[Magictour] Re: various points arising

GPJ:
>Has anyone done a count of 4x8 semi-magic tours (in the sense of
>adding to the magic constant in all rows or all files but not both)?
>I find 66 magic in the 8-cell lines and 8 magic in the 4-cell lines
>(not counting reversals), but Murray (1917) claimed 67 of the first =
type.

GS: I found 62176 knight tours on the 4*8 .
544 were semimagic in the 8-cell lines
64 were semimagic in the 4-cell-lines
unless some of these are symmetric, (I found none) this gives
68 and 8 symmetry-classes .

GPJ: Thanks for that. I rechecked my search and located the two missing =
tours.

-----
On the completion of the 8x8 magic tours search: I've now updated the =
pages in Knight's Tour Notes, including the history and catalogue =
introduction. At John Beasley's request I also wrote an article on =
"Recent Advances in Magic Knight's Tours" to appear in Variant Chess.

-----
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).

Here is a simple magic rectangle by the way (from my special issue of =
Chessics1986 on Magic Tours):

01 15 03 08 13
09 02 11 12 06
14 07 10 04 05

-----
On non-intersecting paths: If you want some other shaped boards to =
investigate here is one (or several): On a sufficiently large array of =
unit square cells draw a circle of diameter 10 units that contains the =
maximum possible number of whole cells. Different types of "circular" =
board are generated depending on where the circle is centred. The =
13-unit case is also tricky.

----
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?

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.

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!

George Jelliss





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Received on Thu Dec 04 14:24:18 2001