We tend to apply names from Greek Myth to pieces that combine moves in all the three directions used in chess, namely those of rook, bishop and knight (RBN). The combined wazir-fers-knight (WFN) is the simplest piece of this type, making a single step of each type, making it Knighted King. Since it combines Man and Horse it also has the individual name of Centaur. The Knighted Queen is called an Amazon among Problemists. In variant chess it turns up under numerous different names.
Some other particular cases with restricted queen moves have their own individual names. Pieces with shorter moves can also be named according to the letters assigned to those moves (W for Wazir, F for Fers, D for Dabbaba, N for Knight, A for Alfil, T for Threemover, X for Tripper). These can be listed from shorter to longer or vice versa which gives alternative names for the same piece.
| T | C | Z | X |
| D | N | A | Z |
| W | F | N | C |
| O | W | D | T |
The tours are arranged here according to the number of different moves employed, usually excluding the closure move that joins the end cell to the initial cell. All these Amazomian tours necessarily use at least three different move types.
The most recent update [17 October 2022] adds a number of 6×6 magic tours derived from semimagic king tours. These are in the Centaur, and King + Knight + Dabbaba sections below.
2×3 board. There is an axially symmetric closed centaur tour that can be numbered in two ways to be magic in the files (magic in the ranks of course is impossible). There are also four other closed tours. Two rotary and two asymmetric.
2×4 board There is a biaxial centaur tour that is semi-magic, shown above.
3×3 board. Some history of the 3×3 magic square is outlined in the Theory of Magic page (and PDF #1). It is an open centaur tour.
6×6 board. Magic Centaur Tours can be derived from Walter Trump's list of semimagic king tours with six ranks and four files magic. The non-magic files can be made magic by transposing pairs of complements, but this introduces extra knight or dabbabba moves. Here we show some of those using knght moves.
First we show thirty tours with a single pair of knight moves, in batches of ten.
A second batch of ten.
A third batch of ten.
Now ten tours with two parallel pairs of knight moves.
Lastly two tours with two crrossing pairs of knight moves.
More examples may be added here in due course when further king tours in the list have been considered.
See later on this page for tours using {1,5} moves and tours using knight + dabbaba moves. See also the Queen section for tours using just dabbaba moves.
8×8 board. Alternating Knight + King open tour (¶50) by the Rajah of Mysore (d.1868).
Three diagonal magic tours using king and knight moves by a Monsieur Galtier of La Mastre published in the
'Jeux D'Esprit' column in Le Gaulois (¶309 14 Apr 1882, ¶448 2 Mar 1885, ¶468 30 Mar 1885).
The dark links connect the quarters of the tour.
32×32 board. Robert Bosch has published a series of 32×32 centaur tours on Twitter which are supposedly impressionistic representations of classic works of art, such as the Mona Lisa, darker cells being those where more knight moves congregate.
4×4 board. Among the Frénicle 4×4 diamagic tours there are two open tours of WAN type both axially symmetric (17, 436) and having {0,3} as the closure move. There is also one WAN asymmetric closed tour, with four magic numberings (31, 237, 420, 582).
2×4 board. A semi-magic biaxial closed tour by TFN mover.
2×3 board. These are the four moves possible on the 2×3 board. Tours using all four moves, one rotary, three asymmetric: are possible.
The symmetric tour is "arithmic", the files adding to 6, 7, 8 (cd=1) and ranks 10, 11.
3×3 board The arrangement of the first nine numbers in a square so that all eight lines of three add to the total 15 is the oldest known magic square. See above and PDF #1. This is an open centaur tour. To close the tour requires a fourth type of move {0,2}.
6×6 board. Here we show a batch of ten more magic tours derived from the semimagic king tours, similar to the centaur tours show above. These use one pair of knight moves and one pair of dabbaba moves. The dabbaba moves are shown here as a broken V-shape line, since a curve is more difficult to draw.
4×4 board. This magic square was reported to A. H. Frost (1819-1907) by Robert Shortrede (1800-1868) as having copied it in 1841 from an inscription in an old temple in the hill fort of Gwalior, said to bear the date 1483. It is 304 in the Frénicle list.
6×6 board. Tours of the Falkener type (based on 2×2 King paths) can easily be transformed to give other magic tours in which the {0,1} and {1,1} king moves are replaced by (0,3) and (3,3) moves. This transform applied to a tour using five move-types (shown later) produced this diamagic tour that uses only four types of move. Showing it in diagram form is difficult because of the overlap of moves. Only the ends of the three-unit moves are shown, and the {0,1} moves are curved.
3×7 board. This pattern, which fancifully resembles two chess players, was derived from the tour below, using {0,4} move, by interchange of two files at each end. The {0,3} moves overlap. The reverse numbering is also shown.
3×5 board. Magic rectangle tour by Marian Trenkler (1999), asymmetric with middle row a permuted arithmetic progression (numbering reversed).
3×7 board. The second numbering shown will be seen to be related to the first by inversion of the second, fourth and sixth files. The tour however is the same as the reverse numbering of the diagram (then reoriented so the 1 is at the top left).
