A383154 The number of 2n-by-2n fers-wazir tours.

2, 2, 22, 1620, 882130, 3465050546

Offset: 1

Author

Don Knuth, Apr 18 2025

Keywords

• nonn
• more

Comments

The simplest fairy chess pieces, going back to 9th-century Persia, are the fers -- a (1,1) leaper -- and the wazir -- a (1,0) leaper. (A king combines the moves of a fers and a wazir.) A fers-wazir tour visits every cell of a board exactly once, making fers and wazir moves alternately, and returns to the starting cell.

Such tours exist only when the number of rows is even and the number of columns is even.

Examples

			For n=2 the a(2) = 2 solutions are transposes of each other:
.
  0-f 4-3   0 e-d b
   X   X    |X   X|
  e 1-2 5   f 1 a c
  |     |     | |
  d a-9 6   4 2 9 7
   X   X    |X   X|
  b-c 7-8   3 5-6 8
		

References

• D. E. Knuth, Hamiltonian paths and cycles, Section 7.2.2.4 of The Art of Computer Programming (to appear).

Links

• George Jelliss, Introducing Knight's Tours, has a 9th century example of a fers-knight tour due to As-Suli.

Crossrefs

Diagonal of A383153.

Cf. A140519.