One object I've pondered recently is a 3x3x3 cube
with the center cube removed. That makes 26
cubes, nicely fitting in with a 26 letter alphabet.
How many 3-letter words are possible in a knight's
tour of a centerless cube?
--Ed Pegg Jr.
no knight's tour is possible. (provided I have no bug)
I assume that the 3d-knight is a (0,1,2)-leaper able to move to 24 cells =
on a sufficiently large board.
Guenter
That's right, no tour is possible: "chequer" the 3D board and there are =
14 cells of one colour and 12 of the other, an excess of two. This is =
the same argument that proves there is no knight tour on a chessboard =
with two opposite corners missing; a well known old puzzle. However ... =
suppose the cube is a "hypertorus" (I presume there is such a thing or =
else I've just invented it) where each face connects to the opposite =
face. Then a knight can move between cells of the same colour. Such a =
move is equivalent to a fers move (one step diagonally) and now a tour =
is possible!
GPJ
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Received on Thu Dec 04 14:24:18 2001
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