	07	46	27	50	61	06

#45
56	05	24	35	14	57	44	25
11	62	37	32	49	12	19	38
64	59	42	47	02	07	22	17
15	52	45	26	55	04	29	34
50	13	20	39	10	61	36	31
01	06	23	18	63	58	43	48
54	03	28	33	16	53	46	27
09	60	41	30	51	08	21	40

#46
52	29	08	41	04	51	30	45
47	06	55	26	43	02	49	32
12	37	64	17	60	11	38	21
23	62	15	34	19	58	09	40
42	03	50	31	46	07	56	25
53	28	01	48	05	54	27	44
18	59	10	39	22	63	16	33
13	36	57	24	61	14	35	20

#47 diagonals magic
42	21	52	11	26	41	16	51
31	46	05	56	37	32	47	06
08	03	64	35	58	61	02	29
27	40	17	50	43	20	53	10
38	25	48	15	22	45	12	55
57	62	01	30	07	04	63	36
34	19	60	09	28	33	18	59
23	44	13	54	39	24	49	14

#48
36	21	58	25	12	35	16	57
23	46	05	38	55	32	47	14
28	03	64	41	52	61	02	09
11	34	17	50	43	20	59	26
54	31	48	15	22	45	06	39
37	62	01	24	13	04	63	56
42	19	60	27	10	33	18	51
29	44	07	40	53	30	49	08

#49
59	14	27	02	47	54	35	22
08	49	40	61	20	09	32	41
05	52	37	64	17	12	29	44
46	55	34	23	58	15	26	03
19	10	31	42	07	50	39	62
60	13	28	01	48	53	36	21
57	16	25	04	45	56	33	24
06	51	38	63	18	11	30	43

#50
27	42	55	02	47	26	07	54
44	49	04	29	52	45	32	05
37	24	09	64	17	40	57	12
14	19	34	59	22	15	62	35
51	46	31	06	43	50	03	30
28	41	56	01	48	25	08	53
21	16	61	36	13	20	33	60
38	23	10	63	18	39	58	11

#51
19	14	61	06	43	20	35	62
12	49	38	57	24	11	32	37
47	40	55	64	17	26	09	02
42	21	34	29	52	15	60	07
23	44	31	36	13	50	05	58
18	25	10	01	48	39	56	63
53	16	27	08	41	54	33	28
46	51	04	59	22	45	30	03

#52
55	14	27	44	05	54	35	26
08	49	42	19	62	07	32	41
47	52	37	64	17	12	29	02
04	59	34	25	56	15	22	45
61	06	31	40	09	50	43	20
18	13	28	01	48	53	36	63
57	16	23	46	03	58	33	24
10	51	38	21	60	11	30	39

#53
27	46	13	58	35	22	47	12
40	15	56	05	32	41	16	55
37	20	03	64	29	44	61	02
34	23	48	11	26	51	08	59
31	42	17	54	39	14	57	06
28	45	62	01	36	21	04	63
25	50	09	60	33	24	49	10
38	19	52	07	30	43	18	53

#54
25	38	51	44	05	30	11	56
02	49	46	23	58	03	32	47
41	28	13	64	17	36	53	08
06	61	34	55	26	15	20	43
59	04	31	10	39	50	45	22
24	37	52	01	48	29	12	57
63	16	19	42	07	62	33	18
40	27	14	21	60	35	54	09

#55
29	38	51	44	05	30	11	52
02	49	46	23	58	03	32	47
41	26	09	64	17	40	55	08
06	61	34	53	28	15	20	43
59	04	31	12	37	50	45	22
24	39	56	01	48	25	10	57
63	16	19	42	07	62	33	18
36	27	14	21	60	35	54	13

#56
12	05	58	47	42	63	04	29
31	44	09	40	49	32	45	10
52	15	38	19	22	51	26	37
17	64	03	54	35	06	57	24
48	01	62	11	30	59	08	41
13	50	27	46	43	14	39	28
34	21	56	25	16	33	20	55
53	60	07	18	23	02	61	36

#57
13	18	41	22	59	08	63	36
38	49	10	61	20	39	32	11
51	46	25	34	15	56	03	30
44	07	64	53	28	17	42	05
21	58	01	12	37	48	23	60
14	19	40	31	50	09	62	35
27	16	55	04	45	26	33	54
52	47	24	43	06	57	02	29

#58
09	18	41	22	59	08	63	40
38	49	12	61	20	37	32	11
51	46	29	34	15	52	03	30
44	07	64	55	26	17	42	05
21	58	01	10	39	48	23	60
14	19	36	31	50	13	62	35
27	16	53	04	45	28	33	54
56	47	24	43	06	57	02	25

The above enumeration of magic fiveleaper tours has recently (February 2023) been confirmed in a study by Yann Denef who lists the 67 arithmetically distinct solutions, rather than the 58 geometrically distinct. The nine cases that count twice are numbers 1, 3, 4, 5, 6, 12, 15, 26, 33 in Marlow's list, as noted in the introductory paragraph to this section.

é Fiveleaper Pseudotours 8×8

The following results are by Tom Marlow, October 1990. They enumerate the ways of forming two or four paths that by reflection or rotation can be placed to cover the 8×8 board, using each cell once, as in Vandermonde's method for the knight. The following is Marlow's own text:

Chains of 16 squares that reflect or rotate to cover the whole board.

All chains start at square 1. Two-fold chains end one move from square 64 and so can then be repeated after rotation by 180 degrees to make closed loops of 32 squares. Four-fold chains end one move from square 1, so form closed loops. Squares 36 and 29 are one move from both corner squares, i.e. 1 and 64. Consequently chains that end on either can be used in two-fold or four-fold form.

Reflective loops can then be reflected about a vertical axis (and a horizontal axis of four-fold) to cover the whole board. Similarly rotatory loops can be turned, 180 degrees if two-fold, or in three steps of 90 degrees if four-fold, to cover the whole board.

Two-fold reflective (3 cases):
01, 41, 46, 11, 39, 04, 32, 52, 23, 63, 28, 56, 21, 50, 30, 59
01, 41, 46, 18, 53, 25, 60, 20, 55, 27, 07, 36, 16, 11, 39, 59
01, 41, 21, 50, 30, 02, 37, 09, 14, 43, 23, 52, 32, 04, 39, 59
Four-fold reflective (3 cases):
01, 41, 46, 11, 39, 04, 33, 13, 42, 02, 37, 09, 44, 15, 35, 06
01, 41, 46, 18, 53, 25, 05, 45, 10, 38, 58, 29, 49, 54, 26, 06
01, 41, 21, 50, 30, 02, 37, 09, 14, 43, 23, 52, 32, 51, 26, 06
Two-fold and Four-fold reflective (12 cases):
01, 41, 46, 06, 35, 15, 44, 09, 14, 34, 05, 40, 12, 47, 07, 36
01, 41, 46, 06, 35, 15, 44, 09, 14, 34, 05, 25, 53, 18, 58, 29
01, 41, 46, 11, 16, 45, 10, 38, 03, 31, 60, 40, 12, 47, 07, 36
01, 41, 46, 11, 16, 45, 10, 38, 03, 31, 60, 25, 53, 18, 58, 29
01, 41, 21, 50, 30, 02, 42, 13, 33, 61, 26, 06, 46, 11, 16, 36
01, 41, 21, 50, 30, 02, 42, 13, 33, 04, 39, 59, 19, 54, 49, 29
01, 41, 12, 40, 05, 45, 10, 38, 58, 18, 46, 06, 26, 54, 49, 29
01, 41, 12, 40, 05, 34, 62, 27, 55, 20, 49, 54, 19, 47, 07, 36
01, 41, 12, 40, 05, 34, 14, 09, 44, 15, 35, 06, 46, 18, 58, 29
01, 41, 12, 40, 60, 31, 03, 38, 10, 45, 16, 11, 46, 18, 58, 29
01, 41, 12, 40, 60, 31, 51, 56, 21, 50, 30, 59, 19, 47, 07, 36
01, 41, 12, 40, 60, 20, 55, 27, 07, 47, 19, 59, 39, 11, 16, 36
Two-fold reflective and rotatory (1 case):
01, 41, 46, 11, 16, 36, 07, 47, 12, 40, 35, 15, 44, 04, 39, 59
Two-fold and Four-fold rotatory (3 cases):
01, 41, 46, 06, 26, 61, 21, 50, 30, 25, 53, 18, 23, 63, 58, 29
01, 41, 12, 40, 35, 15, 44, 04, 39, 59, 19, 54, 14, 09, 49, 29
01, 41, 12, 40, 35, 15, 44, 04, 39, 59, 19, 47, 42, 02, 07, 36

04	23	46	27	42	11	02
25	40	29	36	19	44	09
16	13	34	07	38	17	14
21	32	01		05	22	31
48	43	10	03	24	47	28
37	18	45	26	41	12	35
06	39	30	15	20	33	08


54	21	14	31	18	11	28	15	52
23	50	07	46	33	04	49	08	45
36	43	26	39	02	35	42	25	38
19	12	29	16	53	20	13	30	17
32	05	48	09	22	51	06	47	10
01	34	41	24	37	44	27	40	03

19	35	35	60	05	30	11	46
44	22	22	17	48	43	62	21
57	32	09	14	51	56	33	08
06	29	12	39	26	53	36	59
49	42	63	20	45	02	23	16
18	55	34	61	04	31	10	47
27	52	37	58	07	28	13	38
40	01	24	15	50	41	64	25

57	34	63	20	13	50	35	64
22	15	60	43	06	29	48	55
39	10	27	46	53	40	09	26
12	51	36	01	24	17	62	37
05	30	49	56	33	04	19	44
58	41	08	21	14	59	42	07
23	16	61	38	11	28	47	54
32	03	18	45	52	31	02	25

10	21	40	29	52	09	20	63
61	50	13	06	43	32	49	12
04	57	24	37	16	03	56	27
45	34	19	64	59	46	35	18
42	31	54	11	62	41	30	53
07	22	39	28	51	08	23	38
60	47	14	05	44	33	48	15
01	58	25	36	17	02	55	26

60	31	34	17	08	57	30	33
19	36	53	14	47	22	37	52
10	45	26	41	50	11	44	27
07	56	03	64	61	06	55	16
48	23	38	29	32	35	24	39
59	12	43	18	09	58	13	42
20	05	54	15	46	21	04	51
01	62	25	40	49	02	63	28

03	48	29	22	11	04	49	30
64	27	20	41	56	13	06	33
25	18	51	58	39	46	15	08
10	53	60	31	02	37	44	23
55	12	05	34	63	28	21	42
40	47	14	07	26	19	50	57
01	38	45	24	09	52	59	32
62	17	36	43	54	61	16	35

04	23	46	27	42	11	02
25	40	29	36	19	44	09
16	13	34	07	38	17	14
21	32	01		05	22	31
48	43	10	03	24	47	28
37	18	45	26	41	12	35
06	39	30	15	20	33	08


54	21	14	31	18	11	28	15	52
23	50	07	46	33	04	49	08	45
36	43	26	39	02	35	42	25	38
19	12	29	16	53	20	13	30	17
32	05	48	09	22	51	06	47	10
01	34	41	24	37	44	27	40	03

19	35	35	60	05	30	11	46
44	22	22	17	48	43	62	21
57	32	09	14	51	56	33	08
06	29	12	39	26	53	36	59
49	42	63	20	45	02	23	16
18	55	34	61	04	31	10	47
27	52	37	58	07	28	13	38
40	01	24	15	50	41	64	25

57	34	63	20	13	50	35	64
22	15	60	43	06	29	48	55
39	10	27	46	53	40	09	26
12	51	36	01	24	17	62	37
05	30	49	56	33	04	19	44
58	41	08	21	14	59	42	07
23	16	61	38	11	28	47	54
32	03	18	45	52	31	02	25

10	21	40	29	52	09	20	63
61	50	13	06	43	32	49	12
04	57	24	37	16	03	56	27
45	34	19	64	59	46	35	18
42	31	54	11	62	41	30	53
07	22	39	28	51	08	23	38
60	47	14	05	44	33	48	15
01	58	25	36	17	02	55	26

60	31	34	17	08	57	30	33
19	36	53	14	47	22	37	52
10	45	26	41	50	11	44	27
07	56	03	64	61	06	55	16
48	23	38	29	32	35	24	39
59	12	43	18	09	58	13	42
20	05	54	15	46	21	04	51
01	62	25	40	49	02	63	28

03	48	29	22	11	04	49	30
64	27	20	41	56	13	06	33
25	18	51	58	39	46	15	08
10	53	60	31	02	37	44	23
55	12	05	34	63	28	21	42
40	47	14	07	26	19	50	57
01	38	45	24	09	52	59	32
62	17	36	43	54	61	16	35

04	23	46	27	42	11	02
25	40	29	36	19	44	09
16	13	34	07	38	17	14
21	32	01		05	22	31
48	43	10	03	24	47	28
37	18	45	26	41	12	35
06	39	30	15	20	33	08


54	21	14	31	18	11	28	15	52
23	50	07	46	33	04	49	08	45
36	43	26	39	02	35	42	25	38
19	12	29	16	53	20	13	30	17
32	05	48	09	22	51	06	47	10
01	34	41	24	37	44	27	40	03

19	35	35	60	05	30	11	46
44	22	22	17	48	43	62	21
57	32	09	14	51	56	33	08
06	29	12	39	26	53	36	59
49	42	63	20	45	02	23	16
18	55	34	61	04	31	10	47
27	52	37	58	07	28	13	38
40	01	24	15	50	41	64	25

57	34	63	20	13	50	35	64
22	15	60	43	06	29	48	55
39	10	27	46	53	40	09	26
12	51	36	01	24	17	62	37
05	30	49	56	33	04	19	44
58	41	08	21	14	59	42	07
23	16	61	38	11	28	47	54
32	03	18	45	52	31	02	25

10	21	40	29	52	09	20	63
61	50	13	06	43	32	49	12
04	57	24	37	16	03	56	27
45	34	19	64	59	46	35	18
42	31	54	11	62	41	30	53
07	22	39	28	51	08	23	38
60	47	14	05	44	33	48	15
01	58	25	36	17	02	55	26

60	31	34	17	08	57	30	33
19	36	53	14	47	22	37	52
10	45	26	41	50	11	44	27
07	56	03	64	61	06	55	16
48	23	38	29	32	35	24	39
59	12	43	18	09	58	13	42
20	05	54	15	46	21	04	51
01	62	25	40	49	02	63	28

03	48	29	22	11	04	49	30
64	27	20	41	56	13	06	33
25	18	51	58	39	46	15	08
10	53	60	31	02	37	44	23
55	12	05	34	63	28	21	42
40	47	14	07	26	19	50	57
01	38	45	24	09	52	59	32
62	17	36	43	54	61	16	35