Princely Tours

© by George Jelliss (July 2014)

Here we gather together results concerning tours that use both knight moves and diagonal moves. We call these Princely tours since the Knight + Fers is known as a Prince, and the Knight + Bishop as a Princess (on the same principle as the distinction beteeen King and Queen, Emperor and Empress).

The first princely tours, indeed the first tours by two-pattern leapers, were the alternating knight-fers and knight-alfil tours given by As-Suli (c.900). These were asymmetric tours. Symmetric examples were constructed by P. B. van Dalfsen who called them crippled knight tours (Fairy Chess Review problem 9601, June 1953, p.74, solutions October 1953, p.93).

To avoid ambiguity it has been necessary to curve the alfil moves in the above diagrams. In any alternating closed tour of the 8×8 board in which one of the component moves is the fers (1,1) or alfil (2,2) or commuter (4,4) the diagonal moves must form a fixed pattern of crosses. In fact these patterns remain fixed on any board 2hx2k as can be proved by starting from the corners and working along the edges. However the coordinates of the digonal moves must divide the board size; thus on the 6×6 we replace (4,4) by (3,3).

I sent the following results to Stefanos Pantazis for the US Probem Bulletin on 14 June 1993, but I don't think they were ever published there. I found that it is possible to construct a pair of knight-fers and knight-alfil tours that are not only symmetric but both use the same pattern of knight's moves! (as shown in the third diagram).

Also from the same letter is an alternating knight-commuter tour (in numerical form because of difficulty in showing it graphically). But using different knight connections.

In the first online issue of The Games and Puzzles Journal #19, January-April 2001, Professor D. E. Knuth improved on the above result by finding two symmetric knight formations that will combine with alternating fers, alfil and commuter to give tours (and with other properties - but see the article).