[Magictour] Re: various points arising

GPJ:

>I've been arguing with Eric W. Weisstein about "the meaning of magic"
>since his headlines in MathWorld said there are now known to be
>"no 8x8 magic knight's tours"! He means of course none with diagonals
>also magic. I maintain that a square is a special case of a rectangle
>(one with equal sides) and therefore a magic square is a special case
>of a magic rectangle. No diagonal properties are normally required of
>magic rectangles. And anyway all the people who have constructed
>"magic tours" have always (since Beverley) taken this to mean the
>rows and columns add to the magic constant, and any diagonal properties
>are incidental (Beverley emphasised the magic properties of the 2x2 and
>4x4 blocks in his construction).

I'm not so much obliged to history and who,when called it what and why.
IMO it just makes sense to have a name for the "row+col sum = const. things".
More important than for the "row+col+main-diag. sum = const. things".
I wouldn't call them "magic" though. You could call many things "magic"
without this being a definition. Like amazing,remarkable,interesting,...
These words aren't usually used for definitions either.
However, as you say, when there are magic squares and magic rectangles,
we would like those rectangles which happen to be squares, to
satisfy both magic conditions.
 
>----
>If anyone is interested in doing other searches, an enumeration
>I would like the answer to (but non-magic) is how many 8x8 knight's
>tours are there that are of "squares and diamonds" type?

we will be searching regular 12*12 tours soon, provided it doesn't
take too long. So I think we can do this easily.
(not sure about the definitions, though)
 
>I did some enumerations a few years back, still not published since
>they need checking (though I will put up a page on the subject shortly),
>which found 10,145,864 "Rogetian" tours (i.e. those with three slants),
>1,003,600 of these being reentrant, and thus giving 250,900 closed tours
>with four slants.

I never heard of these. I will have to look up the definitions first.
 
>Of these three groups I found that 2688, 368 and 92 respectively were
>of squares and diamonds type.
>
>Then of course no one has yet checked Brendan Mackay's count of all
>the 8x8 tours!

that's McKay,BTW., not to be confused with our Mackay.
And he only counts the closed tours, so that's just one of our
136 ranges. He gets 1.3e13 tours , with our 200 cycles
per knight-move that would be 2.6e15 cycles or 30 days.
But that's only for the maximum nodes in the tree.
Fei Lu gets 4e4 tours per second with 300MHz , so that's
1.3e5/s*GHz or 1 year with a 3GHz Computer, so it's doable.
Hmm, maybe Fei Lu is counting open tours too, I'll have to check.

Anyway... JC, Hugues , is that a project for us ?!?

BTW. I made a page about latin leaper tours :
http://magictour.free.fr/latin.htm

tell me, what you think and what I should add/change/remove.
I should also include some links to related pages,
now that we get some hundred hits for the page per day !


Guenter



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Received on Thu Dec 04 14:24:18 2001