[Magictour] Re: on rectangles

JCM asked: "Since tours on 4m x 4n+2 are unlikely to exist, is one magic =
tour on 4x8 or 6x10 known ? Also, does an (odd)x(odd) magic tour exist =
?" and: "George, can you give us a list of rectangular / square boards =
where magic tours may exist, but are not known?"

Most of these questions are answered on the first page on 'General =
Theory' of magic tours on the KTN website. No magic knight tours are =
possible on oddxodd boards (since alternate lines add to odd and even =
sums). No magic knight tours are possible on boards 4m+2 by 4n+2, such =
as 6x10 (this I recently proved - see the site cited).

However I notice that I have made an error in stating that magic knight =
tours are known to exist on all boards 4m by 4n except 4x4.
I believe Murray established that there are no magic knight tours 4x8 =
(by constructing all possible semimagic ones). However I don't think the =
cases 4x12, 4x16 etc have been properly examined. There are magic tours =
on all boards 4m by 4n larger than 8x8.

Whether there are magic tours 4m by 4n+2 remains unknown, though Tim =
Roberts and myself have eliminated the cases smaller than 8x10 and a few =
others.

GS wrote: "I see from the other proofs on your webpage, that these only =
depend on the values mod 4 in the magic rectangle."

In fact I think it's not quite as simple as this. The argument to prove =
the 4m+2 by 4n+2 case is quite subtle, involving a special partitioning.

GS also wrote: "So, is it true that you suspect, that any knight's tour =
on a 8*10 is impossible, where the corresponding rectangle with all =
move-numbers mod 4 has constant row and column (rank and file-) - sums =
mod 4?"

I'm afraid I can't quite follow what this means (hence my delay in =
replying). It may have become garbled in translation, or I may just be =
dense - I've never been keen on "mod". I do suspect that an 8x10 magic =
tour is impossible, but could be wrong.

When I try to construct an 8x10 magic knight tour it almost works but =
it's just impossible to get the final connections. The connections =
become possible if wazir moves (single-step rook moves) are allowed. The =
key to the problem I think lies in Roget's LEAP labelling of the cells. =
The knight can make moves of types LL, EE, AA, PP (straights) and LE, =
LA, PE, PA (slants) but not moves LP, EA. On the other hand a wazir =
easily makes LP or EA moves.

George Jelliss
25 June 2003




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Received on Thu Dec 04 14:24:18 2001