This page of tours [August 2022] is made up from the previous "Imperial Tours" page and from the 2019 PDF #10 on "Augmented Knight and Leaper Tours", which followed research into 19th century French newspaper columns. Some duplicate tabular presntation of tours is retained.
Considerable new work has been reported by Yann Denef (France) in enumerating 11194 "two-knight tours" on the 8×8 board in which the link 32-33, between cells of opposite colour, can be any move type. Among these are 3560 in which the connection is a wazir move, making it an Emperor tour. I hope eventually to include diagrams of selected tours here and on the Empress and Augmented Leaper pages. Meanwhile the reader can explore many diagrams on Yann's web pages: http://ydenef.free.fr/english/2-kt.htm.
Sections on this page: — Introduction — Emperor Tours on Small Boards — Alternating and Figured 8×8 Emperor Tours — Closed Two-Knight Magic Emperor Tours 8×8 — Open Two-Knight Magic Emperor Tours 8×8 — Four-Knight Magic Tours 8×8 — A Magic Eight-Knight Emperor Tour — Emperor on a Larger Board
A piece that can move either as Knight or Wazir (i.e. a single-step rook) is known as an Emperor. The Emperor is the simplest piece in the Imperial (Rook + Knight) family. A piece capable also of longer rook moves is called an Empress. At least these are the names used by Problemists. Being a combination of two free-leapers, Wazir {0,1} and Knight {1,2}, the Emperor has considerable freedom and ability to make tours. It is now a sufficiently important piece to deserve its own page, like the King. Our main sections concern magic tours on the 8×8 board. This is preceded by various results on smaller rectangular boards.
A class of tours that has received intensive study on the 8×8 board are those that consist of chains of knight moves, connected by only a small number of rook moves. These are also known, perhaps misleadingly, as "Two-Knight Tours" and "Four-Knight Tours". In particular magic tours are easily constructed with two or four rook moves, all the others being knight moves. Some results of this type were collected by H. J. R. Murray and are included in his manuscripts (1942 chapter XIV) and (1951 chapter X). He wrote that: "The problem was popular with French composers between 1880 and 1914 when nearly 300 two-chain magic tours were published in the Paris chess columns." It seems he was unaware of earlier work by "E.H." published in the Glasgow Weekly Herald from 1873 onwards. I was able to consult the 1873-4 volume at the British Library Newspaper depository when it was at Colindale [on 25 October 1996] and include some results of this, here and in the Empress page, but further work needs to be done in checking this source. Subsequently by online research I was able to locate many of these tours in the French newspaper columns, notably the one run by M. A. Feisthamel in Le Siècle.
Among the knight tours in other pages of the Knight's Tour Notes many open tours can be found whose end-points are on edge-adjacent cells, i.e. a wazir move apart. If the ends are connected to make a closed tour the result can be described as an Emperor tour. In particular among the "Magic One-Knight Tours" there are nine of this type: (01a) to (01i) shown in our catalogue of magic knight tours (and in PDF #9). Three were found by Ligondès 1883-4, one by Grossetaite 1896, two by Murray 1939, two by Marlow 1987 and the last by Roberts 2003.
Many of these results were first shown in Chessics #26, Summer 1986, the special issue on magic tours.
2×3 Board. The Emperor has two semi-magic tours 2×3. When closed these open tours give only one geometrically distinct tour.
There are also single tours with biaxial and rotary symmetry.
2×4 Board. There are three magic emperor tours, one giving a biaxially symmetric closed tour.
2×8 Board. There are 11 magic emperor tours, three of which are biaxial when closed.
4×4 Board. A diagonally magic emperor open tour is found among the catalogue of magic squares drawn up by
Frénicle de Bessy around 1660. The magic constant is 34.
That this is the only two-pattern solution does not seem to have been noticed until I reported it in Chessics #26 1986.
Also in Chessics #26 I noted that there are two other emperor nondiagonal magic tours that show biaxial symmetry when the two end-points of the path are joined up to give a closed path. One of these is among the alternating tours shown earlier.
We show here 24 symmetric emperor tours with alternating wazir and knight moves on the 4×4 board. The first 4 are biaxial and the other 20 are rotary.
4×6 Board
Since the moves of the emperor always take it to a cell of opposite colour to that on which it stands, the results of the theorems, regarding magic knight tours on even sided boards apply to it. The question of the possibility or not of magic knight's tours on rectangles 4m by 4n + 2, led me to consider emperor tours on such boards. As expected, they are possible. The examples here and below were first published in The Games and Puzzles Journal #26 March-April 2003.
The following two are the only "two-knight" emperor magic tours on the 4×6 board. The magic constants are 50 and 75. These tours can be renumbered cyclically from the half-way point (i.e. 12-13 becomes 24-1 and vice versa) and remain magic but are then Empress tours since the 12-13 rook-move link is a three-cell move.
14 7 22 3 18 11 23 4 13 12 21 2 8 15 6 19 10 17 5 24 9 16 1 20 |
23 4 13 12 21 2 14 7 22 3 18 11 5 24 9 16 1 20 8 15 6 19 10 17 |
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The knight half-tours (see PDF #4) can be paired with themselves to give semi-magic emperor tours adding to 50 in the files, I found 15 of this type, or to 75 in the ranks, I found 4 of this type.
On this the board there are also "Four-Knight" magic tours which use four wazir moves. I found the following three biaxial examples on 10 Aug 1991. The first of these is also given by Trenkler (1999). The ranks consist of three pairs of complements adding to 25. Each quarter-path covers six cells, one in each file. If numbered from the other ends of the quarters the results are semi-magic with files adding to 50, but the ranks having two different sums. Since the diagrams are clear it is left to the reader to put in the numbers if desired.
4×8 Board Here are ten magic emperor tours with biaxial symmetry found 16 Oct 2014. The ranks consist of four pairs of complements adding to 29 (sum 116). Each quarter-path covers eight cells one in each file (sum 58).
6×8 Board (Jelliss 2003) The first was consciously constructed using 'contiguous contraparallel chains', but the others were constructed for their visual symmetry. In the first tour if the files are divided into three pairs then all these pairs add to 49 (file total therefore 3×49 = 147. In the ranks pairs related by reflection in the horizontal median add to 25 or 73 which together equal 98 (twice 49) thus ensuring the rank sum is 4×49 = 196. This is the same in the second tour, but in the third tour the constants in the ranks are 37 and 61. ['Emperor Magic Tours' The Games and Puzzles Journal #26]
34 15 46 3 32 17 47 2 33 16 5 44 14 35 4 45 18 31 1 48 13 36 43 6 24 25 12 37 30 19 11 38 21 28 7 42 26 23 40 9 20 29 39 10 27 22 41 8 |
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15 32 45 4 17 34 44 3 16 33 46 5 31 14 1 48 35 18 2 43 36 13 6 47 23 30 37 12 19 26 42 11 24 25 38 7 29 22 9 40 27 20 10 41 28 21 8 39 |
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42 5 36 13 44 7 35 22 43 6 27 14 4 41 12 37 8 45 21 34 23 26 15 28 40 3 38 11 46 9 33 20 25 24 29 16 2 39 18 31 10 47 19 32 1 48 17 30 |
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Tours on the 8×8 board with alternating wazir and knight moves interested Indian composers. Here is an example from Harikrishna (1871), attributed to the Rajah of Mysore, two from Naidu (1922) and a symmetric tour of my own (Jelliss 1984).
Figured Emperor Tours were considered by W. E. Lester (Fairy Chess Review vol.3 #8 Oct 1937 p.86 ¶2930-31, #9 Dec 1937 p.99 and #10 Feb 1938 p.110 ¶3035-36). He shows Emperor open tours with square numbers along the ranks, without intersection of move lines. The first scheme will work on boards of any size n×n. The third includes a 4×4 subtour.
These consist of two or four paths of knight moves connected by wazir moves. Most of these tours are of the biaxial type described in the Magic Theory section (and in PDF #1). They remain magic when renumbered from 32-33 as 1-64.
The examples shown here are mainly from the early years of Le Siècle. Later years I have not yet fully explored.
These examples are not diagonally magic. They are shown in their original orientation.
The first tour here (and another seven that follow) are from the "Knightly Peripatetics" in the Glasgow Weekly Herald 1873-4.
These tours were set as problems, and the solution was published a week or two later, hence the two dates.
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EH VII
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The EH tour is also shown in alternative tabular form (and inverted). The coloured cells highlight the eight numbers 1, 16, 17, 32, 33, 48, 49, 64 that mark the ends of the four quarters of the tour. In two-knight tours the links 16-17 and 48-49 are knight moves. To assist in recognition of these tours, the conventions are observed that we number the tour in such a way that the 32-33 move is as short as possible (a wazir move in the case of these Emperor tours), that the principal diagonal consists of even numbers, and the cell numbered 01 is on the left side of the board.
The next four are diagonally magic.
(1) Asymmetric published by 'Adsum' (Bouvier) in the 'Passe-Temps Hebdomadaire' column in Le Galois ¶553 (16 Apr 1883).
(2) Le Siècle ¶2062 (8 Jun 1883).
(3) Le Siècle ¶2140c (7 Sep 1883) found independently by both Palamede (Ligondes) and Adsum (Bouvier)
published in a three-part problem with similar Empress tours.
(4) Le Siècle ¶2158 (28 Sep 1883) by M. E. Reuss Strasbourg. Three very similar of double half-board type.
The above are mainly axial. Some examples are more rotary (described as 'angulaire' in the French text). One is diagonally magic.
(1) Le Siècle ¶934 (31 Oct 1879) irregular with non-complementary diagonals 296 and 272.
(2) Le Siècle ¶2320 (4 Apr 1884) by Palamede (Ligondes) diagonals 260.
(3) Le Siècle ¶2356 (16 May 1884) by Adsum (Bouvier) diagonals 296 and 256.
(4) Le Siècle ¶2368 (30 May 1884) by Reuss diagonals 236 and 240.
Further rotary
(5) Le Siècle ¶2374 (6 Jun 1884) by Béligne, diagonals 292 and 240.
(6) Le Siècle ¶2410 (18 Jul 1884) by Béligne, diagonals 272 and 280.
(7) Le Siècle ¶2422 (1 Aug 1884) by Palamede (Ligondes) diagonals 208 and 300.
(8) Le Siècle ¶2434 (15 Aug 1884) by Adsum (Bouvier) diagonals 260±24.
The first sixteen collected here are of the nondiagonal type.
(1) E. H. GWH ¶13 (16/30 Aug 1873) 260±28.
(2) E. H. GWH ¶24 (20 Dec 1873/3 Jan 1874) 260±4.
(3) E. H. GWH ¶9 (24/31 May 1873) 260±4. This uses two different 16-move paths.
(4) is from Lehmann (1932) 260±4.
EH XIII
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EH XXIV
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EH IX
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LEHMANN
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(5) and (6) are from Murray (1951) 260±52 and 260±20.
(7) Celina (Francony) Le Siècle ¶982 (26 Dec 1879) 260±8.
(8) Celina Le Siècle ¶952 (21 Nov 1879) 260±20.
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MURRAY (5)
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MURRAY (6)
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Here are another eight from the early years of Le Siècle.
(9) ¶1138 (25 Jun 1880) 260±124.
(10) ¶1150 (9 Jul 1880) 260±12.
(11) ¶1162 (23 Jul 1880) 260±8.
(12) ¶1168 (30 Jul 1880) 260±56.
(13) ¶1174 (6 Aug 1880) 260±64.
(14) ¶1186 (20 Aug 1880) 260±80.
(15) ¶1384 (8 Apr 1881) 260±60.
(16) para;1450 (24 Jun 1881) 260±84 by L. de Croze à Marseille.
There are many others of this {0,3} type in Le Siècle in later years.
The following five examples are diagonally magic. Murray (1951) gives these five tours (lettered A, B, G, H, K in his list
- the other ten being Empress tours) attributing them to Count Ligondès and other French composers,
but at least four appear in the chess column by E. H. in Glasgow Weekly Herald 1873-4.
(1) EH GWH ¶XII (19 July/2 Aug 1873) = Le Siècle ¶1054 (19 Mar 1880).
(2) Le Siècle ¶1030 (20 Feb 1880).
(3) EH GWH ¶XXV (10/24 Jan 1874) = Le Siècle ¶1090 (30 Apr 1880).
(4) EH GWH ¶XXI (18 Oct/1 Nov 1873) = Le Siècle ¶1078 (16 Apr 1880).
(5) EH GWH ¶XXII (8/27 Nov 1873) = Le Siècle ¶1102 (14 May 1880).
The last three in Le Siècle are by X a Belfort (Reuss).
The first four have the quarter points all in the same pattern, and the path 1-16 is non-intersecting.
We show these five in alternative tabular form (and differently oriented).
MURRAY (A)
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MURRAY (B)
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MURRAY (G)
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MURRAY (H)
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MURRAY (K)
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Murray notes that the tours A B G H are generated by paths without self-intersection, and tours G H have similar central pattern.
(1) Le Siècle ¶868 (15 Aug 1879) 260±20.
(2) Le Siècle ¶898 (19 Sep 1879) 260±20.
(3) by Béligne Le Siecle ¶904 (26 Sep 1879) 260±68.
(4) by Béligne Le Siècle ¶976 (19 Dec 1879) 260±54.
(5) Le Siècle ¶988 (3 Jan 1880) 260±28.
(6) Le Siècle ¶1018 (6 Feb 1880) 260±80.
(7) Le Siècle ¶1036 (27 Feb 1880) 260±12.
(8) Le Siècle ¶1240 (22 Oct 1880) 260±56. Many more are possible.
These take the form of four knight paths, each of 15 moves, connected by four rook moves (or by three if the closure move connecting 1 to 64 is omitted). The results here are mainly my own work. They were first published in The Games and Puzzles Journal #26 March-April 2003 together with the results on rectangles given above.
The next diagram shows that the same plan as for the first 8×6 tour above works on the 8×8 board, the right-hand side being geometrically like the Beverley tour, but not arithmetically. The horizontal pairs all add to 65, and the vertical pairs to 97 and 33 which together equal 130. (The diagonal sums are DO = 360, DE = 288).
46 19 62 3 44 21 6 59 63 2 45 20 5 60 43 22 18 47 4 61 24 41 58 7 1 64 17 48 57 8 23 42 32 33 16 49 40 25 10 55 15 50 29 36 9 56 39 26 34 31 52 13 28 37 54 11 51 14 35 30 53 12 27 38 |
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3 30 45 52 5 28 43 54 46 51 4 29 44 53 6 27 31 2 49 48 25 8 55 42 50 47 32 1 56 41 26 7 15 18 33 64 9 24 39 58 34 63 16 17 40 57 10 23 19 14 61 36 21 12 59 38 62 35 20 13 60 37 22 11 |
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The second tour was formed after a study of the structure of the first magic knight tour published by William Beverley in 1848. The righthand half of his tour is symmetrical (with horizontal axis). My idea was that perhaps Beverley started from a biaxially symmetric pattern (the H ‘crosspatch’ pattern) and fiddled around with the lefthand side until he found the solution using what we now call ‘Beverley quartes’. My tour replaces the four knight moves in the braid on c3d3-c6d6 by four wazir links. The result is a magic emperor tour whose right-hand side is the same as the Beverley tour, both arithmetically and geometrically. (DO = 240, DE = 280, adding to 520, twice the magic constant.)
There is no proof that Beverley found his tour by this method. Another, equally possible method based on ‘Contiguous Contraparallel Chains’ was described by H. J. R. Murray. Alternatively it may be that Beverley found his tour by some sort of search method, since his is the first of the regular tours to be encountered when they are arranged in sequence according to the Frénicle method for magic squares (i.e. oriented with the smallest number at the top left corner and the number to its right less than the number below it).
The following is the text of an article I published in The Problemist July 1986 (p.196).
A Unique Magic Tour. In 1882 the Abbé Jolivald (alias "Paul de Hijo") published an enumeration of all the ways of covering the chessboard with four 16-move knight paths in rotary or reflective symmetry. T. W. Marlow has recently (Chessics 24, p.92) corrected his count for the reflective case, finding 368 instead of 301. [Examination of the de Hijo work later found that all the paths were listed, but there was an error in the statement of the total.] If one move in the same relative position in each quarter is deleted the ends can be reconnected to form Empress or Emperor tours. For these imperial tours to be magic in ranks and files, when numbered from the start of a quarter, each quarter must occupy 2 squares in each rank (or 2 in each file). Of the 368 cases, only 30 fulfil this condition. Of the 61 Emperor tours derived from these 30 cases, only one also adds up to the magic constant (260) in the diagonals as well as the ranks and files. This unique case is illustrated. It happens that the curiously shaped 16-move knight circuit is one of the 7 that have no self-intersection.
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The other examples shown here each use two different 16-cell paths and are from Le Siècle ¶2290 (29 Feb 1884) by 'M. D. Zibetta à St Ouen', which includes two double-Beverley quads, and Le Siècle ¶4144 (21 Feb 1890) by 'M. A. F' (i.e. Feisthamel himself). The paths used in these are also non-crossing.
The tour below was found as an offshoot from a study of maximum number of 3-move knight-lines in tours of squares and diamonds type. The squares and diamonds are joined to make 12 three-unit lines, thus forming four knight-paths. When the ends of the knight paths are joined by wazir moves this gives a magic emperor (wazir + knight) tour. (Not diagonally magic: DO = 288, DE = 240.)
6 27 64 33 32 1 38 59 63 34 5 28 37 60 31 2 26 7 36 61 4 29 58 39 35 62 25 8 57 40 3 30 14 19 56 41 24 9 46 51 55 42 13 20 45 52 23 10 18 15 44 53 12 21 50 47 43 54 17 16 49 48 11 22 |
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This tours consists of four paths of knight moves connected by three wazir moves and a three-leap closure. This example (Jelliss undated, but probably not original).
Derived from Beverley's tour by four interchanges.
This tour constructed 27 Apr 2018 while writing this note is axial but is not of the complemented type.
Ranks consist of pairs adding to 33 and 97. Files are of pairs adding to 17 and 113 on the left and 49 and 81 on the right.
The diagonals add to 260±4.
Tours of '16-knight' type can easily be constructed but I haven't found a magic example. The alternating tours can be regarded as '32-knight' tours.
Emperor tours of other types can of course also be constructed, for example formed mainly of wazir paths with knight-move connections.
The same plan as in the first 8×6 tour above can be extended to any boards 8×2n (with n >2). See below for an 8×8 example. Here is an 8×10 example: the horizontal pairs all add to 81 and the vertical pairs add to 41 and 121 giving the magic constants: row sum R = 5×81 = 405 and file sum S = 4×81 = 324.
58 23 78 3 26 55 74 7 28 53 79 2 57 24 75 6 27 54 73 8 22 59 4 77 56 25 10 71 52 29 1 80 21 60 5 76 51 30 9 72 40 41 20 61 36 45 70 11 32 49 19 62 37 44 65 16 31 50 69 12 42 39 64 17 46 35 14 67 48 33 63 18 43 38 15 66 47 34 13 68 |
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