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: — IntroductionFiveleaper Tours on Small BoardsDouble 8×8 ToursOther Fiveleaper Tours 8×8
Magic Fiveleaper Tours 8×8: NotesThe 42 Closed Magic ToursThe 16 Open Magic ToursFiveleaper 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.

04234627421102
25402936194409
16133407381714
213201052231
48431003244728
37184526411235
06393015203308


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

542114311811281552
235007463304490845
364326390235422538
191229165320133017
320548092251064710
013441243744274003

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
4033165962133447
4526510629442352
6411561938030857
3114353841321536
2861225346276021
3904175863120518
4225500730432449
0110552037020954
5-leaper double tour, part 2
0663581524050859
4138292661483936
1851104334195053
2312450255322116
1447040764573025
2762494037280960
4233201752114435
0156312213460354

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
2047625506214663
3142575011342944
0259160952033617
1322392619142338
5405284564615607
5148353043584910
3241243712334025
0160150853042718
5-leaper double tour, part 2
4637601156490463
3924315227402332
5415062134291613
5708036419365910
2641504538254251
4728611255480562
2035305314072233
0118435809021744

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
2831341308273033
1136571647103756
5419502344532049
0726036461064114
4609382932355839
1752211255185122
2405421548250443
0162594045026360
5-leaper double tour, part 2
0629225112591633
0310571431244136
6447264308556239
4520173405282118
5053603702495213
0730234011581532
0409566346254235
0148274419546138

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.

1935356005301146
4422221748436221
5732091451563308
0629123926533659
4942632045022316
1855346104311047
2752375807281338
4001241550416425

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.

5734632013503564
2215604306294855
3910274653400926
1251360124176237
0530495633041944
5841082114594207
2316613811284754
3203184552310225

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.

1021402952092063
6150130643324912
0457243716035627
4534196459463518
4231541162413053
0722392851082338
6047140544334815
0158253617025526

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.

6031341708573033
1936531447223752
1045264150114427
0756036461065516
4823382932352439
5912431809581342
2005541546210451
0162254049026328


G. P. Jelliss, unpublished, constructed 22 March 2018.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

0348292211044930
6427204156130633
2518515839461508
1053603102374423
5512053463282142
4047140726195057
0138452409525932
6217364354611635

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 *
1152214207602938
6419160940336217
5514310247503526
2459043712452257
4106612853204308
1051346318153039
0146495625320348
5413442358053627
#2 «
3548251221084962
6427105146230633
2918416037561504
2043583102395413
4522073463261152
3647240528095061
0138551419425932
3017405344571603
#3 *
1942611435200762
6425163712335617
2722051039445954
5231084702415023
1334571863241536
3843045526210611
0140492853320948
4623045130455803
#4 *
1362352009246136
6433221558391811
5106274655420528
0825485302314449
5740171263342116
1459381910236037
0132435007264754
5203304556410429
#5 * diagonals magic
0338592211303760
6425084552332409
1948154255184914
1231366102395821
5334290463260744
4617502310471651
0140572013324156
6227064354352805
#6 *
0338515823301146
6425160544335617
6128130841365320
2231104702395059
4334551863261506
0437525724291245
0140496021320948
6227140742355419
#7 «
5148290825044946
6427342310630633
1354215839124320
5603601518373041
0962055047283524
5211440726532245
0138314255025932
1417365740611619
#8 «
5738291419045940
6417342112631633
0954235047104324
5203600726373045
1362055839283520
5611421518552241
0148314453024932
0827365146610625
#9 «
1948133661204914
6411342706632233
0958254255083924
6003441518533037
0562215047123528
5607402310572641
0154313859024332
4617522904451651
#10 «
2340115245225710
6427341518630633
2948036037304904
4621580924391251
1944075641265314
3617620528351661
0138315047025932
4225541320430855
#11 «
2417463562511609
1164216037123322
4027583102390657
2952154255184504
3613502310472061
2538073463265908
5401440528533243
4148193003144956
#12 «
0443244750095429
2764075245263306
6221403102571235
1916556037421714
4649100528234851
0344253463085330
3801581320393259
6122411815561136
#13 «
0449145344194829
2764074255263306
6209403102572435
1120476037501322
5445180528155243
0356253463084130
3801582310393259
6116511221461736
#14 «
5217462508511645
1164233859103322
6205363102612835
0756154453184126
5809502112472439
0360293463043730
5401422706553243
1348194057144920
#15 «
0843244750095425
2764055245283306
6221363102611235
1916555839421714
4649100726234851
0344293463045330
3801601320373259
5722411815561140
#16 «
4027622112350657
1964154255183314
2437044552296009
1148315839024922
5417340726631643
4128612013360556
4601502310473251
2538034453305908
#17 «
4617620528351651
1164235839103322
2025565344410813
2736315047026106
3829341518630459
4540091221245752
5401420726553243
1948036037304914
#18 «
3627622112350661
1964154255183314
2439084552255809
1148316037024922
5417340528631643
4126572013400756
4601502310473251
2938034453305904
#19 «
4417520726451653
2364093661243310
2847145938195005
0340315443025730
6225341122630835
3718510627461560
4201562904413255
2148135839204912
#20
3603425346350441
2764551417443306
0849223958095025
4720316037021152
1845340528635413
5716432607561540
3801105148213259
2962231219306124
#21 «
6003460924513037
1164215839123322
0649185542154827
2552313661024508
4013342904632057
5916471023501738
5401440726533243
0562195641143528
#22 «
6003382508593037
1364194255143320
2217505344471611
0958313661023924
5607342904632641
4348151221184954
5201462310513245
0562274057063528
#23 «
6003381122593037
0964255245083324
1449185542154819
2158313661023912
4407342904632653
5116471023501746
5601401320573241
0562275443063528
#24 «
4603501122473051
0564275641063328
2025585344390813
2348313661024910
4217342904631655
4540071221265752
6001380924593237
1962155443183514
#25 «
5403440726533043
0564195641143328
1017505938471623
2552313661024508
4013342904632057
5548150627184942
6001460924513237
1162215839123522
#26 *
4603300750435427
2564335205240948
2061365716211237
5942551839023114
0623104726633451
4504290849445328
4001321360415617
1962355815221138
#27 «
5304590825382944
1417645542331619
1162315047023522
2439284552055809
4126372013600756
5403341518633043
5148011023324946
1261065740273621
#28 «
0946253859085124
4417642904331653
5562312211023542
6007501320472637
0558155245183928
1003344354633023
2148013661324912
5619402706571441
#29 «
1158236235103922
6017641320331637
5308410429562544
4631502706470251
1934153859186314
1257246136094021
0548015245324928
5407420330552643
#30 «
2346116235225110
4017640528331657
5920532607441338
3631500924470261
2934155641186304
0645123958215227
2548016037324908
4219540330431455
#31 «
2346116235225110
0817643760331625
2720535839441306
0431504156470229
6134152409186336
3845120726215259
5748012805324940
4219540330431455
#32 «
2946034057305104
3817642310331659
4320532607441354
5631500528470241
0934156037186324
2245123958215211
2748014255324906
3619622508351461
#33 *
1358033047125938
5009643320491025
2344613641220528
4617543914570231
1948112651086334
4221042924436037
1556013245165540
5207623518530627
#34 «
2948055839284904
3419640924331463
2142115245225512
4027500330470657
2538156235185908
4423541320431053
3146015641325102
3617600726371661
#35 «
1156273661064122
3417641320331663
5308393003582544
4605502310472851
1960154255183714
1257263562074021
3148015245324902
5409382904592443
#36 «
0756273661064126
3417641320331663
5310433003542344
4605502508472851
1960154057183714
1255223562114221
3148015245324902
5809382904592439
#37 «
0619543562431427
2548096433244908
6045122904215237
3142155839185502
3423500726471063
0520533661441328
4017560132411657
5946113003225138
#38 «
2039283562055813
1148236433104922
4625561518410851
3104594453382902
3461062112273663
1940095047245714
5417420132551643
4526373003600752
#39 «
1049600330374823
2158136433203912
5615462508511841
3136275443066102
3429381122590463
0950194057144724
4407520132452653
5516056235281742
#40 «
3819540330431459
2548096433244908
2845126136215205
3142155839185502
3423500726471063
3720530429441360
4017560132411657
2746116235225106
#41 «
5805401320572839
1148236433104922
0261305344033631
1956153859184114
4609502706472451
6304351221622934
5417420132551643
0760255245083726
#42 «
6205401320572835
1148236433104922
0259265344073831
1956153661184114
4609502904472451
6306391221582734
5417420132551643
0360255245083730

é The 16 Open Magic Tours

#43
0239585130231047
4528053249445304
6425081336415617
1522116219385934
5043540346270631
0140575229240948
2037603316211261
6326071435425518
#44
1453581938150459
4130170255404332
6411082334496209
3726216013364720
2839440552291845
0154574231160356
2435486310252233
5112074627506106
#45
5605243514574425
1162373249121938
6459424702072217
1552452655042934
5013203910613631
0106231863584348
5403283316534627
0960413051082140
#46
5229084104513045
4706552643024932
1237641760113821
2362153419580940
4203503146075625
5328014805542744
1859103922631633
1336572461143520
#47 diagonals magic
4221521126411651
3146055637324706
0803643558610229
2740175043205310
3825481522451255
5762013007046336
3419600928331859
2344135439244914
#48
3621582512351657
2346053855324714
2803644152610209
1134175043205926
5431481522450639
3762012413046356
4219602710331851
2944074053304908
#49
5914270247543522
0849406120093241
0552376417122944
4655342358152603
1910314207503962
60132801483621
5716250445563324
0651386318113043
#50
2742550247260754
4449042952453205
3724096417405712
1419345922156235
5146310643500330
2841560148250853
2116613613203360
3823106318395811
#51
1914610643203562
1249385724113237
4740556417260902
4221342952156007
2344313613500558
1825100148395663
5316270841543328
4651045922453003
#52
5514274405543526
0849421962073241
4752376417122902
0459342556152245
6106314009504320
1813380148533663
5716234603583324
1051382160113039
#53
2746135835224712
4015560532411655
3720036429446102
3423481126510859
3142175439145706
2845620136210463
2550096033244910
3819520730431853
#54
2538514405301156
0249462358033247
4128136417365308
0661345526152043
5904311039504522
2437520148291257
6316194207623318
4027142160355409
#55
2938514405301152
0249462358033247
4126096417405508
0661345328152043
5904311237504522
2439560148251057
6316194207623318
3627142160355413
#56
1205584742630429
3144094049324510
5215381922512637
1764035435065724
4801621130590841
1350274643143928
3421562516332055
5360071823026136
#57
1318412259086336
3849106120393211
5146253415560330
4407645328174205
2158011237482360
1419403150096235
2716550445263354
5247244306570229
#58
0918412259086340
3849126120373211
5146293415520330
4407645526174205
2158011039482360
1419363150136235
2716530445283354
5647244306570225

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


Back to top
© G. P. Jelliss and contributing authors.