[Magictour] Re: latin knight tours

GS: when we use octal numbers and start with 0 instead of 1,
then 132 of the SMKTs are semimagic with respect to both of their octal =
digits.

GPJ: Do you have a list of which tours these are? (Preferably using my =
coding system, so that I can mark them in the catalogue.) I suspect they =
will be among those I have listed in the "rhombic" and "beverley" =
sections of the catalogue, since these are constructed on the basis that =
when expressed in base 4 the unit digits in the 4x4 quarters form latin =
squares.

GS: From two orthogonal n*n latin squares you can make a semimagic =
square, by just concatenating the octal numbers from the latin-squares. =
Can knight tours, numbered octally, make such orthogonal latin squares ? =
I checked the known 8*8 - SMKTs and there is none among them, but maybe =
8*8 torus tours ? These should be easier to search by computer, since we =
have strong constraints early in the rows/cols and not only when the sum =
exceeds a value as with SMKTs. What about n*n latin squares, where all =
their n digits form knight-paths of length n ?

GPJ: I don't think they can make a pair of latin squares, because the =
number of "satins" (one cell in each rank and file) in which the cells =
make a path of knight moves are restricted to passing from one quarter =
to the opposite quarter across the centre of the board (as in the =
Jaenisch tour 00a). But something might be possible on the torus where =
movement is much freer. Also "bisatin" paths (two cells in each rank and =
file) may be worth investigating.

GS: if this was already examined, where can I find it ? I just searched =
google for "knight tour" and "latin square" - nothing,)

GPJ: I've thought along these lines a bit, but not got very far, though =
it's an approach obviously worth looking into sometime. There are some =
results on latin squares in books and papers dealing with "semigroups" =
(i.e. groups without the associative axiom), since for these the =
"operation table" is a latin square.




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Received on Thu Dec 04 14:24:18 2001