Euler presented the first 10×10 knight's tour in 1759, it has quaternary symmetry, being formed from four 5×5 parts. Other pre-20th century examples are illustrated, but none of these others has quaternary symmetry.
The next 10×10 tours with quaternary symmetry that I am aware of are two given by Ernest Bergholt in the magazine Queen 1916. He preceded these by two others, both showing approximate quaternary direct symmetry, and formed from two 50-move axially symmetric paths, one the reflection of the other, by deleting one move in each circuit and connecting the loose ends to form in the first case an open tour and in the second a closed tour with diametral symmetry. This was a precursor of his tours with mixed quaternary symmetry, since it mixes four moves in oblique quaternary symmetry with 96 in direct quaternary symmetry.
H. J. R. Murray, in a note dated 1917, calculated that there are 28 tours of Euler's type, repeating a 5×5 tour by 90° rotation, with four methods of linking the quarters (typically e1-f3, e1-g2, e5-f3, e5-g4).
There follows an assortment of six tours by Murray in quaternary symmetry from his 1942 manuscript on The Knight's Problem. His particularly nice third example has just four 'slants' (moves, like a2-c3, that cross two of the 'grid' lines that divide the board into 2×2 blocks). Two other border braid examples by Kraitchik and Alban are inserted. Kraitchik also gave another example joining four 5×5 tours. The one by 'Alban' (Will Scotland) was used by him for a crossword that appeared in the magazine of the Crossword Club. (There is more on this in the Cryptotours page).
W. H. Cozens published two articles on ‘Cyclically Symmetric Knights Tours’ in Mathematical Gazette (Dec. 1940 and 1960?). I quote three that Cozens gave in there and three others. These are from a set of notebooks left by Mr Cozens, containing numerous diagrams of tours of this type, though many of them are repetitive since he seems to have been attempting to enumerate all tours with a particular central configuration. He estimated the total of geometrically distinct 10×10 tours in quaternary symmetry to be around 200,000, but the true figure is now known to be more than twice this amount, namely 415,902, found by T. W. Marlow (1998). (See the end of this page)
My open tour shown above is a piece-wise symmetric tour derived from a pseudotour. If this is numbered, diametrally opposite cells will be found to differ by 10 throughout. The last tour shown above, from Chessics 22 (1985), is in binary symmetry and has no successive knight moves at right angles.
The examples that follow are all results of my own research. The family of 10×10 tours with four slants and in quaternary symmetry, has 408 geometrically distinct members (2×32 + 3×34 + 2×52 + 2×86) by my calculations (not independently checked). The following diagrams show one tour for each of the 8 workable positions of the four slants. The enumeration is made reasonably simple, since each 25-cell knight path joining the slants is equivalent to a wazir tour of a 5×5 board; its ends occuring on the pair of cells determined by the slants.
In the first issue of Chessics (1976) I proved that every 8×8 knight's tour contains a right angle, but in Chessics 22 (1985) I showed that this result does not hold true on the 10×10 board, by giving a tour, in binary symmetry, with no right angles (see above). Subsequently I have enumerated 40 examples in quaternary symmetry, of which I give a selection, one for each of the central angle formations; apart from the first two which are the only ones not including the d8-h6, h7-f3, g3-c5, c4-e8 square.
The following eight examples were constructed just for the interest of their patterns. The central St John or Maltese Cross is impossible to achieve on the 8×8 board.
The last tour shown above is an example of a 'celtic tour', so named by Prof. D. E. Knuth; that is, a tour in which no 'size 1' triangles occur. Size 1 triangles are the smallest possible formed by knight moves, for example by a1-c2, b2-d1, b1-c3, and have an area 1/120 of a board-square. The triangle formed by a1-c2, b1-c3, c1-b3 is size 2, which has an area 1/30.
New Results by T.W.Marlow.
Tom Marlow (1998) applied his computer to calculating the number of 10×10 knight's tours with quaternary symmetry, finding a total of 415902 geometrically distinct solutions. This is more than twice W. H. Cozen's estimate of 200,000 made in his Mathematical Gazette article in the 1960s. The process of enumeration is described by him as follows:
The method of search was to start a tour at c2, proceed to a1 and then to b3, numbering these squares as 1, 2, 3. The chain was then extended to a length of 25 cells and at each step the corresponding cells, after repeated 90° rotation, were marked as unavailable. At this stage a check was made as to whether a link was possible to h5, and if so a tour recorded. After this, or at any previous dead end, the programme systematically back-tracked and tried a new step forward until all steps from b3 had been tried. The total was then recorded. The actual total recorded was 831804 but each geometrically distinct tour can be diagrammed in two forms in each case, one a reflection of the other in the vertical median. The routine described counts both versions, so the totals must be halved. (Because of the 180° rotational symmetry of the tours, reflection in the horizontal median has the same effect as reflection in the vertical median.)
The same count was found by a second (more time-consuming) check as follows. The start was made at b1 and extended by the same method until a link to j2 could be sought. A link to a9 would produce a tour that is a 90° rotation of a tour linking at j2 so is not geometrically different and should not be counted. This method produced the same count.
Tom Marlow also notes: "Tours fall into two types. The first links b3 to b8 and then, via a10, c9 to h9, i8 to i3 and h2 to c2 — call this a circular tour. The second type runs b3 to i3, h2 to h9, i8 to b8 and c7 to c2 — call this a loop tour since it loops the loop at each corner. The count of the two sorts is as follows: circular tours 206937, loop tours 209065, total 415902.
This interesting distinction between circular and loop tours led George Jelliss to consider the general case of tours (not necessarily symmetric) on rectangular boards. There appear to be 13 types of connection, as shown below. Only the three types in the first column can occur in tours with Eulerian symmetry; the first two are the circular and loop types mentioned above.
Only the types in the second column can occur in tours with Bergholtian symmetry, and only those in the third column in those with Sulian symmetry. The dark lines represent sequences of connecting moves that can be quite irregular. Two connections shown by dark lines of the same move-pattern do not necessarily represent congruent sequences of moves.
Much of the material on this page first appeared in The Games and Puzzles Journal issue 16 (1999), pages 286-288.
Here is Tom Marlow's 10×10 semi-magic tour with oblique quaternary symmetry (i.e. 90 degree rotational symmetry) which appeared in The Games and Puzzles Journal, issue 25, March 2003, where a geometrical diagram is also given. The yellow coloured squares are those containing numbers of the forms 4n + 2 or 4n + 3: there must be an even number of these in a line for it to be magic.
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It has been proved that a magic tour is impossible on any board 4n+2 × 4m+2, and the 10×10 is a particular case of this general result.