Quaternary Pseudotours
formed of four copies of a 16-move knight path

by G. P. Jelliss, April 2016 (based on work by P. de Hijo 1882 and T. W. Marlow 1985).

All possible ways of covering the 8×8 board with four congruent 16-move paths to form quaternary symmetric pseudotours were listed by the Abbé Philippe Jolivald under the alias of 'Paul de Hijo' (1882). For the cases of oblique symmetry he found 140 arrangements of four 16-move circuits and 150 arrangements of two 32-move circuits. For the cases of direct quaternary symmetry he found 368 arrangements of four 16-move circuits and 378 arrangements of two 32-move circuits. An error in his book misprints the 368 total as 301 and this figure was copied by later writers, but his count is otherwise correct. The correct totals were confirmed in a computer study by T. W. Marlow reported in Chessics (#24 Winter 1985 p.92).

A square pattern is said to have quaternary symmetry if it is unaltered by four of the transformations that preserve the square. In other words by rotations and reflections of the square it can be shown in only two distinct forms. Quaternary symmetry is of two types, direct where it is changed by 90 degree rotation, and oblique where it is changed by reflection. Here we study ways of arranging 16-move knight paths to form pseudotours with quaternary symmetry on the 8×8 board.

The scheme for coding knight paths used here, which I call centrifugal numbering, is applicable to odd and even boards of any sizes. The numbering is row by row from the centre cell or cells outwards, beginning with 0. In the case of the 7×7 and 8×8 board there are ten differently placed cells, and so this system conveniently labels them with the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

This scheme has the advantage that any sequence of numbers that represents a path of knight moves on a small square board also represents a path of knight moves on any larger square board of the same parity. For example on any even square board the eight moves 1-1 and the eight moves 0-2 form the two squares and two diamonds in the central 4×4 area.

Using this coding the 41 possible knight moves on the 8×8 are as listed here. (There are 42 possible moves in any given direction, but there are two labelled 1-1, parallel to each other.)
0 → 1², 2², 3², 4²
1 → 0, 1², 3, 4, 5, 6, 7
2 → 0², 3², 6², 8²
3 → 0, 1, 2, 4, 7, 8
4 → 0, 1, 3, 6, 7, 9
5 → 1², 6²
6 → 1, 2, 4, 5
7 → 1, 3, 4, 8
8 → 2, 3, 7
9 → 4²

The superscript ² indicates a choice of two moves to cells with the same number.

A pair of numbers u-v in most cases indicates a single unambiguous move from any of the cells with the first number u. Ambiguity occurs only in the case of the diagonal cells 0, 2, 5, 9, ... and the special moves 1-1 which (on even boards) form four-move circuits round the centre.

In paths and tours sequences of the form u-v-u are possible only when v is a diagonal cell (e.g. 4-9-4 through the corner cell of the 8×8 board): if v is not a diagonal cell u-v-u (e.g. 9-4-9) represents a 'switchback', using the same cell u twice, which by definition is not allowed in a tour.

In applying our coding to enumerate 16-move paths in quaternary symmetry we need to note that the diagonal cell numbers are used once and the others twice, and that the two occurrences of each off-diagonal number must have a diagonal number (0, 2, 5, 9) somewhere between them.

As a matter of historical interest, various ways of labelling the cells have been proposed for purposes of enumerating tours on particular boards, or for constructing tours with a particular symmetry. The works of Vandermonde (1771), Lavernède (1939), Roget (1840), de Hijo (1882), Bergholt (1918) and Murray (1942) all used different methods. A ten digit scheme for the 8×8 board was used by Murray (1942) but he numbered the diagonal cells inwards 0, 1, 2, 3 and the other cells 4, 5, 6, 7, 8, 9 row by row. This method has the disadvantage that sequences of cells that indicate moves on smaller boards do not also indicate similar moves on larger boards.

The following tables of the numbers of paths between particular squares are adapted from Marlow's results. It is assumed that the path starts in a corner (9)-(4). The next move is then to any of the cells numbered (0), (1), (3), (6), (7) and the path continues until the 15th cell reached is also one of the cells numbered (0), (1), (3), (6), (7) that is a knight's move from the appropriate (4) that connects either to the initial corner (for closed 16-move circuits), or the opposite corner (for open paths which combine to form 32-move circuits). An 'appropriate' (4) is one related to the initial (4) by reflection in a diagonal.
-Direct closed (Dc)Direct open (Do)Oblique closed (Oc)Oblique open (Oo)
-total 368total 378total 140total 150

As an example to illustrate the four cases we use the circuits used by A-T. Vandermonde in his 1771 article that began the mathematical study of this subject. In terms of our cell coding the four 16-move corner-to-corner paths, both closed and open, are represented by the same sequence 9478310-2656138749. The path is closed or open according to which (2) the knight moves to following the (0). There is also an apparent choice of (6) following the (2), but the wrong choice here leads the knight into the final corner from the wrong direction.

A peculiarity of the circuits chosen by Vandermonde is that by reflecting them in a diagonal they can also be arranged in oblique quaternary symmetry, as noted by H. E. Dudeney (Amusements in Mathematics 1917). For it to be possible to arrange four identical 16-move paths in both direct and oblique quaternary formation, the pattern of cells occupied by two diametrally paired paths must be reflectible in a diagonal, as well as being superposable by a half-turn. These two properties combine to ensure that it must also be reflectible in the other diagonal, i.e. the pattern of cells (but not the pattern of moves) has both diagonals as axes of symmetry. There are ten other formulae that give four pseudotours related in the same way as in Vandermonde's example.

The catalogue that follows shows diagrams of all quaternary pseudotours formed of four 16-move paths. The border of a pseudotour is shown as a broken line to distinguish it from a tour. I believe this work is the first in which all the pseudotours have been shown in pictorial form. Other quaternary pseudotours are of course possible, for example formed of eight 8-move paths.

Catalogue of Quaternary Pseudotours

8-fold: There are 2 formulae that give 8 pseudotours, all in direct symmetry, 4 in which the 16-move path is closed and 4 in which it is open (4Dc, 4Do). Recall that each formula should be preceded by 94 and followed by 49, indicating the corner moves.

The eight cases can be distinguished if required by means of the underlining convention: the code number for a diagonal cell, 0, 2, 5, is underlined if the move does not pass through the diagonal; a 1-1 move is underlined if it does not cross the diagonal through the initial corner:

For example, the first formula has the eight forms: 6203873871156, 6203873871156, 6203873871156, 6203873871156, 6203873871156, 6203873871156, 6203873871156, 6203873871156.

4-fold:. There are 62 formulae giving 4 pseudotours. They form four classes 4a, 4b, 4c, 4d.

4a: 20 formulae giving one of each type (1Dc, 1Do, 1Oc, 1Oo).
These include the 11 of Vandermonde type, marked (V). 4a: formulae 1-5:

4a formulae 6-10

4a formulae 11-15

4a formulae 16-20

4b: 39 formulae giving pseudotours in direct symmetry, 2 closed paths and 2 open paths (2Dc, 2Do).

Of  these, 37 include move 1-1 which allows only direct symmetry. The other two contain 13873871 or its reverse.

When all formulae are put in numerical order, the first six are 4b type.
4b formulae 1-5:

4b formulae 6-10.

4b formulae 11-15.

4b formulae 16-20.

4b formulae 21-25.

4b formulae 26-30.

4b formulae 31-35.

4b formulae 36-39.

4c: 1 formula (giving 2Dc, 2Oc).

4d: 2 formulae (giving 2Do, 2Oc).

2-fold: There are 235 formulae giving two pseudotours. They form ten classes.

Three classes give direct symmetry (2a, 2b, 2c). Three give oblique symmetry (2d, 2e, 2f). Four give direct and oblique (2g, 2h, 2i, 2j).

2a: 111 (1Dc, 1Do).

2a formulas 1-4

2a formulae 5-14.

2a formulae 15-24.

2a formulae 25-34

2a formulae 35-44

2a formulae 45-54

2a formulae 55-64

2a formulae 65-74

2a formulae 75-84

2a formulae 85-94

2a formulae 95-104

2a formulae 105-111

2b: 4 (2Dc): 2b formulae 1-2

2b formulae 3-4

2c: 8 (2Do). 2c formulae 1-8.

2d: 23 (1Oc, 1Oo).

2d formulae 1-10:

2d formulae 11-20:

2d formulae 21-23:

2e: 1 only (2Oc).

2f: 2 only (2Oo).

2g: 20 (1Dc, 1Oc). 2g formulae 1-10, cases of one path.

2g: 20 (1Dc, 1Oc). 2g formulae 10-20, cases of two different paths:

2h: 25 (1Dc, 1Oo). 2h formulae 1-10.

2h formulae 11-20:

2h formulae 21-25:

2i: 11 (1Do, 1Oc).

2i formulae 1-4:

2i formulae 5-11:

2j: 30 (15 cases of one path and 15, marked *, of two different paths) (1Do, 1Oo).

2j formulae 1-10:

2j formulae 11-20:

2j formulae 21-30:

1-fold: There are 302 formulae that yield just one pseudotour. They form four classes 1a 96 Dc, 1b 100 Do, 1c 58 Oc, 1d 48 Oo.

1a: 96 (Dc). 1a formulae 1-20:

1a formulae 21-40:

1a formulae 41-60:

1a formulae 61-80:

1a formulae 81-96:

1b: 100 (Do). 1b formulae 1-20:

1b formulae 21-40:

1b formulae 41-60:

1b formulae 61-80:

1b formulae 81-100:

1c: 58 (Oc). 1c formulae 1-20:

1c formulae 21-40:

1c formulae 41-58:

1d: 48 (Oo). 1d formulae 1-20:

1d formulae 21-40:

1d formulae 41-48:

Table of numbers of formulae and pseudotours

Key: D = direct, O = oblique, c = closed, o = open.