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Computing Magic Knight Tours
[1]
in Deutsch : magische Springertouren [2]
+------------------------------+
| Computing Magic Knight Tours |
+------------------------------+
August 5th 2003 : we have finished . All 136 ranges have been checked.
There are 140 magic knight's tours on the chessboard and none of these
is diagonally magic.
This project was to enumerate all the magic knight's tours
("MKTs") on an 8*8 chessboard.
A knight's tour (or knight tour) is an n*n matrix a(n,n) containing
the numbers 1 to n*n exactly once and consecutive numbers are a
chessknight's move apart :
a(u,v)=1+a(x,y) implies (u-x)*(u-x)+(v-y)*(v-y) = 5.
A knight's tour a(,) is magic, iff all rows and columns of a(,)
sum to (n*n+1)*n/2, the magic constant of the MKT.
A knight's tour a(,) is called diagonally magic, iff it is magic
and the two main diagonals sum to the magic constant too.
This project showed that no diagonally magic knight tour exists on
the 8*8 board.
A knight's tour a(,) is called pandiagonal iff all the 2*n broken
diagonals add to the magic constant too.
that means : SUM {j=1..n} of a(j,(i+j) mod n) = n*n+1)*n/2 for i=0..n-1
and SUM {j=1..n} of a(j,(i-j) mod n) = n*n+1)*n/2 for i=0..n-1
Sometimes MKTs are call "semimagic" and diagonally magic tours
are called magic. But we follow Jelliss' nomenclature,
indicating that the two main diagonals are not so important.
In the following the assume n=8.
Prior to this project 2128 MKTs were known,from 133
symmetry-classes (see below). 112 new MKTs were found during
this project from 7 symmetry classes.
for a complete list see :
http://www.ktn.freeuk.com/mc.htm [3]
you can download a complete list of 2240 MKTs, 140 symmetry classes
and 108 geometry-classes in computer-readable form here : (674KB)
mkts [4]
Example of a MKT:
+------+----+----+----+----+----+----+----+----+
| 8- | 01 | 30 | 47 | 52 | 05 | 28 | 43 | 54 |
+------+----+----+----+----+----+----+----+----+
| 7- | 48 | 51 | 02 | 29 | 44 | 53 | 06 | 27 |
+------+----+----+----+----+----+----+----+----+
| 6- | 31 | 46 | 49 | 04 | 25 | 08 | 55 | 42 |
+------+----+----+----+----+----+----+----+----+
| 5- | 50 | 03 | 32 | 45 | 56 | 41 | 26 | 07 |
+------+----+----+----+----+----+----+----+----+
| 4- | 33 | 62 | 15 | 20 | 09 | 24 | 39 | 58 |
+------+----+----+----+----+----+----+----+----+
| 3- | 16 | 19 | 34 | 61 | 40 | 57 | 10 | 23 |
+------+----+----+----+----+----+----+----+----+
| 2- | 63 | 14 | 17 | 36 | 21 | 12 | 59 | 38 |
+------+----+----+----+----+----+----+----+----+
| 1- | 18 | 35 | 64 | 13 | 60 | 37 | 22 | 11 |
+------+----+----+----+----+----+----+----+----+
| | a | b | c | d | e | f | g | h |
+------+----+----+----+----+----+----+----+----+
| Beverley,1848 a8-c1 |
+------+----+----+----+----+----+----+----+----+
this is the first found MKT.
The start-square is the square containing "1" and the end-square
is the square containing "64". These are a8 and c1 in the example above.
For a history of MKT-search see: http://www.ktn.freeuk.com/1d.htm [5]
Every MKT gives 15 other MKTs in the same symmetry-class by rotation,
reflection and reverse numbering.
A tour is called closed, if startsquare and endsquare are a knight's
move apart, otherwise it is called open.
A closed MKT sometimes gives new MKTs in the same geometry-class, by
adding a constant c to each entry:
b(x,y) = ( (a(x,y)-1+c) modulo 64 ) +1
there are 2240 MKTs from 140 symmetry classes and
108 geometry-classes,
1232 open MKTs from 77 symmetry classes and 77 geometry-classes, and
1008 closed MKTs from 63 symmetry classes and 31 geometry-classes.
We estimated that an exhaustive computer-search for all MKTs would
take less than 6 months on a 2GHz computer.
Indeed, after JC Meyrignac had optimized his program a lot
only 2 months were needed !
the program can be downloaded here :
MagicTour program [6]
The program runs at idle priority on Windows. You can stop the program,
but the current computation is lost (there is no save/load function).
The program is heavily optimized, the source-code is long and hard
to understand. Here is a C-program by Fei Lu with explanations:
knight.c [7]
We are also interested in general Hamilton-path-programs to eventually
extend this search to other boards and pieces later.
April 27th 2003: First version of the web site.
May 25th 2003: Second version of the web site
(improvement-suggestions are welcome)
June 19th 2003: Third version of the web site
June 16th 2003: computation of ranges had started
June 18th 2003: Success: after 2 days of computation, Hugues Mackay
had found the first new magic knight's tour !
6 other new MKTs were found during the project.
here are the 7 new MKTs :
+------+----+----+----+----+----+----+----+----+
| 8- | 59 | 30 | 35 | 24 | 57 | 22 | 15 | 18 |
+-----------+----+----+----+----+----+----+----+
| 7- | 36 | 25 | 58 | 29 | 16 | 19 | 56 | 21 |
+------+----+----+----+----+----+----+----+----+
| 6- | 31 | 60 | 27 | 34 | 23 | 54 | 17 | 14 |
+------+----+----+----+----+----+----+----+----+
| 5- | 26 | 37 | 32 | 49 | 28 | 13 | 20 | 55 |
+------+----+----+----+----+----+----+----+----+
| 4- | 39 | 04 | 61 | 12 | 33 | 48 | 53 | 10 |
+------+----+----+----+----+----+----+----+----+
| 3- | 62 | 01 | 38 | 07 | 50 | 11 | 44 | 47 |
+------+----+----+----+----+----+----+----+----+
| 2- | 05 | 40 | 03 | 64 | 45 | 42 | 09 | 52 |
+------+----+----+----+----+----+----+----+----+
| 1- | 02 | 63 | 06 | 41 | 08 | 51 | 46 | 43 |
+------+----+----+----+----+----+----+----+----+
| | a | b | c | d | e | f | g | h |
+------+----+----+----+----+----+----+----+----+
|Mackay/Meyrignac/Stertenbrink 2003/06/18 b3-d2| |
+------+----+----+----+----+----+----+----+----+
+-----+----+----+----+----+----+----+----+----+
| 8- | 11 | 46 | 51 | 40 | 09 | 38 | 31 | 34 |
+-----+----+----+----+----+----+----+----+----+
| 7- | 52 | 41 | 10 | 45 | 32 | 35 | 08 | 37 |
+-----+----+----+----+----+----+----+----+----+
| 6- | 47 | 12 | 43 | 50 | 39 | 06 | 33 | 30 |
+-----+----+----+----+----+----+----+----+----+
| 5- | 42 | 53 | 48 | 01 | 44 | 29 | 36 | 07 |
+-----+----+----+----+----+----+----+----+----+
| 4- | 55 | 20 | 13 | 28 | 49 | 64 | 05 | 26 |
+-----+----+----+----+----+----+----+----+----+
| 3- | 14 | 17 | 54 | 23 | 02 | 27 | 60 | 63 |
+-----+----+----+----+----+----+----+----+----+
| 2- | 21 | 56 | 19 | 16 | 61 | 58 | 25 | 04 |
+-----+----+----+----+----+----+----+----+----+
| 1- | 18 | 15 | 22 | 57 | 24 | 03 | 62 | 59 |
+-----+----+----+----+----+----+----+----+----+
| | a | b | c | d | e | f | g | h |
+-----+----+----+----+----+----+----+----+----+
| Jelliss 2003/06/19 d5-f4 |
+-----+----+----+----+----+----+----+----+----+
+------+----+----+----+----+----+----+----+----+
| 8- | 34 | 51 | 32 | 13 | 62 | 37 | 20 | 11 |
+------+----+----+----+----+----+----+----+----+
| 7- | 15 | 30 | 35 | 50 | 19 | 12 | 61 | 38 |
+------+----+----+----+----+----+----+----+----+
| 6- | 52 | 33 | 14 | 31 | 36 | 63 | 10 | 21 |
+------+----+----+----+----+----+----+----+----+
| 5- | 29 | 16 | 49 | 44 | 05 | 18 | 39 | 60 |
+------+----+----+----+----+----+----+----+----+
| 4- | 48 | 53 | 04 | 17 | 64 | 43 | 22 | 09 |
+------+----+----+----+----+----+----+----+----+
| 3- | 01 | 28 | 45 | 56 | 25 | 06 | 59 | 40 |
+------+----+----+----+----+----+----+----+----+
| 2- | 54 | 47 | 26 | 03 | 42 | 57 | 08 | 23 |
+------+----+----+----+----+----+----+----+----+
| 1- | 27 | 02 | 55 | 46 | 07 | 24 | 41 | 58 |
+------+----+----+----+----+----+----+----+----+
| | a | b | c | d | e | f | g | h |
+------+----+----+----+----+----+----+----+----+
|Mackay/Meyrignac/Stertenbrink 2003/06/21 a3-e4|
+------+----+----+----+----+----+----+----+----+
+------+----+----+----+----+----+----+----+----+
| 8- | 18 | 39 | 64 | 09 | 58 | 41 | 24 | 07 |
+------+----+----+----+----+----+----+----+----+
| 7- | 63 | 10 | 17 | 40 | 23 | 08 | 57 | 42 |
+------+----+----+----+----+----+----+----+----+
| 6- | 38 | 19 | 36 | 61 | 16 | 59 | 06 | 25 |
+------+----+----+----+----+----+----+----+----+
| 5- | 11 | 62 | 13 | 20 | 33 | 22 | 43 | 56 |
+------+----+----+----+----+----+----+----+----+
| 4- | 50 | 37 | 32 | 35 | 60 | 15 | 26 | 05 |
+------+----+----+----+----+----+----+----+----+
| 3- | 31 | 12 | 49 | 14 | 21 | 34 | 55 | 44 |
+------+----+----+----+----+----+----+----+----+
| 2- | 48 | 51 | 02 | 29 | 46 | 53 | 04 | 27 |
+------+----+----+----+----+----+----+----+----+
| 1- | 01 | 30 | 47 | 52 | 03 | 28 | 45 | 54 |
+------+----+----+----+----+----+----+----+----+
| | a | b | c | d | e | f | g | h |
+------+----+----+----+----+----+----+----+----+
|Mackay/Meyrignac/Stertenbrink 2003/06/24 a1-c8|
+------+----+----+----+----+----+----+----+----+
+------+----+----+----+----+----+----+----+----+
| 8- | 54 | 13 | 50 | 23 | 10 | 19 | 44 | 47 |
+------+----+----+----+----+----+----+----+----+
| 7- | 51 | 24 | 53 | 12 | 45 | 48 | 09 | 18 |
+------+----+----+----+----+----+----+----+----+
| 6- | 14 | 55 | 22 | 49 | 20 | 11 | 46 | 43 |
+------+----+----+----+----+----+----+----+----+
| 5- | 25 | 52 | 15 | 64 | 37 | 42 | 17 | 08 |
+------+----+----+----+----+----+----+----+----+
| 4- | 56 | 01 | 26 | 21 | 16 | 63 | 36 | 41 |
+------+----+----+----+----+----+----+----+----+
| 3- | 27 | 30 | 59 | 04 | 33 | 38 | 07 | 62 |
+------+----+----+----+----+----+----+----+----+
| 2- | 02 | 57 | 32 | 29 | 60 | 05 | 40 | 35 |
+------+----+----+----+----+----+----+----+----+
| 1- | 31 | 28 | 03 | 58 | 39 | 34 | 61 | 06 |
+------+----+----+----+----+----+----+----+----+
| | a | b | c | d | e | f | g | h |
+------+----+----+----+----+----+----+----+----+
|Mackay/Meyrignac/Stertenbrink 2003/07/01 b4-d5|
+------+----+----+----+----+----+----+----+----+
+------+----+----+----+----+----+----+----+----+
| 8- | 54 | 45 | 18 | 23 | 42 | 51 | 12 | 15 |
+------+----+----+----+----+----+----+----+----+
| 7- | 19 | 24 | 53 | 44 | 13 | 16 | 41 | 50 |
+------+----+----+----+----+----+----+----+----+
| 6- | 46 | 55 | 22 | 17 | 52 | 43 | 14 | 11 |
+------+----+----+----+----+----+----+----+----+
| 5- | 25 | 20 | 47 | 64 | 05 | 10 | 49 | 40 |
+------+----+----+----+----+----+----+----+----+
| 4- | 56 | 01 | 26 | 21 | 48 | 63 | 36 | 09 |
+------+----+----+----+----+----+----+----+----+
| 3- | 27 | 30 | 59 | 04 | 33 | 06 | 39 | 62 |
+------+----+----+----+----+----+----+----+----+
| 2- | 02 | 57 | 32 | 29 | 60 | 37 | 08 | 35 |
+------+----+----+----+----+----+----+----+----+
| 1- | 31 | 28 | 03 | 58 | 07 | 34 | 61 | 38 |
+------+----+----+----+----+----+----+----+----+
| | a | b | c | d | e | f | g | h |
+------+----+----+----+----+----+----+----+----+
|Mackay/Meyrignac/Stertenbrink 2003/07/01 b4-d5|
+------+----+----+----+----+----+----+----+----+
+------+----+----+----+----+----+----+----+----+
| 8- | 34 | 23 | 42 | 47 | 18 | 31 | 38 | 27 |
+------+----+----+----+----+----+----+----+----+
| 7- | 43 | 48 | 33 | 24 | 37 | 28 | 17 | 30 |
+------+----+----+----+----+----+----+----+----+
| 6- | 22 | 35 | 46 | 41 | 32 | 19 | 26 | 39 |
+------+----+----+----+----+----+----+----+----+
| 5- | 49 | 44 | 21 | 36 | 25 | 40 | 29 | 16 |
+------+----+----+----+----+----+----+----+----+
| 4- | 06 | 53 | 04 | 45 | 20 | 61 | 12 | 59 |
+------+----+----+----+----+----+----+----+----+
| 3- | 01 | 50 | 07 | 56 | 09 | 58 | 15 | 64 |
+------+----+----+----+----+----+----+----+----+
| 2- | 54 | 05 | 52 | 03 | 62 | 13 | 60 | 11 |
+------+----+----+----+----+----+----+----+----+
| 1- | 51 | 02 | 55 | 08 | 57 | 10 | 63 | 14 |
+------+----+----+----+----+----+----+----+----+
| | a | b | c | d | e | f | g | h |
+------+----+----+----+----+----+----+----+----+
|Mackay/Meyrignac/Stertenbrink 2003/08/01 a3-h3|
+------+----+----+----+----+----+----+----+----+
Here are the 136 ranges which have been checked :
+---------+--------+----------+------+-----------+------------------+
| from-to | MKTs | symmetry-| new | new | checked by |
| | known | classes | MKTs | symmetry- | |
| | before | | | classes | |
| | June'03| | | | |
+---------+--------+----------+------+-----------+------------------+
| a1-a2 | 0 | 0 | 0 | 0 | Mackay |
| a1-a4 | 0 | 0 | 0 | 0 | Mackay |
| a1-a6 | 0 | 0 | 0 | 0 | Mackay |
| a1-a8 | 0 | 0 | 0 | 0 | Mackay |
| a1-b3 | 0 | 0 | 0 | 0 | Mackay |
| a1-b5 | 0 | 0 | 0 | 0 | Mackay |
| a1-b7 | 0 | 0 | 0 | 0 | Mackay |
| a1-c4 | 0 | 0 | 0 | 0 | Mackay |
| a1-c6 | 0 | 0 | 0 | 0 | Mackay |
| a1-c8 | 8 | 8 | 1 | 1 | Mackay |
| a1-d5 | 0 | 0 | 0 | 0 | Mackay |
| a1-d7 | 0 | 0 | 0 | 0 | Mackay |
| a1-e6 | 0 | 0 | 0 | 0 | Mackay |
| a1-e8 | 0 | 0 | 0 | 0 | Mackay |
| a1-f7 | 0 | 0 | 0 | 0 | Mackay |
| a1-g8 | 0 | 0 | 0 | 0 | Mackay |
| a2-a3 | 0 | 0 | 0 | 0 | Thomasson,Mackay |
| a2-a5 | 0 | 0 | 0 | 0 | Thomasson,Mackay |
| a2-a7 | 0 | 0 | 0 | 0 | Thomasson |
| a2-b2 | 0 | 0 | 0 | 0 | Thomasson |
| a2-b4 | 0 | 0 | 0 | 0 | Thomasson |
| a2-b6 | 0 | 0 | 0 | 0 | NovaTec |
| a2-b8 | 0 | 0 | 0 | 0 | NovaTec |
| a2-c1 | 0 | 0 | 0 | 0 | NovaTec |
| a2-c3 | 0 | 0 | 0 | 0 | Tyman |
| a2-c5 | 0 | 0 | 0 | 0 | Tyman |
| a2-c7 | 0 | 0 | 0 | 0 | Tyman,NovaTec |
| a2-d2 | 0 | 0 | 0 | 0 | Schauer |
| a2-d4 | 0 | 0 | 0 | 0 | Schauer |
| a2-d6 | 0 | 0 | 0 | 0 | NovaTec |
| a2-d8 | 0 | 0 | 0 | 0 | NovaTec |
| a2-e1 | 0 | 0 | 0 | 0 | NovaTec |
| a2-e3 | 0 | 0 | 0 | 0 | NovaTec |
| a2-e5 | 0 | 0 | 0 | 0 | NovaTec |
| a2-e7 | 0 | 0 | 0 | 0 | Marlow |
| a2-f2 | 0 | 0 | 0 | 0 | Marlow |
| a2-f4 | 0 | 0 | 0 | 0 | NovaTec |
| a2-f6 | 0 | 0 | 0 | 0 | NovaTec, |
| a2-f8 | 0 | 0 | 0 | 0 | NovaTec, |
| a2-g3 | 0 | 0 | 0 | 0 | Euler |
| a2-g5 | 0 | 0 | 0 | 0 | Rottmann |
| a2-g7 | 0 | 0 | 0 | 0 | GrafZahl |
| a2-h2 | 0 | 0 | 0 | 0 | Rottmann |
| a2-h4 | 7 | 7 | 0 | 0 | Mackay |
| a2-h6 | 0 | 0 | 0 | 0 | Euler,Schauer |
| a3-a4 | 1 | 1 | 0 | 0 | Mackay |
| a3-a6 | 0 | 0 | 0 | 0 | Mackay |
| a3-b3 | 0 | 0 | 0 | 0 | Mackay |
| a3-b5 | 8 | 8 | 0 | 0 | Mackay |
| a3-b7 | 0 | 0 | 0 | 0 | Mackay |
| a3-c2 | 1 | 1 | 0 | 0 | Mackay |
| a3-c4 | 0 | 0 | 0 | 0 | Mackay |
| a3-c6 | 3 | 3 | 0 | 0 | Mackay |
| a3-c8 | 1 | 1 | 0 | 0 | Mackay |
| a3-d1 | 0 | 0 | 0 | 0 | Mackay |
| a3-d3 | 0 | 0 | 0 | 0 | Mackay |
| a3-d5 | 4 | 4 | 0 | 0 | Mackay |
| a3-d7 | 0 | 0 | 0 | 0 | Mackay |
| a3-e2 | 0 | 0 | 0 | 0 | Mackay |
| a3-e4 | 1 | 1 | 1 | 1 | Mackay |
| a3-e6 | 0 | 0 | 0 | 0 | GrafZahl,Mackay |
| a3-e8 | 0 | 0 | 0 | 0 | Mackay,GrafZahl |
| a3-f3 | 0 | 0 | 0 | 0 | Mackay,GrafZahl |
| a3-f5 | 0 | 0 | 0 | 0 | Mackay |
| a3-f7 | 0 | 0 | 0 | 0 | Euler |
| a3-g2 | 1 | 1 | 0 | 0 | Mackay |
| a3-g4 | 0 | 0 | 0 | 0 | Euler |
| a3-g6 | 0 | 0 | 0 | 0 | Schauer |
| a3-h3 | 0 | 0 | 2 | 1 | Mackay |
| a3-h5 | 4 | 4 | 0 | 0 | Mackay |
| a4-a5 | 6 | 3 | 0 | 0 | Mackay |
| a4-b2 | 0 | 0 | 0 | 0 | Schauer,Mackay |
| a4-b4 | 0 | 0 | 0 | 0 | Stertenbrink |
| a4-b6 | 19 | 19 | 0 | 0 | Mackay |
| a4-c3 | 0 | 0 | 0 | 0 | Schauer,Mackay |
| a4-c5 | 3 | 3 | 0 | 0 | Mackay |
| a4-c7 | 0 | 0 | 0 | 0 | Mackay |
| a4-d2 | 2 | 2 | 0 | 0 | Mackay |
| a4-d4 | 3 | 3 | 0 | 0 | Mackay |
| a4-d6 | 4 | 4 | 0 | 0 | Mackay |
| a4-d8 | 1 | 1 | 0 | 0 | Mackay |
| a4-e3 | 1 | 1 | 0 | 0 | Mackay |
| a4-e5 | 3 | 3 | 0 | 0 | Mackay |
| a4-e7 | 5 | 5 | 0 | 0 | Mackay |
| a4-f2 | 0 | 0 | 0 | 0 | Mackay |
| a4-f4 | 0 | 0 | 0 | 0 | Mackay |
| a4-f6 | 1 | 1 | 0 | 0 | Mackay |
| a4-g3 | 0 | 0 | 0 | 0 | Mackay |
| a4-g5 | 0 | 0 | 0 | 0 | Mackay |
| a4-g7 | 0 | 0 | 0 | 0 | Mackay |
| a4-h4 | 0 | 0 | 0 | 0 | Mackay |
| b2-b3 | 0 | 0 | 0 | 0 | Mackay |
| b2-b5 | 0 | 0 | 0 | 0 | Mackay |
| b2-b7 | 2 | 1 | 0 | 0 | Mackay |
| b2-c4 | 2 | 2 | 0 | 0 | Mackay |
| b2-c6 | 0 | 0 | 0 | 0 | Mackay |
| b2-d5 | 0 | 0 | 0 | 0 | Mackay |
| b2-d7 | 0 | 0 | 0 | 0 | Mackay |
| b2-e6 | 1 | 1 | 0 | 0 | Mackay |
| b2-f7 | 0 | 0 | 0 | 0 | Mackay |
| b3-b4 | 2 | 2 | 0 | 0 | Mackay |
| b3-b6 | 6 | 3 | 0 | 0 | Mackay |
| b3-c3 | 1 | 1 | 0 | 0 | Mackay |
| b3-c5 | 6 | 6 | 0 | 0 | Mackay |
| b3-c7 | 0 | 0 | 0 | 0 | Mackay |
| b3-d2 | 4 | 4 | 1 | 1 | Mackay |
| b3-d4 | 8 | 8 | 0 | 0 | Mackay |
| b3-d6 | 3 | 3 | 0 | 0 | Mackay |
| b3-e3 | 0 | 0 | 0 | 0 | Mackay |
| b3-e5 | 0 | 0 | 0 | 0 | Mackay |
| b3-e7 | 0 | 0 | 0 | 0 | Mackay |
| b3-f4 | 0 | 0 | 0 | 0 | Mackay |
| b3-f6 | 0 | 0 | 0 | 0 | Mackay |
| b3-g3 | 0 | 0 | 0 | 0 | Mackay |
| b3-g5 | 0 | 0 | 0 | 0 | Mackay |
| b4-b5 | 0 | 0 | 0 | 0 | Mackay |
| b4-c4 | 0 | 0 | 0 | 0 | Mackay |
| b4-c6 | 0 | 0 | 0 | 0 | Mackay |
| b4-d3 | 0 | 0 | 0 | 0 | Mackay |
| b4-d5 | 0 | 0 | 2 | 2 | Mackay |
| b4-d7 | 0 | 0 | 0 | 0 | Mackay |
| b4-e4 | 1 | 1 | 0 | 0 | Mackay |
| b4-e6 | 0 | 0 | 0 | 0 | Mackay |
| b4-f3 | 0 | 0 | 0 | 0 | Mackay |
| b4-f5 | 0 | 0 | 0 | 0 | Mackay |
| b4-g4 | 12 | 6 | 0 | 0 | Mackay |
| c3-c4 | 2 | 2 | 0 | 0 | Mackay |
| c3-c6 | 0 | 0 | 0 | 0 | Mackay |
| c3-d5 | 6 | 6 | 0 | 0 | Mackay |
| c3-e6 | 1 | 1 | 0 | 0 | Mackay |
| c4-c5 | 0 | 0 | 0 | 0 | Mackay |
| c4-d4 | 0 | 0 | 0 | 0 | Mackay |
| c4-d6 | 2 | 2 | 0 | 0 | Mackay |
| c4-e5 | 0 | 0 | 1 | 1 | Mackay |
| c4-f4 | 0 | 0 | 0 | 0 | Mackay |
| d4-d5 | 0 | 0 | 0 | 0 | Mackay |
+---------+--------+----------+------+-----------+------------------+
| total: | 146 | 133 | 8 | 7 | |
+---------+--------+----------+------+-----------+------------------+
Thanks to
Hugues Mackay, Dan Thomasson, Guenter Stertenbrink, Tom Marlow, NovaTec,
Hendrik Tyman, Werner Schauer, Thilo Rottmann, GrafZahl, Euler
for computing ranges and helping to finish the project.
Special thanks to Hugues Mackay, who computed more than 100 ranges alone !
Not to forget JC Meyrignac for writing the fast program and helping to
set up this webpage
as of August,05.,2003 136 ranges have been checked
5305279.08 seconds = 61.40 days of computation time were needed
CPUs went through 11_945038 Gigacycles = 138.25 days with 1GHz
52_280100_180652 knightmoves were done, that's 228.5 CPU-cycles per move(=node)
between 7.0e10 and 3.1e12 nodes per range, average 3.8e11, deviation 5.4e11
nodecounts of the different ranges : nodecounts [8]
graphics of the complexity of the ranges : NODES2.GIF [9]
computer-readable list of all MKTs : (674KB) mkts.txt [10]
picture of the 108 geometry-classes: MKTS108B.GIF [11]
our page on latin torus leaper tours : latin torus-leaper-tours [12]
our oncoming(?) project to (re)count all 8*8 knight tours :
computing knight tours [13]
here is the mathworld article about the project : mathworld article [14]
other pages:
Jelliss, knight tour notes [15]
[ most comprehensive webpage about knight tours ! ]
Velucchi, knight-links [16]
[ special site devoted to knight tour links ]
Friedel,F.:"The Knight's Tour." [17]
[ entertaining and curious ]
mathworld-MagicTour: [18]
[ introduction,definitions,examples,links
note, that E.Weisstein at Mathworld uses different definitions than we and
Jelliss:
"magic tour" (Mathworld) = "diagonally magic tour" (Jelliss)
"semimagic tour" (Mathworld) = "magic tour" (Jelliss)
"semimagic tour" (Jelliss) = [only the rows add to the magic constant]
some links to German sites : Heise Artikel [19]
[ article about this project with discussion ] A.Conrad [20]
[ how to find a tour on large boards ] Dan Thomasson [21]
[ the same, with another method, nice graphics ] PDF-paper by Lee [22]
[ about fast algorithms to count knight tours ]
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| last updated: January/14/2004 | c 2003 Guenter Stertenbrink |
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