[Magictour] Re: Perhaps too easy

GS: I found lots of (semi)magic 3*3*3 cubes, but no magic torus knight =
tours. Do the theorems 1-4 magically extend into 3 dimensions ?

GPJ: As regards the general theorems:
Theorem 1 no longer quite applies on magic rectanguloids (hxwxd). =
Instead it shows that they must have (a) all sides even (b) all sides =
odd or (c) one side odd and two sides even. Theorem 2 still applies on =
bounded boards, and on evevxevenxeven torus boards, but not on toral =
boards with an odd dimension, because the knight can then move to a =
same-colour cell. Theorem 3 still seems to apply in cases where two =
dimensions are even. Theorem 4 does not seem to carry over to three =
dimensions.

I've not previously thought at all about 3D magic tours (though I have =
constructed 3D knight tours with geometrical properties). So I looked in =
W.S.Andrews "Magic Squares and Cubes" (1917, reprinted by Dover 1960), =
and his first example is:

10 26 06 23 03 16 09 13 20
24 01 17 07 14 21 11 27 04
08 15 19 12 25 05 22 02 18

This is constructed on the torus principle, though he doesn't use the =
term, saying: "when a move is to be continued upward from the top square =
it is carried around to the bottom square, and when a move is to be made =
downward from the bottom square it is carried round to the top square, =
the conception being similar to that of the horizontal cylinder used in =
connection with odd magic squares".

The tour is constructed on what I call the "step-side-step method" =
except that there are two sidesteps. He calls them "breakmoves". Most of =
the moves are knight moves (0,1,2) which become fers moves (0,1,1) when =
taken "round the bend". The first breakmoves are sprite moves (1,1,1) =
occurring at 3-4, 6-7, 12-13, 15-16, 21-22, 24-25 (i.e. at multiples of =
3). The second breakmoves are wazir moves (0,0,1) occurring at 9-10, =
18-19, 27-1 (i.e. at multiples of 9).

He assumes that a "magic cube" requires the four "great diagonals" (i.e. =
connecting opposite corners) to add to the magic constant, but not the =
face diagonals, so that the faces and cross-sections are not themselves =
"magic squares" (although the middle ones are here, because of the =
symmetric construction - which is also responsible for the =
space-diagonals being magic).

This just points up how arbitrary the convention is that "magic squares" =
should have their diagonals magic. I wish everyone would adopt the =
logical defnition of "magic" as applying only to the lines parallel to =
the sides and recognise that diagonal properties are just one type of =
"add-on". Then we can get rid of the horrible nomenclature of calling =
magic knight's tours "semimagic", which treats the diagonal magic =
property as if it were an important feature (half the magic) and not a =
mere incidental.

George Jelliss




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Received on Thu Dec 04 14:24:18 2001