ç Knight's Tour Notes Index

# Introducing Knight's Tours

Sections on this page: The Earliest Knight's ToursKnight's Tour as Conjuring TrickThe Squares and Diamonds MethodSymmetry and Shape in Knight's ToursCryptotoursEnumeration of Knight's ToursMagic Knight's ToursFigured ToursOther Piece Tours.
The following notes describe, to the best of my current knowledge, the main results (and a few sideshows) that have been achieved in the study of knight's tours and related questions. The treatment given here is introductory. For much more detail go to main Knight's Tour Notes index. Definitions of terms are inserted in the text as the need for more explanation arises.

## éThe Earliest Knight's Tours

The earliest surviving knight's tour that can be given a reasonably definite date is this one by al-Adli ar-Rumi, who flourished in Baghdad around 840 and is known to have written a book on Shatranj (the early form of chess played in the Middle East). His work survives in the form of extracts in later manuscripts. For instance, the later master of Shatranj as-Suli based his works on those of al-Adli, which he criticised. The tour is given in two diagrams in a manuscript scribed c.1350 by Abu Zakariya Yahya ben Ibrahim al-Hakim, with the title Nuzhat al-arbab al-'aqulfi'sh-shatranj al-manqul {The delight of the intelligent, a description of chess}. In one diagram the cells are numbered, in the other words are written on the squares, to be read in the order that gives a poem printed elsewhere in the manuscript. [Main source of this information: H. J. R. Murray A History of Chess (1913) pp.336-337.] Adli's tour is asymmetric but reentrant i.e. its last and first cells are a knight's move apart. (The geometrical diagram formed by joining up the ends of a reentrant tour is termed a closed tour. Note that one asymmetric closed tour of N moves gives rise to N different reentrant tours, by deleting each of the N moves in turn, so the distinction is advisable to avoid confusion when counting tours.)

Two other knight's tours are due to the later Shatranj expert Abu-Bakr Muhammad b.Yahya as-Suli. The first of these incorporates a 30-move tour in the left halfboard which (when closed) shows perfect axial symmetry with no cells on the axis (which I call Sulian symmetry). The other is composed of two half-board tours. Murray describes two arabic manuscripts, one in Constantinople (as it then was) and the other in Cairo. These are of the same work Kitab ash-shatranj mimma'l-lafahu'l-`Adli was-Suli wa ghair-huma {Book of chess; extracts from the works of al-'Adli, as-Suli and others}. The Turkish ms is written by Abu Ishaq Ibrahim b. al-Mubarak b.'Ali al-Mudhahhab al Baghdadi and has the Muslim date 535 (i.e. 1140ad). The Egyptian ms was dated as written about 1370ad. The two authors mentioned in the title are known, from a bibliography by b.Ishaq an-Nadim the Kitab al-fihrist, compiled 988ad, to have written books on chess, but these have not survived except as extracted in these and other later manuscripts. Murray (1930) describes the solution to the second of as-Suli's tours as “given in two poems, one by Tahir al-Basri, the other by ibn Duraid (died 934). Each poem consists of 64 lines, and the first two letters in each line give the name of the square in the literal notation used by the Muslim chessplayers.” Two other remarkable tours by Suli are shown in the later section on Other Piece Tours.
A number of other knight's tours appear in mediaeval manuscripts of chess problem collections. Most of these cover only half the board, i.e. 4×8, or join two such half-board tours. The subject then seems to have been forgotten until it was rediscovered in 1722. Three open tours of the 8×8 board were supplied to the editor of the 1725 edition of Jacques Ozanam's Récréations Mathématiques et Physiques (vol.1, pp.260-269) by Jean-Jacques de Mairan (Directeur de l'Academie Royale des Sciences, Paris). One tour was by himself and the others by the well known mathematicians Raimond de Montmort and Abraham de Moivre. The first edition of this popular compilation had appeared in 1694, but Ozanam died in 1717. The date 1722 is given in the margin. However, these tours did not reach the same level of achievement reached by al-Adli and as-Suli eight centuries before. For instance, the much-quoted tour by de Moivre follows the same plan, of beginning in a corner, keeping close to the edges as far as possible, and then touring the remaining central cells, as a tour by Ali bin Mani given in the mediaeval manuscript that contains the al-Adli tour.
The first thorough scientific study of knight's tours was presented by the prolific mathematician Leonhard Euler (1707-1783) in 1759, though not printed until 1766 as: "Solution d'une question curieuse qui ne paroit soumise a aucune analyse" {Solution of a curious question that does not seem to have been subject to any analysis}, in the Memoirs de l'Académie Royale des Sciences et Belles Lettres. Année 1759. Tome XV. pp.310-337. His method of presenting the tours is, as in al-Adli, by numbering the successive cells occupied by the knight. If the final cell is a knight's move from the initial cell he calls the tour reentrant (rentrante in the French). He explains a general method, apparently suggested to him by L. Bertrand of Geneva who also later (1778) issued an account of it, whereby knight's paths can be joined and new ones generated. The Euler/Bertrand construction method simply notes that a sequence of cells connected by knight moves and numbered say a...m in which an end cell, m, is a knight move from an internal cell of the path, say e, can be renumbered in the sequence a...em...f. Then if the end of another sequence of moves n...z (or an unused cell n) is a knight move from f then n...z can be incorporated into a...m by a...em...fn...z. Using this method Euler derived two series of 19 and 35 irregular tours of the 8×8 board (of which 5 and 4 respectively are reentrant). The Euler/Bertrand method is a general method of completing partial tours and deriving new tours from existing tours, but the initial tours are formed in a random manner. In this paper Euler also investigated symmetries possible in tours, and enumeration of tours on smaller boards. These results are detailed in later sections.
The first book to show knight's tours in the form of line diagrams was printed in 1766 by Lelio dalla Volpe in Bologna, with the title Corsa del Cavallo per tutt'i schacchi dello Scacchiere. Presumably previous workers must have drawn tour diagrams for their own interest, but no hand drawn examples seem to have survived. His frontispiece is the first true geometrical diagram of a closed tour, with no points identified as the start or finish.
The mathematician Alexandre-Théophile Vandermonde (1735-1796) published a paper "Remarques sur les Problèmes de Situation", in L'Histoire de l'Académie des Sciences avec les Mémoires, Année 1771, Paris 1774, Mémoires, pp.566-574 and plate I. He begins by covering the 8×8 board with four similar circuits which together form a pattern reflectible in the horizontal and vertical medians. He then links opposite pairs of these circuits together, by deleting a pair of parallel moves and joining the loose ends, to give two congruent circuits each with 180° rotational symmetry, which together still form a pattern with biaxial symmetry. Finally he links together these two circuits by the same method to form a true tour, which however does not preserve the symmetry. In fact it is possible to link Vandermonde's four circuits to form a symmetric tour by deleting only one move in each circuit; there are four solutions. Also the same four circuits can be arranged in rotatory symmetry (instead of reflective), which can similarly be linked to form symmetric tours.
The early history of knight's tours, before 1725, appears in the work of chess historians but still seems largely unknown to mathematicians. The main sources are: Antonius van der Linde Geschichte und Litteratur des Schachspiels (Springer, Berlin 1874), in which "Das Problem des Rösselsprungs" pp.101-111, is a bibliography of the subject and pp.292-295 give diagrams. There is also some material in his later publication Quellenstudien zur Geschichte des Schachspiels 1881 pp.196-198. H.J.R. Murray A History of Chess (Oxford University Press, 1913), contains many details of the mediaeval manuscripts, including knight's tours. He also left an unpublished manuscript The Early History of the Knight's Tour, 60 pages plus bibliography, written c.1930. This is now in the Bodleian Library, Oxford University.

## éKnight's Tour as Conjuring Trick

The first chess playing 'automaton', called the Turk, was built in 1769 by Farkas Kempelen (1734-1804), also known as Wolfgang von Kempelen. This was not of course a true machine but was the first 'cabinet illusion', the cabinet beneath the figure being apparently empty or occupied by machinery, but in fact concealing an operator. Besides playing chess with members of the public it was also able to demonstrate a knight's tour, from any square chosen by a spectator.
One frequent spectator, Karl Gottlieb von Windisch, in a book he published in 1783 (English translation in 1819), describes how the knight's tour was exhibited. “The leap of the knight, which this machine makes traverse all over the board, is too remarkable not to be mentioned. It is this; as soon as all the chessmen are removed, one of the spectators places a knight on any one of the squares he thinks proper; the Automaton immediately takes it, and commencing from that square, and strictly observing the move of the knight, he makes it traverse the sixty-four squares of the chessboard, without missing one, and without touching any of them a second time; this is proved by the counter, which the spectator himself places on each square which the knight has touched, observing to put a white counter on the one from which he first begins and red counters on all those which he afterwards touches in succession. Try to do as much yourself with your chess-board, perhaps you will succeed better than I have done; all my attempts for that purpose have been unsuccessful.”
The Edinburgh Philosophical Journal 1821 reported: "The Automaton Chess Player of M[onsieur] de Kempelen was introduced into England by its inventor in 1783 and has during the last two years been exhibited in various parts of England and Scotland, under the direction of M[onsieur] Maelzel." Johan Nepomuk Maelzel, inventor of the metronome, subsequently took the Turk to North America. After his death in 1838 a group of citizens from Philadelphia bought the machine and exhibited it until it was destroyed by fire in 1854. One of the group was Dr S. Weir Mitchell, who presented a template for use in the automaton to the Chess Library of George Allen (1808-1876), which was acquired by the Library Company of Philadelphia, in which collection it still apparently survives. It is 13 inches square, creased to be folded into quarters and pierced to be hung on a hook. Inside the automaton it fitted over an array of 64 plugs on what I suppose can be called the operator's 'control panel'. A photo of this artefact is in the Good Companion Chess Problem Club Folders (1917) p.140. It shows Euler's first closed tour.
Euler's 1759 paper, which was not printed until 1766, was reviewed in the Journal Encyclopédique February 1767, Tome I pp.121-122, quoting his first two tours. Later that year Cosimo Alessandro Collini (1727-1806), who was at that time personal secretary to the Elector Palatine, submitted an article that appeared over three issues of the journal (September-October 1772, Tome VI pp.453-462, Tome VII pp.112-118, 284-290) in which he stated the problem of finding a tour between given squares and described a method of solution. Collini's method is to start from the pattern of eight circuits consisting of four 4-move circuits in the central 4 by 4 area and four 12-move circuits in the surrounding border. He then seeks to delete one move from each circuit (and as few others as possible) and to join up the ends to achieve the required route. The next year he published a more detailed account in his book Solution du Problème du Cavalier au Jeu des Echecs (Mannheim 1773). In the book Part I prescribes the initial square, Part II the final square, Part III a reentrant tour and Part IV different start and finish squares. In the journal only one tour is shown, on p.116, and the same tour is Table (5) in the book. He gives 20 tours (Tables 5, 7-9, 11, 13-21, 23-28) of which four are reentrant (11, 13, 15, 25). Only in one tour (20) does he connect two inner circuits in succession; the others all alternate outer and inner circuits. Of the 20 tours 12 use the minimal number of 8 deletions (and 7 insertions). To solve the problem when the start and finish squares are on the same circuit takes more than 8 deletions. The corner-to-corner tour (17) is one where he makes more deletions than necessary. Puzzle: Construct a corner-to-corner tour by Collini's method but making only 8 deletions.
Various writers have tried to devise a simple rule that, if followed step by step, will generate a tour or tours. The Mani and De Moivre tours for example follow the rule of starting in a corner and keeping close to the edges before completing the central area. A more general rule proposed by H.C. von Warnsdorf in Des Rösselsprunges einfachste und allgemeinste Lösung (Schmalkalden 1823), is to make the next move to the cell from which there are fewest exits. However it is difficult to follow this rule accurately with any rapidity. Where there is a choice of moves fulfilling the criterion the assumption is that any can be taken.
A practical solution to presenting the knight's tour rapidly was given by Peter Mark Roget in his article: "Description of a Method of moving the Knight over every square of the Chessboard, without going twice over any one; commencing at any given square, and ending at any other given square of a different colour" in The London and Edinburgh Philosophical Magazine and Journal of Science April 1840 (vol.XVI pp.305-309 and plate facing p.305). Roget was Secretary of the Royal Society at that time but had not yet published his famous Thesaurus. His method divides the board into "four separate systems, of 16 squares each" marked with letters of the word LEAP as below (actually he used L e a p). The knight's moves are then either of the types LL, EE, AA, PP (these moves form four 4×4 nets, each of 24 moves), or EL, AL, EP, AP (these each consist of six strands of three moves). Moves LP or EA (connecting different consonants or vowels) are not possible. Roget's method is to traverse each of the four nets, L, E, A, P, separately, as far as possible, so that the tour is in four parts, except when the start and finish points are in the same net or when the ends of the tour are L and P, or A and E, when it is necessary to traverse one of the nets in two parts. He gives three example tours; of types L-A, L-L and L-P. Roget's method satisfactorily solves the problem of presenting a tour as a conjuring trick, since, with practice, a tour can readily be drawn between any two cells of opposite colour quite quickly, even blindfold! Later accounts of the method, with minor refinements, are given by W. H. Browne (1870) and R. C. Read (1959).
H.J.R. Murray introduced the useful terminology of straights for moves of types LL, EE, AA, PP and slants for moves of types EL, AL, EP, AP. This terminology is applicable to the description of any 8×8 tours, Rogetian or not, and can be extended to other even-sided boards. In this terminology, Roget's method is to construct a tour between the given squares using the minimum number of slants. In retrospect it can be seen that since Collini's circuits are composed entirely of straights, his twelve examples in which minimal deletions are employed to connect the circuits are tours of this Rogetian type.

## éThe Squares and Diamonds Method

1825. F.P.H. The first published 8 by 8 tours formed by the "squares and diamonds method" (though not explicitly so named) appear in an appendix (pages 533-6) to the sixth edition (1825) of Studies of Chess (based on the French work by the chessmaster Philidor which first appeared in 1749). One of Euler's symmetric tours combining two half-board tours is first given with commentary. Then there follow three tours by F.P.H. (identity otherwise unknown) in which the numbers on diametrally opposite cells differ by 32, 16 and 8 respectively.
```56 35 52 41 62 37 60 47   18 57 28 53 24 61 30 51   27 14 59 44 11 30 63 46
53 42 55 36 59 48 63 38   27 54 19 60 29 52 23 62   58 43 28 13 62 45 10 31
34 57 44 51 40 61 46 49   58 17 56 25 64 21 50 31   15 26 41 60 29 12 47 64
43 54 33 58 45 50 39 64   55 26 59 20 49 32 63 22   42 57 16 25 48 61 32  9
32  7 18 13 26  1 22 11    6 47 16 33  4 43 10 39    1 24 53 40 17  8 49 34
17 14 29  8 19 12 25  2   15 34  5 48  9 40  1 42   56 39  4 21 52 33 18  7
6 31 16 27  4 23 10 21   46  7 36 13 44  3 38 11   23  2 37 54  5 20 35 50
15 28  5 30  9 20  3 24   35 14 45  8 37 12 41  2   38 55 22  3 36 51  6 19
```
Besides the constant differences of 16 and 8, the second and third tours also satisfy the condition that: "If the board were divided into quarters, i.e. viewing each of these sections of sixteen squares separately, the difference between any two squares through whose centres a line drawn would cut that quarter in half, to be 2". This numerical property is precisely equivalent to the geometriocal property that the knights paths through each quarter take the form of squares and diamonds. The tour with difference 32 is not quite squares and diamonds type. [I am grateful for a photocopy of this appendix, provided by M.Sheehan in 1991]
The numerical property of numbers in opposite cells differing by 32 simply reflects the geometrical symmetry of the tour, i.e. when 64 is joined to 1 it forms a closed path that looks exactly the same when rotated 180°, as was pointed out by Euler. Similarly the constant difference of 16 means that the tour consists of two symmetric circuits, each covering 32 cells, one link being deleted in each circuit and their ends joined to make an open tour. Likewise the constant difference of 8 indicates that the tour consists of four 16-cell symetric circuits linked up in this way. I call such tours piece-wise symmetric tours.
1826. Clemens Rudolph Ritter von Schinnern. Ein Dutzend mathematischer Betrachtungen {A Dozen Mathematical Contemplations} (Geistinger, Wien, 1826). The eighth item "Die Formeln für den geometrisch-arithmetischen Rösselsprung" (pages 16-29) of this curious little 36-page booklet appears to be the first full exposition of the squares and diamonds method. The formulas referred to indicate the sequences in which the quarters are visited and whether the moves within them follow a square (quadrat) or diamond (rhombus). He gives 11 figures of which 4-11 are tours, but 5 is the reverse-numbering of 4 and 8 is a reflection of 4 in the principal diagonal, while 7 is the reverse of 6. All five tours are semi-magic (i.e. their ranks or their files all add to 260 but not both ranks and files). (4) and (6) are magic in the ranks, while (9), (10) and (11) are magic in the files.
```(4)                       (6)
11 22 55 42  7 26 39 58   10 39 26 55 42  7 58 23
54 43 10 23 40 57  6 27   27 54  9 40 57 24 43  6
21 12 41 56 25  8 59 38   38 11 56 25  8 41 22 59
44 53 24  9 60 37 28  5   53 28 37 12 21 60  5 44
13 20 45 52  3 30 61 36   36 13 30 49  4 45 64 19
48 51 14 17 64 33  4 29   29 52 33 16 61 20  3 46
19 16 49 46 31  2 35 62   14 35 50 31 48  1 18 63
50 47 18 15 34 63 32  1   51 32 15 34 17 62 47  2
(9)                       (10)                      (11)
10 27 38 53 14 51 34 31   10 27 38 53 36 29 14 51   23 42 11 54 13 46 17 50
39 54 11 28 35 32 15 50   39 54 11 28 13 52 33 30   10 55 22 43 20 49 14 47
26  9 56 37 52 13 30 33   26  9 56 37 32 35 50 15   41 24 53 12 45 16 51 18
55 40 25 12 29 36 49 16   55 40 25 12 49 16 31 34   56  9 44 21 52 19 48 15
42 57  8 21 48 17  2 63   42 57  8 21  4 61 48 17    7 40 25 60 29 36  1 62
7 24 41 60  1 62 47 18    7 24 41 60 45 20  3 64   26 57  8 37  4 61 32 35
58 43 22  5 20 45 64  3   58 43 22  5 62  1 18 47   39  6 59 28 33 30 63  2
23  6 59 44 61  4 19 46   23  6 59 44 19 46 63  2   58 27 38  5 64  3 34 31
```
Tours (9) and (10) are in fact geometrically the same closed tour; they differ only in the choice of initial square and direction of numbering. The last tour (11) has only two non-magic ranks, the first and fourth, which add to 260±4. Short of the real thing, it is impossible to get any closer than this to constructing a magic tour. And the real thing took another 22 years! [I am grateful for a photocopy of this booklet provided by David Singmaster in 1996.]
1836. Teodoro Ciccolini, Marchese di Guardiagrele. Del Cavallo Degli Scacchi. Paris 1836. One reentrant tour formed by the squares and diamonds method is given, followed by a large number derived from it by Euler's method, which spoils the rhombic structure; the aim being to solve the Collini problem of finding tours connecting any two squares of opposite colour.
```38 35 16 11 22 63 20 55
13 10 37 34 17 54 23 62
36 39 12 15 64 21 56 19
9 14 33 40 53 18 61 24
46 41  8  1 32 25 52 57
5  2 47 44 49 60 31 28
42 45  4  7 26 29 58 51
3  6 43 48 59 50 27 30
```
For example the first tour derived from this one is presented in Euler's mannner, as 1-25, 64-43, 26-42. At the end Ciccolini gives two 10 by 10 tours, one formed by adding a braid along two edges of the original 8 by 8 tour, the other joining four repetitions of a 5 by 5 tour to form a symmetric (but not quatersymmetric) tour. He also shows the same 8 by 8 tour on a circular board formed by idebtifying the a and h sides of the board, and reducing the a8-h8 edge to the central point.
Many subsequent writers have mistakenly attributed the squares and diamonds method to Ciccolini. This myth seems to have originated in Q. Poirson-Prugneaux Introduction pratique du jeu des echecs 1849, whence it was quoted in other influential sources, such as Lucas 1895, Ahrens 1901.
1839. J.E.Thomas de Lavernède. "Problème de situation". Memoires de l'Académie Royale du Gard 1858-9, pp.151-179. Lists 256 knight tours of the 8 by 8 board, all of squares and diamonds type; using a rather over-elaborate notation consisting of letters a,A,b,B, (naming the principal and secondary diagonals in each quarter) marked with subscripts (-,1,2,3, naming the four quarters) and superscripts (-,','',''') naming the four ranks in the quarter; the purpose being to show a tour connecting any cell to any other of opposite colour; his first example is shown in numbered form:
```50  7 36 29 54  9 38 27
33 30 51  8 37 28 55 10
6 49 32 35 12 53 26 39
31 34  5 52 25 40 11 56
62  3 48 17 60 13 42 23
47 18 61  4 41 24 57 14
2 63 20 45 16 59 22 43
19 46  1 64 21 44 15 58
```

1842. E.Troupenas. "Problème du Cavalier, parcourant toutes les cases de l'echiquier", Le Palamède vol.2 (October, November and December issues 1842) pp.166-171, 221-226, 268-277. This article in the first ever chess magazine gives an exposition of the methods of Euler, of Vandermonde and of squares and diamonds. The following squares (carrés) and diamonds (lozanges) tour appears on p.275.
```42 59 24  7 46 61 26 11
23  6 43 60 25 10 47 62
58 41  8 21 64 45 12 27
5 22 57 44  9 28 63 48
40 55 20  1 36 49 30 13
19  4 37 56 29 16 33 50
54 39  2 17 52 35 14 31
3 18 53 38 15 32 51 34
```

The above squares and diamonds tour, rotated a half-turn, was given wider circulation in Aaron Alexandre's Beauties of Chess, published simultaneously in French, German and English editions in 1846. It also appears in W.W.Rouse Ball's Mathematical Recreations and Essays (11th edition 1939, reprint 1956, p.182) where it is misleadingly described as "Roget's solution".
1845. Charles Tomlinson. Amusements in Chess, London 1845. Chapter VII "The Knight's Move" pp114-128 and Lesson II "The Moves" pp.152-155. Gives tours from de Mairan (172), Euler (1759) and Willis (1821) but his account of Roget's method confuses it with the squares and diamonds method. The diagrams are shown by white lines on a black background. The first tour shown here is said to follow the rule "keep as far from the centre of the board as possible", and breaks this rule only at one point (move 37-38). The second is a piecewise symmetric tour with constant difference 16 (see FPH 1825) that has near-quaternary symmetry (reflective type). The third is a closed tour of squares and diamonds type in which the files all add to 260 but the rank totals vary from 242 to 278.
```31 28  9 24 45 40  7 22   17  8 43 38 15 10 45 36   22 11 36 53 20 13 38 51
10 25 30 41  8 23 44 39   42 39 16  9 44 37 14 11   35 54 21 12 37 52 17 14
29 32 27 46 63 42 21  6    7 18 41 48  5 12 35 46   10 23 56 33 16 19 50 39
26 11 62 59 54 49 38 43   40 49  6 19 34 47  4 13   55 34  9 24 49 40 15 18
33 58 47 50 61 64  5 20   29 20 63 50  3 22 33 56   26  7 48 57 32  1 42 63
12 15 60 55 48 53  2 37   62 51 28 21 64 57  2 23   47 58 25  8 41 62 31  2
57 34 17 14 51 36 19  4   27 30 53 60 25 32 55 58    6 27 60 45  4 29 64 43
16 13 56 35 18  3 52  1   52 61 26 31 54 59 24  1   59 46  5 28 61 44  3 30
```

Similar accounts of "Roget's Method" based on Tomlinson appeared subsequently in numerous magazines. For instance The Westminster Papers (a chess magazine, not political) June 1969 and April 1871 (the above semi-magic tour is reproduced on p.187). Edward Falkener Games Ancient and Oriental 1892, also conflates Roget's method with the squares and diamonds, and on page 313 reproduces the semimagic tour, saying that Roget "had a card printed for circulation among his friends ... having the squares in black and the diamonds in red on the diagram ... and underneath this "Key to the Knight's Move as a Magic Square" was printed - "With best compliments of the author". However, Tomlinson does not attribute the tour to Roget, so unless someone can come up with a copy of Roget's supposed card I am doubtful of the truth of Falkener's story (particularly as his accounts are unreliable in other respects).

## éSymmetry and Shape in Knight's Tours

1759 Leonhard Euler. Having explained a way of finding many tours, Euler goes on to consider tours of special types using the method. First he constructs symmetric tours. As he points out, these can be recognised in the numerical form by the property that the numbers in diametrally opposite cells have the constant difference 32 (half the total number of cells). However, symmetry is primarily a geometrical property and is only made visible by joining up the centres of successive cells by straight lines, including the move from the last to the first cell. This results in a geometrical closed tour. Its symmetry is its invariance to 180° rotation. These symmetric tours Euler forms by constructing two diametrally opposite paths simultaneously, joining in any unused cells and then arranging for the end of each path to be a knight's move from the start of the other. He gives five tours of this type. Apart possibly from the Nilakant-ha tour, whose date is uncertain (see 1787), these are the first fully symmetric tours composed. Next he turns to symmetric tours composed of two equal 4×8 tours and presents five of this type also.
Euler also gives the first ever examples of tours exhibiting quaternary symmetry (though he does not explicitly use this term). These consist of two tours of cross-shaped boards in which the symmetry is by reflection in the two diagonals, and a 10×10 tour formed by repeating the same 5×5 tour in each quarter, related by a 90° rotation.
1776. Monneron. Writing from the East Indies, this author supplied two tours to the Nouveau Dictionaire, Pancoucke, Paris, 1776. The second of these is ascribed to Nilakant-ha. I have not seen the actual source; both tours are quoted in Hoffmann (1893).
```Nilakant-ha
1  6 51  8 11 60 57 54
50 13  2 61 52 55 10 59
5 64  7 12  9 58 53 56
14 49 62  3 16 47 36 31
63  4 15 48 35 30 17 46
24 21 26 41 44 39 32 37
27 42 23 20 29 34 45 18
22 25 28 43 40 19 38 33
```

1842. E.Troupenas. This includes an attractive 10 by 10 closed tour shown in diagram form which incorporates Euler's 12-cell cross tour in the centre:
```41 54 43 50 83 52 81 74 85 72
44 67 40 53 38 69 84 71 36 75
55 42 49 68 51 82 37 80 73 86
48 45 66 39 10  7 70 35 76 79
99 56 47 12  5  2  9 78 87 24
46 65 00  3  8 11  6 25 34 77
57 98 13 92  1  4 17 88 23 26
64 95 58 15 60 29 90 27 20 33
97 14 93 62 91 16 31 18 89 22
94 63 96 59 30 61 28 21 32 19
```

1842. Victor Käfer. Vollstandige Anweisung zum Schachspiel. This work has an appendix: "Vom Rösselsprunge" pp.191-193 plus a plate with 24 knight tours on the 8 by 8 board and 16 other construction diagrams. The emphasis is on construction of tours with marked quaternary symmetry, either rotatory or reflective (either of which are, in strict terms, impossible on the 8 by 8 board).
1849. Carl Wenzelides. "Bemerkungen über den Rösselsprung", Schachzeitung, Vol.4, 1849 pp.41-93. (Dated at end 1 October 1848.) This is an account of knight's tour construction, including symmetric and near-magic examples, with 72 line diagrams. It is followed by the article:
1871. Victor Gorgias. The Westminster Papers. vol.III. The frontispiece is a knight's tour of the 160-cell four-handed chess board showing near-quaternary symmetry (reflective type). The same author published other near-symmetric open tours on the 8 by 8 board in The Gentlemen's Journal August 1871, reproduced in The Leisure Hour 1873 (see below).
1892. Edward Falkener Games Ancient and Oriental and How to Play Them, Longmans, Green and Co, 1892. (Reprint, Dover Publications, New York, 1961). Pages 267–356 deal with magic squares and knight's tours, including magic knight's tours pp.319–336. See the note under Roget (1840). The magic tours quoted are Beverley's one, another by Wenzelides, seven by Jaenisch (three of which he claims as his own), eight by Palamède (Ligondès), and the one by Caldwell. He gives a collection of 8×8 tours and tours of the four-handed chessboard with striking patterns, however close examination shows that most of them deviate quite considerably from true symmetry.
1922. S.Rangiah Naidu. Feats in Chess, published in India. Contains 100 tours or partial tours, numbered 1-64 plus a supplement numbered 1-36. Numbers 17, 24 and 26 are the Wenzelides magic tour 27b in various orientations, while 37 is the Mysore magic tour 00b. Many of the tours cover an irregular area defined by the outline of an animal drawn on the chessboard diagram. Some of the tours are the same as in the Harikrishna (1871) work.

## éCryptotours

1844. Jules de Poilly. Possibly the earliest examples of what I call cryptotours in which the 64 words or syllables of a verse are printed on the cells of a chessboard diagram and are to be read in the sequence of a knight's tour. Le Palamède vol.IV October 1844 p.466 solution p.517 and vol.VI September 1846 p.418 solution p.476 [Photocopy kindly supplied by K.Whyld] Tour puzzles of this type later became popular in Germany in the 1850s and 1860s and in England in the 1870s and 1880s. See Staunton 1871.
1867. A. Canel. "Chequered Poem" pp.274-5 in Recherches sur les jeux d'esprit, les singularites et les bizarreries litteraires, principalement en France, 2 vols, Evreux 1867. Syllables on the cells of the chessboard are to be read in the sequence of a knight's (open asymmetrical) tour. This was reproduced in Hugh Haughton (editor) The Chatto Book of Nonsense Poetry, 1988.
1870. Howard Staunton. The last few years (1870-74) of his chess column in Illustrated London News featured 16 knight's tour puzzles of the type I call cryptotours (See Poilly 1844) in which words or syllables of a poem are shown on the cells of a chessboard to be read along the route of a knight's tour. These puzzles, at a time when crosswords had not been invented, were very popular, judging by the very long lists of solvers. Most of them quoted well known verse from writers such as Shakespeare and Walter Scott, and even included French (de Musset), Italian (Dante) and macaronic verse (Porson). The tours are mainly from Jaenisch and Tomlinson. The first puzzle to appear is one of the best of its kind and reveals a squares-and-diamonds tour and a chess-related poem that is quite amusing:
```PUZZLE
sor    to     king   good   say    luck   loy    eth
and    moth   a      soon   dis    our    to     bad
place  ry     church his    force  is     hat    al
er     queen  him    wight  he     to     may    truth
man    his    and    and    chess  es     knight op's
a      sneer  the    and    un     lawn   of     tates
cas    that   at     less   pawn   no     bish   lant
eth    faith  tles   hath   the    gal    in     love

TOUR SOLUTION             VERSE SOLUTION
14 55 22 37 12 51 18 35   The man that have no love of chess
23 38 13 54 17 36 11 50   Is, truth to say, a sorry wight,
56 15 40 21 52  9 34 19   Disloyal to his king and queen.
39 24 53 16 33 20 49 10   A faithless and ungallant knight;
2 57 28 41  8 61 32 47   He hateth our good mother church,
25 42  1 60 29 48  7 62   And sneereth at the bishop's lawn;
58  3 44 27 64  5 46 31   May bad luck force him soon to place
43 26 59  4 45 30 63  6   His castles and estates in pawn!
```

1896. H. Eschwege. The Knight's Tour. In a continuous and uninterrupted Ride over 48 Boards or 3072 Squares. Adapted from Byron's ‘Mazeppa’. Silsbury Brothers, Shanklin, Isle of Wight. Dedicated to Sir George Newnes, Bart, President of the British Chess Club. December 31st 1986. Byron's poem is presented on a series of chessboards, one word to a square, commencing at f8 and proceeding to e2, whence to leap to f8 on the next board, and so on. The same tour, supplied by H.E. Dudeney, is used throughout, except on the 48th board where it is modified to end at h1. [British Library shelfmark 11647 ee37, catalogued under Byron, G.G.N.]

## éEnumeration of Knight's Tours

By the problem of enumeration of tours I mean proving the existence or nonexistence of tours on given boards or fulfilling other specified conditions, as well as the counting of all solutions possible to these problems. Accurate enumerations of even some of the simplest cases have proved very difficult. Modern computer methods have of course led to an explosion of results, but no claim should be taken as definitive unless it can be cross-checked by an independent method.
1759. Leonhard Euler. In the later sections of his paper, Euler turns to tours on smaller boards which, apart from the 4×8 which was popular in mediaeval times, seem not to have been considered before. He notes that no tours are possible on squares 2×2, 3×3 or 4×4, though on the latter he gives a 15-cell open path omitting a corner cell. He notes the general property that, since odd and even numbered cells follow alternately, a reentrant tour is not possible on any board with an odd number of cells, since the numbers on the first and last cells are odd. On 3-rank boards he gives the three possible tours on the 3×4 board (or four in numerical form: the fourth being the reverse numbering of the asymmetric tour). He notes that no tours are possible 3×5 or 3×6, and gives two examples 3×7.
Unfortunately at one point he states incorrectly that reentrant tours are not possible on boards with side less than five cells, and his influence was so great that no-one seems to have questioned his statement for 150 years! For instance Jaenisch (1862 Tome II p.46) and Ahrens (1910 p.350) repeat the assertion. Ernest Bergholt (1918) showed the error by constructing closed tours 3×10 and 3×12.
On 4-rank rectangular boards Euler gives single examples 4×5, 4×6 and 4×7. On the 5×5 board he notes that every tour must commence or finish at a corner cell, and gives formulae for 34 others. These include all eight symmetric cases (all corner-to-corner type) which are also given separate diagrams. On the 5×6 board he gives an asymmetric closed tour and on the 6×6 board a symmetric closed tour.
1821. Robert Willis. An Attempt to Analyze the Automaton Chess Player of Mr De Kempelen. To which is added a copious collection of The Knight's Moves over the chessboard (London 1821). No author's name is given, but Tomlinson (1845) says it was by Robert Willis, later Professor of Mechanics at Cambridge. "Observing that the Automaton, under the direction of Mr Maelzel, occasionally traversed half the board, I was induced to pursue the subject, and found that the move might be performed on any parallelogram consisting of twelve squares and upwards with the exception of fifteen and eighteen squares." He gives 38 tours, 18 on small boards 20 on the 8 by 8, all in geometrical form. [British Library shelfmark 1040.d.26(1); catalogued under Kempelen]
1873. J.B.D. and Heinrich Meyer. The Leisure Hour, "The Knight's Tour"; September 13th 1873, pp.587-590 is an article by J.B.D. (identity otherwise unknown) with an account of "Roget's Method" and some original tours (derived from the H-shaped braid pattern), together with an attempt to count the number of tours between specific end-points, and also the number of types of tour classified by end-points. An addendum p.752 tries to correct these figures.
December 20th pp.813-815 is a response by Meyer who gives a brief history of the subject with bibliography and example tours, including three by Victor Gorgias (see 1871). He then gives a correct count of the number of types of tour classified by end-points, finding 21 reentrant and 115 non-reentrant, total 136.
In Chessics 22 1985 p.66 I expressed these results by a formula: if the ends are an {r,s} move apart (with r+s odd to ensure the ends are on opposite colours, and r < s by convention) then there are (8–r)(8–s)/2 relative positions when r is not zero, while there are 18 – 2s in the {0,s} case. The results are: {0,1} 16, {0,3} 12, {0,5} 8, {0,7} 4, {1,2} 21, {1,4} 14, {1,6} 7, 2,3} 15, {2,5} 9, {2,7} 3, {3,4} 10, {3,6} 5, {4,5} 6 {4,7} 2, {5,6} 3, {6,7} 1; total 136.
1877. C. Flye Sainte-Marie. "Note sur un problème relatif à la marche du cavalier sur l`echiquier" (Séance du 18 Avril 1877) Bulletin de la Société Mathématique de France, Année 1876-77, vol.5, pp.144-150. Analysis of the 4×8 board, applicable to 4-rank boards of any length. He proves that any tour must be open, must start and end on the first or fourth ranks, must be in two halves joined by a single move on the two inner ranks, the two halves being on two fixed families of squares (the outer white and inner black, and the outer black and inner white) and shows how the number of tours can be calculated from the numbers of half-tours. For the 4×8 board he correctly counts the half-tours 118 a2, 32 c3, 54 e2, 42 g3 and the full tours 118×32 + 32×54 + 54×42 = 7772. Does not diagram any actual tours.
1880. E.M. Laquière. "Géometrie de l'Echiquier, Solutions Regulières du Problème d`Euler sur la Marche du Cavalier ...", Bulletin de la Société Mathématique de France, Paris, vol.8, pp.82-102, 132-158. Roget's method restricted to closed tours of the nets; such tours are 16-move paths of H or C shape. Laquière seems to have been the first to recognise the role of the linkage polygon formed by the alternate deleted and inserted moves. Applies squares and diamonds to the 4×8 board.
1881. E.M. Laquière. "Note sur le Nombre des Marches Rentrentes du Cavalier ...", Bulletin de la Société Mathématique de France, Paris, vol.9, pp.11-17. Extends the figures of Sainte-Marie (1877) to find the number of 8×8 reentrant tours formed of two 4×8 tours. Gives total 31,054,144, which is the number of tours presented in numerical form, i.e. 16 times the generic number which is 1,940,884. Murray (1918) claimed this total was wrong, but I checked it (1997) by an independent method, and find it correct.
1882. Paul De Hijo. (an alias of the Abbé Philippe Jolivald), Le Problème du Cavalier des échecs d'après les méthodes qui donnent la symétrie par rapport au centre. Metz 1882. A subtitle boasts that the work contains over 413,000 tours. Pages 23-51, 63-85, 87-97, 99-110, 129-135, 140-153, 158-159, 165-167 are almost entirely taken up by tedious lists of moves, but in between are to be found some interesting results. Chapter II "Les méthodes de Vandermonde" pp.124–155, enumerates all possible ways of covering the 8×8 chessboard with four 16-move paths in quaternary symmetry. On page 154 de Hijo gives the total of 16-move closed paths in direct symmetry as 301, a figure quoted by later authorities, such as Lucas (1895). The correct figure was shown to be 368 by T.W. Marlow (1985). An examination of de Hijo's list however shows that he did not in fact miss out any but merely made an error in stating the total. On pp.128 and 139 are diagrams of the 14 patterns formed at the four central squares in the cases of oblique and direct quaternary symmetry. These are lettered a-k, there being two each lettered c, f, i where two patterns use the same squares (the second is distinguished by using an italic letter). The formations j and k are the same in each case, having octonary symmetry. He also gave the formations descriptive names. On p.126 the special case of four circuits in oblique quaternary symmetry with no move cutting another (the solution to 'Aladdin's conundrum') is mentioned but not diagrammed. On the p.127 there is a digression in which his methods are adapted to the 6×6 board and the five quatersymmetric solutions are listed, though only one is actually diagrammed; the other tours are in coded numerical form. Many authors have subsequently rediscovered this attractive result independently (including Bergholt 1918, Papa 1921, Cozens 1940). [Photocopy supplied by Koninklijke Bibliotheek, Netherlands]
1908. Charles Planck. "Chessboard Puzzles" series in Chess Amateur includes, as Problems 25-26, December 1908 p.83 and February 1909 p.147, a complete enumeration of the knight's tours on a 5 by 5 board, giving the total as 1728 arithmetical forms (consisting of 16 derived from each of the 104 geometrically distinct asymmetric tours and 8 from each of the 8 symmetric tours).
1918. Ernest Bergholt. A series of letters (the first dated 18 November 1917) in The British Chess Magazine pp.7-8, 48, 74, 104, 195, 262. These notes contain: (a) "Perfect quaternary symmetry" by which he means 8by8 tours in which 4k of the moves are in rotational quaternary symmetry, while the remaining 4.(16–k) are in reflective quaternary symmetry. He reports on p.195 having explained his methods for constructing such tours in a series of memoirs entrusted to H.J.R.Murray. These came to light among Murray's papers in 1991 and are now in the Bodleian Library. I hope to include a version of them in these Knight's Tour Notes eventually. (b) Symmetric 8by8 tour with 8 three-move lines, maximum. (c) Closed tours 3by10 and 3by12, disproving Euler's statement that closed tours were possible only on boards with side 5 of more. (d) Diametral tour on 7by8 board in which two moves must cross at the centre point; now known as bergholtian symmetry. Other examples appear in the memoirs.
1927. M. Kraitchik. Le Probleme du Cavalier (l'Echiquier, Bruxelles & Gauthiers-Villars, Paris, 1927). This 100-page booklet reprints Kraitchik's articles first published in the chess magazine L'Echiquier in 1926, with extra material on 3-rank boards.
1920. Colonel Ugo Papa. Article in La Scienza Per Tutti (fortnightly review) 15 August 1920. English translation by H.R.Bigelow, with modifications and additions by T.R.Dawson, F.Douglas, H.A.Adamson and O.T.Blathy, in: "The Problem of the Knight's Tour", Chess Amateur July–October 1922, and July 1923. The original article attempts to provide a systematic method of enumerating tours on small boards, but apart from the smallest boards the results tabulated are inaccurate, so the method or its application was not a success. Dawson provided, or expanded, the historical notes. Douglas, Adamson and Blathy attempted estimates of the number of 8by8 tours.
1942. H.J.R.Murray. The Knight's Problem. This is an unpublished manuscript now in the Bodleian Library Oxford. It consists of 21 chapters of which 1-8, 15 and 20 deal with geometrical properties such as symmetry, and construction methods, particularly with reference to the "straights and slants" method, including in 20 boards with omission of cells; 9-14 deal with numerical properties, mainly magic tours both 8x8 and larger; 16-19 and 21 deal with enumeration of tours on smaller boards.
A few years ago I attempted an enumeration of various Rogetian tours, using a pocket calculator, and found: Tours with 3 slants = 10,145,864. Reentrant tours with 3 slants = 1,003,600. Closed tours with 4 slants = 250,900 (this is a quarter of the reentrant total). Total of 3-slant open tours of squares and diamonds type = 2688. Number of these reentrant = 368. Total 4-slant closed tours of squares and diamonds type = 92. These figures have not yet been independently checked.

## éMagic Knight's Tours

As we have seen in the section on origin of tours, magic squares, that is square arrays of numbers in which the ranks and files all add to the same total, have been known for more than 4000 years, and can be regarded as generated by tours of pieces capable of a variety of different moves. When the cells are numbered in the sequence they are visited in Rudrata's (circa year 900) simple 4×8 rook tour, which goes left to right along the top rank, then right to left along the second rank and so on "boustrophedonally" (i.e. as in ploughing) the eight files all add to the same total 66. This must surely have led knight's tour constructors to wonder if such "magical" effects could also be incorporated in knight's tours. The development of systematic methods of construction of knight's tours, from Euler onwards, led to attempts to construct magic 8×8 knight's tours. Several composers using the squares and diamonds method of construction came close to a solution, notably C.R.R. von Schinnern in 1825, but the first complete solution was not achieved un til 1848.
It seems as well to point out here that Euler DID NOT COMPOSE ANY MAGIC KNIGHT TOURS, since statements to the contrary have been appearing recently in otherwise reputable sources. For example: Graham Flegg (a former President of the British Society for the History of Mathematics!), Numbers Their History and Meaning (Barnes & Noble, New York 1983) p.235 and Jan Gullberg, Mathematics from the Birth of Numbers (W.W.Norton & Co, New York, 1997) p.209. The magic tour quoted by these authors was indeed the first constructed but it is due to William Beverley (1848).
1848. William Beverley, "On the Magic Square of the Knight's March" (letter dated 5 June 1847), The London and Edinburgh Philosophical Magazine and Journal of Science, August 1848, pp.101-105. The article was in the form of a letter dated 5 June 1847 sent from Beverley at 6 Upper Terrace Islington, London, to his friend the mathematician H.Perigal Jr, of 5 Smith St, Chelsea, London. Perigal sent the letter to the editors of the magazine on 24 March 1848 with a Preface that reads: "The knight's march has engaged the ingenuity of many eminent philosophers and mathematicians; but I believe that Mr W.Beverley is the first who has solved the difficult problem of converting it into a magic square." Both the tour and its reverse are shown in numerical form:
``` 1 30 47 52  5 28 43 54
48 51  2 29 44 53  6 27
31 46 49  4 25  8 55 42
50  3 32 45 56 41 26  7
33 62 15 20  9 24 39 58
16 19 34 61 40 57 10 23
63 14 17 36 21 12 59 38
18 35 64 13 60 37 22 11
```
What was most remarkable about this tour was that it was not composed solely of squares and diamonds, but included a new type of four-cell path that we now call a "beverley quarte". The tour quickly became widely known: Staunton published it in his Chess Player's Chronicle November 1848, and Hanstein in the Schachzeitung January 1849. H.J.R.Murray (1936) analysed the structure of Beverley's tour in terms of "contiguous contraparallel chains" and expressed the view that "Beverley must in fact have envisaged that a tour composed in this way must be semi-magic and the possibility that by a suitable arrangement of his pairs of contraparallel chains it might be made magic." However, since Beverley's tour comes first in the list of all regular tours, when arranged according to Frenicle's method for magic squares as in our catalogue, it seems more likely to me, that Beverley found his tour by a more laborious systematic search and that its analysis in terms of contraparallel chains is really Murray's own work.
The Dictionary of National Biography (Supplement 1901) lists a William Roxby Beverley (born at Richmond, Surrey c.1814, died at Hampstead, London 17 May 1889) who is probably the knight's tour Beverley though I have not been able to establish a definitive proof of this. Upper Terrace Islington no longer exists; it was where Islington Town Hall now stands. I was not able to find Beverley's name recorded at that address in the 1851 census. W.R.Beverley was a scene painter and designer of theatrical effects, and travelled round the country quite a lot in the course of this work. He is recorded as being in London from 1846 onwards, working at the Princess's Theatre, the Lyceum, Covent Garden and Drury Lane, and exhibited water colours at the Royal Academy. He was part of a theatrical family, including three older brothers, Samuel, Henry and Robert. Their father William Roxby (1765-1842) was an actor-manager and adopted Beverley as a stage name, after his home town, the old capital of the East Riding of Yorkshire. Henry Roxby Beverley (1796-1863) controlled the Victoria Theatre, London for a short while and "died at 26 Russell Square, the house of his brother Mr William Beverley the eminent scene painter". This address (not far from Islington) is now an annex of Birkbeck College, University of London).
1849. Carl Wenzelides, "Der Rösselsprung in höchster Kunstvollendung" {The knight's tour in its highest perfection"}, Schachzeitung 1849 pp.94-97. This gives the following closed and symmetric magic knight's tour (12a in our catalogue), and a poem in praise of its author. The article is signed at the end "Hn" (i.e. Hanstein, the editor). This tour is recorded in his 1850 article as having been constructed on 19-20 February 1849. Another tour was given in Vol.4, p.242 (this is presumably his variant of Beverley's tour, 27b).
``` 2 11 58 51 30 39 54 15
59 50  3 12 53 14 31 38
10  1 52 57 40 29 16 55
49 60  9  4 13 56 37 32
64  5 24 45 36 41 28 17
23 48 61  8 25 20 33 42
6 63 46 21 44 35 18 27
47 22  7 62 19 26 43 34
```
The name of the composer of this tour, and author of the preceding article on knight's tours construction, is given only in the form "...l ...s in P". His full name is given in other sources, for example in Jaenisch (1859). H.J.R.Murray (1955) says Carl Wenzelides was "a pensioned archivist of the Princes of Diedrichstein who lived in Nicolsburg, Hungary. He was an invalid, confined for many years to his couch." D.E.Knuth wrote (24 May 1994): "I found Karl Wenzelides, 'polyhistorian', listed in Biographisches Lexicon des Kaiserthums Osterreich by Wurzbach, 1856-1891; this is almost certainly our man! Born September 1770 in Troppau (now Opava in the Czech republic), died 6 May 1852 in Nikolsburg (now Mikulov). He wrote poetry and music, besides works on the Bronze Age, etc; many of his books and letters were in the Troppauer Museum."
1849. A.F.Svanberg. Murray (1951) writes: "In the same year (1849) the Schachzeitung announced that A.F.Svanberg, Professor of Mathematics in the Stockholm University, had also discovered four magic tours 'as a result of mathematical reasoning', but these were never published and are not to be found among Svanberg's papers preserved in Stockholm. All that we know of them is that they were found later than Wenzelides's first tour and that one was 'concordant' with that tour, whatever that may mean."
1850. Carl Wenzelides. Schachzeitung Vol.5, 1850 pp.212, 230. Further on magic knight's tours, describing his discovery of the "quartes" system and his attempts to obtain magic tours with diametral symmetry. He announced p.238 the discovery of six more symmetrical tours. Three more were published in 1858 but three others were lost. This one is quoted in Jaenisch 1859, it is 12b in our catalogue:
``` 2 43 58 19 30 39 22 47
59 18  3 44 21 46 31 38
42  1 20 57 40 29 48 23
17 60 41  4 45 24 37 32
64  5 56 13 36  9 28 49
55 16 61  8 25 52 33 10
6 63 14 53 12 35 50 27
15 54  7 62 51 26 11 34
```

1852. Krishnaraj Wadiar. (1790-1868) Rajah of Mysore. Following the annexation of Mysore by the British in 1799, an infant son of the former ruling family was made rajah, this was Krishnaraj. During his minority (1799-1810) the state was run by a minister. Because of "misgovernment" the British assumed direct administration in 1831, retaining the maharajah as titular sovereign. [For more details see Sir Roper Lethbridge, The Golden Book of India, 1893 pp.362-8] Thus the maharajah had plenty of opportunity from 1831 onwards to devote himself to other pursuits, and one of these was knight's tours. In particular he constructed the following magic tour, 00b in our catalogue.
```11 46 21 52 13 44 19 54
22 49 12 45 20 53 16 43
47 10 51 24 41 14 55 18
50 23 48  9 56 17 42 15
35  8 25 64 29 40 57  2
26 63 36  5 60  1 30 39
7 34 61 28 37 32  3 58
62 27  6 33  4 59 38 31
```
The existence of this tour was not known in the west until it appeared in S.Ranjiah Naidu Feats of Chess 1922, and meanwhile it had been independently constructed by E.Francony in 1881. A collection of tours attributed to the rajah, and also including the Nilakant-ha tour, was compiled by one Pandit Harikrishna Sharma Jyotishacharya in 1871. This was printed in Devanagari script, as part of a encyclopaedia, in 1900, and is reproduced, with English commentary in S.R.Iyer Indian Chess (Nag Publishers, Delhi 1982). The tour has also turned up on several occasions in the form of a (presumably embroidered) silk handkerchief on which it is dated 31 July 1852; H.J.R.Murray (1955) reports this as being exhibited at the Margate Easter Chess Congress in 1938, and Major J.Akenhead (Fairy Chess Review 1947) reported another sighting of this silk (or a duplicate) in a London art shop (A.Hammond, Emil, Burlington Gardens); its present whereabouts are unknown.
Of even greater interest is the inclusion in Harikrishna's collection of the following 12 by 12 magic knight tour. Like the 8 by 8 example, this is also formed on the squares and diamonds principle, and is the earliest known magic tour on a board larger than the 8 by 8 (›Wihnyk 1885).
``` 91 126  53  20 | 93 124  51  22 |107 110  35  38
54  19  92 125 | 52  21  94 123 | 34  37 108 111
127  90  17  56 |121  96  23  50 |109 106  39  36
18  55 128  89 | 24  49 122  95 | 40  33 112 105
----------------|----------------|---------------
87 130  57  16 | 97 120  45  28 |101 116  41  32
58  15  88 129 | 48  25 100 117 | 44  29 104 113
131  86  13  60 |119  98  27  46 |115 102  31  42
14  59 132  85 | 26  47 118  99 | 30  43 114 103
----------------|----------------|---------------
83  12  61 136 | 79 138  67   6 | 69 144  73   2
62 133  84   9 | 66   7  78 139 | 76   3  70 143
11  82 135  64 |137  80   5  68 |141  72   1  74
134  63  10  81 |  8  65 140  77 |  4  75 142  71
```
The snake-like pattern of this tour is similar to that of tour (11) by von Schinnern (1826), so perhaps von Schinnern's little book was in the rajah's library.
1858. Carl Wenzelides. Von Oppen, the new editor of Schachzeitung, published (Vol.13, May 1858 pp.174-5) three more of the magic knight tours which Carl Wenzelides had sent in 1849, but which were not published then, probably owing to the death of the previous editor Hanstein. These are 12e, 12m and 00m in our catalogue.
1859. C.F. Jaenisch. "Solution of the problem of the knight's tour" The Chess Monthly 1859 pp.110-115, 148-151, 176-179, in parallel French and English versions. In the first part he explains the convention he adopts of orienting tours so that the number 1 appears in the octant below the secondary diagonal and left of the vertical median. The two tours he gives are 12o and 12n in our Catalogue.
1862. C.F. de Jaenisch. Traite des Applications de l'Analyse Mathematique au jeu des Echecs, St.Petersburg 1862-3, 3 vols bound as one. Tome I deals with moves and powers of the various pieces. p.229 gives charts of the least number of moves by knight from a1 to each other cell, and of the number of shortest routes. Tome II "Probleme du Cavalier". Calls tours formed by joining two circuits to form one as in Vandermonde's example "three-times reentrant"; the numbers on the deleted and inserted moves are 1, 32, 33, 64. Also on the 8 by 8 we can have four circuits of 16 and when these are joined up with minimal deletions the numbers are 1, 16, 17, 32, 33, 48, 49, 64; Jaenisch called these "five-times reentrant". Tours combining both these properties are "seven-times reentrant". Besides the two tours given previously in 1859, he gives four new magic tours: Two are cyclic, 00a and 00e. Two, 27c, 27d are modifications of Beverley's tour, in which a different symmetrical arrangement of the right-hand half of the tour replaces those used by beverley and Wenzelides. On p.268 he gives the high upper limit of 168C63 for number of tours. Tome III "De la Reaction des pieces de l'echiquier". [Cambridge University Library has a copy. British Library copy was destroyed in the war.]
1873. E.H. (identity otherwise unknown) "Knightly Peripatetics" Glasgow Weekly Herald. This newspaper chess column contains a whole series of results on knight's tours that I've not yet been able to consult fully. It includes the following "two-knight magic tour", which is diagonally magic, but the connecting moves 32-33 and 64-1 are rook moves, thus it is not strictly a knight's tour but an empress (rook + knight) tour.
```21 50 19 42 23 46 15 44
38 41 22 51 14 43 24 27
49 20 39 18 47 26 45 16
40 37 48 13 52 17 28 25
9 12  1 36 29 64 53 56
32 61 10 63  2 55  4 33
11  8 59 30 35  6 57 54
60 31 62  7 58  3 34  5
```

According to H.J.R.Murray (1951), following the publication of Parmentier's catalogues of magic knight's tours in the 1890s "The main interest of French composers turned to the construction of magic two-knight tours in direct quaternary symmetry." However this example shows that work on such tours started much earlier than Murray suggests.
1876. Charles Bouvier. Murray (1951) writes: "Jaenisch's contemporaries regarded his treatment of magic tours as definitive, and no new magic tours were discovered during the next 14 years. Then in 1876, Charles Bouvier, a Frenchman, published [where?] a magic tour [05a] which struck new ground in containing two pairs of irregular quartes."
1876. Dr Exner. Der Rosselsprung als Zauber-quadrat {Knight's tour as magic square}, memoir containing 15 magic tours of which three were new [27i, 34a, 34e] and another gave a fifth arithmetical version of Jaenisch's 00a not previously noticed. His tours are all constructed with consecutive quartes in the same quadrant.
1879. E.C. Caldwell. "The Knight's Tour on the Chessboard as a Magic Square", English Mechanic and World of Science, vol. 29, p.317. Gives two magic knight's tours, one quoted from 'a modern French Cyclopedia' Larousse's Dictionnaire Universel, (this is Jaenisch 12o in our Catalogue) the other 'is original and has never before been published' (05b). Both are of squares and diamonds type. Notes that '...the errors in excess in one section are made to compensate the errors in defect in another section, and vice versa.'
1880-84. M.A. Feisthamel. Murray 1951 writes: "A period of great activity in the composition of magic tours opened in France in 1880, largely stimulated by M.A.Feisthamel in his chess column in Le Siècle {The Age}, 1876-1885, in which he published all the known magic knight's tours, and new ones as they were produced. Two new tours appeared in 1880, six in 1881, 14 in 1882, 39 in 1883 and 2 in 1884. The contributors of these tours all adopted pseudonyms: A = Béligne (2 tours 12c, 27j); Adsum = C.Bouvier (4 tours, 1882 05d, 12i, 12j; 1883 00l); Célina = E.Francony (11 tours 1881 00c, 05f, 27k, 27l; 1882 05c, 05e, 12k, 12l, 23a, 27m; 1883 00k); F = Feisthamel (1 tour, 1884 14d); Paul de Hijo = Abbé Jolivald (4 tours, 1882 12d, 12f, 12g, 12h); Palamède = Count Ligondès of Orleans (33 tours, 1882 14a, 14b; 1883 00f, 00g, 01a, 01b, 01c, 03a, 03c, 03d, 03e, 03f, 05g, 14c, 23c, 23d, 23e, 23f, 23g, 23h, 23i, 23j, 23k, 23l, 23m, 23n, 25a, 34b, 34c, 34d, 34f, 34g; 1884 23b); X à Belfort = Prof. C.E.Reuss (7 tours, 1883 00d, 00j, 27e, 27p, 27q, 27r, 27s). Altogether this group added 65 geometrical magic tours. The majority are composed of [regular] quartes, but they also include tours in which no use is made of [regular] quartes. The first of these was contributed by Charles Bouvier in 1882. None of these composers gave any indication of the methods used to obtain the tours." Another tour composed 1880/81 is anonymous (00i).
1885. M.Wihnyk. Two articles on magic knight's tours in Schachzeitung 1885 pp.98 and 289. The first gave a modification (27f) of Beverley's tour in which irregular quartes were used (similar to Bouvier 1876). The second article showed how Beverley's tour could be extended to give a magic tour on the 16x16 board. (Murray says Wihnyk describes a method for any square board of side 4n (n > 2), but he does not seem to have made an example 12x12.)
1891. General Parmentier. Murray 1951: "The period of activity [on magic tours] in France ended with General Parmentier's paper on the Knight's problem which was read at the Marseilles meeting (1891) of the Association Francais pour l'Avancement des Sciences, which he supplemented at the Pau meeting (1892) [and at Caen (1896)?]. These papers were subsequently published with diagrams of 83 geometrical and 110 arithmetical solutions of magic tours. This remained the standard collection of magic tours for forty years."
1896. Grossetaite. A new magic knight's tour (01e). Source?
1906. Count Ligondes. Le Mode du Petit Journal, Three new magic knight's tours, over several years: 1906 00h, 1910 23o, 1911 23p.
1932. Max Bruno Lehmann. Neue Mathematische Spiele fur die Jugend {New mathematical games for the young}. This seems to be a series title; a second title page reads: Der Geometrische Aufbau Gleichsummiger Zahlenfiguren {The geometrical structure of equal-summed number-figures (i.e. magic arrays)}. In Section L. "Magische Rösselsprünge", pages 301-362, we find a catalogue of all 8 by 8 magic knight's tours and two-chain (zweikettiger) tours known at that time, shown by geometrical diagrams; followed by some magic tours on larger boards.
1933. Max Bruno Lehmann. Sphinx August 1933 a new magic knight tour (16a).
1936. H.J.R.Murray. "Beverley's Magic S-Tour and its Plan" (The Problemist continues to use the German "S" for "Springer" to mean "knight") The Problemist Fairy Chess Supplement Vol.2 No.16 February 1936 p.166 diagrams 2107-2109, and Vol.2 No.17 April 1936 p.177 diagrams 2239-40, and Vol.2 No.18 June 1936 p.187 diagrams 2286-7, and Fairy Chess Review (PFCS renamed) Vol.3 No.1 August 1936 p.3 diagrams 2350-1, and Vol.3 No.2 October 1936 p.18 diagrams 2466-7, and Vol.3 No.3 December 1936 p.29 diagrams 2541-3; these are the 14 tours possible on the "Beverley plan", which Murray later (1942) described as "contiguous contraparallel chains"; diagrams 2108, 2239, 2350 and 2351 were new magic tours: 27g, 27h, 27n, 27o in our catalogue. Another magic tour appeared in FCR: Vol.3 No.4 February 1937 p.41 diagram 2636, but this equals 34d by Ligondes 1883. Four other new magic knight's tours by Murray appeared in FCR: 1939 01d, 01f, 12p; 1940 03g.
1951. H.J.R.Murray. The Magic Knight's Tour; A Mathematical Recreation. This is another unpublished manuscript also in the Bodleian Library Oxford. It covers much the same area as the chapters on magic tours in the 1942 manuscript but in more detail. In the Preface he writes: "My original intention in writing this book was to celebrate the centenary of the composition of the first magic knight's tour by William Beverley and its publication in 1848." Following the appearance of Kraitchik's book in 1927: "I turned in the summer of 1935 to the special study of the magic tours, since Kraitchik had omitted to deal at any length with these tours." In 1939 he succeeded in obtaining Lehmann's 1932 work "which included all the magic tours on the chessboard that were then known. By this time I had perfected my method of constructing magic quarte-tours and had already obtained 49 of the 87 tours in Lehmannn's work. By 1941 I had independently discovered by my methods all the tours given by Lehmann and brought up the number of geometrical solutions to 96, of which eight were entirely new [See Murray 1936]. So far from discovering these tours empirically, I had established the existence of every tour before I placed it on the chessboard." He also reported new methods of constructing magic tours on larger boards.
1988. T.W. Marlow. "Magic Knight Tours" The Problemist January 1988. Using a home computer Tom Marlow made a search for all magic knight tours of quartes type and found the total to be 78. Five of these were new, they are numbers 01g, 01h, 03b, 23q, 25b in the Catalogue. He concluded: "There remains the general problem of how many other magic knight tours [on the 8×8 board] exist and whether a fully [i.e. diagonally] magic tour is possible. This may still be beyond the capacity of any computer yet available."

## éFigured Tours

Figured tours are numbered tours which have less comprehensive numerical properties than magic tours. In retrospect various numerical properties can be found in tours that were not specifically constructed to show them. For example Euler's first 5×5 tour, which is from corner to corner of the board, has the numbers on the corner-to-corner diagonal in arithmetical progression (1, 7, 13, 19, 25).
1852. Krishnaraj Wadiar, Rajah of Mysore. A collection of tours attributed to the rajah, and also including the Nilakant-ha tour, was compiled by one Pandit Harikrishna Sharma Jyotishacharya in 1871. This was printed in Devanagari script, as part of a encyclopaedia, in 1900, and is reproduced, with English commentary in S.R.Iyer Indian Chess (Nag Publishers, Delhi 1982). We give here some examples of what later came to be called figured tours in which certain chosen numbers appear in geometrical formations. In (61) the multiples of 10 lie on a circle round cell 1 (note that since the tour is closed 64 could be counted as 0). In (63) the third rank is an arithmetical progression with common difference 3, while the fifth rank is one with common difference 5. In (67) the multiples of 8 appear along a main diagonal (this anticipates a similar tour by T.R.Dawson).
```(61)                      (63)                      (67)
16 37 46 43 48 27 52 55   54 49 58 63 28 33 44 39   53 20 51 36 55 60 45 64
45 42 15 38 51 54 59 26   57 62 55 50 45 40 27 32   34 37 54 19 46 63 56 61
36 17 44 47 28 49 56 53   48 53 64 59 34 29 38 43   21 52 35 50 59 48  1 44
41 14  5 50 39 60 25 58   61 56 51 46 41 36 31 26   38 33 18 47 40 31 62 57
18 35 40 29  6 57 64 61   52 47 60 35 30 25 42 37   17 22 39 32 49 58 43  2
13 32 21  4  1 62  9 24    1  4  7 10 13 16 19 22   10  7 24  5 14 41 30 27
34 19 30 11 22  7  2 63    8 11  2  5 24 21 14 17   23 16  9 12 25 28  3 42
31 12 33 20  3 10 23  8    3  6  9 12 15 18 23 20    8 11  6 15  4 13 26 29
```

1871. Pandit Harikrishna Sharma Jyotishacharya. A collection of tours attributed to the rajah of Mysore (see 1852), and also including the Nilakant-ha tour. Compiled in 1871. Printed in Devanagari script, as part of a encyclopaedia, in 1900. Reproduced, with English commentary in S.R.Iyer (1982).
1881. G.Carpenter and S.Hertzsprung. Brentano's Chess Monthly, New York, vol.1 (a) issue 1, May, p.36: G.E. Carpenter (problem editor for issues 1 and 2) proposed the problem: “Prize Knight's Tour. To construct a complete knight's tour of the board, that, the squares being numbered consecutively from 1 to 64, the square numbers 1, 4, 9, 16, 25, 36, 49, and 64 will occupy one band or column of squares.” (b) issue 5, September, pp.248-249: S. Hertzsprung, in a letter from Copenhagen dated 17 July 1881, gives four solutions of Carpenter's problem, with the square numbers in order of magnitude along the first, second, third and fourth ranks. He also calculates the number of ways of arranging the square numbers along the ranks. [British Library shelfmark: PP1831.l]
1982. S.R.Iyer. Indian Chess (Nag Publishers, Delhi 1982). Reproduces the Harikrishna Sharma (1871) manuscript which contains work by the rajah of Mysore, see Krishnaraj Wadiar (1852), with English commentary.

## éOther Piece Tours

The 3 by 3 diagonal magic square is said by many sources to have been known in China about 2200bc, though what evidence there is for this date I'm not sure. [Claude Berge Principles of Combinatorics Academic Press 1971, cites F. M. Müller Sacred Books of the East vol.XVI (The Yi-King) Oxford University Press 1882.] The square is usually presented in the orientation shown here, but of course by rotation and reflection (but keeping the number symbols the right way up!) it can appear in eight different orientations. If the successive numbers are joined by straight lines the result is an open tour of the 3 by 3 board by moves with coordinates {0,1}, {1,1} and {1,2}. In variant chess pieces with these moves are known as wazir, fers and knight. A combined wazir + fers + knight is known as a centaur. So the 3 by 3 diagonally magic square can be described as a magic centaur tour of the 3×3 board. This tour is a geometrical pattern which is centrosymmetric, i.e. unaltered by 180° rotation. This symmetry, in an open tour, corresponds to the numerical property that the numbers in diametrally opposite cells add to a constant value, in this case 10.
In Shatranj the piece next to the king was a ferzan (in later mediaeval chess termed fers) which moved only one step diagonally, and the positions next to the king and queen were occupied by pieces called fil (later termed alfil) which moved two steps diagonally. Two manuscripts survive of a work with the title Kitab ash-shatranj mimma'l-lafahu'l-`Adli was-Suli wa ghair-huma {Book of chess; extracts from the works of al-'Adli, as-Suli and others}. One of these manuscripts was scribed by Abu Ishaq Ibrahim ben al-Mubarak ben 'Ali al-Mudhahhab al Baghdadi in 1140. These manuscripts contain four tours from the work of al-Adli's successor Abu-Bakr Muhammad ben Yahya as-Suli (died 946). Two of these are reentrant knight tours like al-Adli's but two others are remarkable tours formed of alternating moves of knight and alfil and of knight and fers. [Source: H. J. R. Murray A History of Chess (1913).] In any tours of these types the pattern of alfil and fers moves is fixed but the pattern of knight moves joining them can vary.
Vandermonde was well ahead of his time in constructing a three-dimensional knight's tour of a 4×4×4 cube. In numbered cell form his tour is:
```63 12 25 34   26 41 54 13    5 62 33 18   42 17  4 55
28 35 64 15   59 16 29 46   38 21  8 51    7 56 43 20
39 24 11 52    6 53 40 19   27 48 61 14   60  3 32 47
10 49 36 23   37 30  1 50   58  9 22 45   31 44 57  2
```

1935. Emile Huber-Stockar. "Le probleme du cavalier generalise". In Congres Internationale de Recreation Mathematique Comptes Rendus (edited by M.Kraitchik). This contains work on tours by compound leapers restricted to moves in given fixed directions. This example given uses moves in four directions: one on knight, two on camel and one on zebra move-lines.
``` 3 16  5 18
24 12  2 14
19  8 21 10
15  4 17  6
11 23 13  1
7 20  9 22
```
The only other diagram shows an asymmetric closed tour of an 11 by 23 board using {1,4}, {3,5}, {3,7}, {2,9} moves each in one fixed direction. A number of other tours of this type by Huber-Stockar were published in Fairy Chess Review subsequently.