recent updates:

Sep.2013
Alex Chernov counted knight's tours

http://alex-black.ru/article.php?content=141

http://magictour.free.fr/enum will be updated

Jan.2009
Awani Kumar has enumerated over 3000 semi-magic knight's tours

on 10x10 board achieving 75% magic ratio, the highest on any

singly-even board till date.

Here is such a tour:
10x10

21.Nov.2008
Awani Kumar has enumerated 208 magic tours of knight in 4x4x4x4 hyperspace.

Sum of all the rows,columns,pillars and files is 514

4x4x4x4,base 4

17.Oct.2008
Awani Kumar has further extended the knight's tour into fifth dimension.

He has constructed a knight's magic tour in a 4x4x4x4x4 hyperspace, the largest till date.

Sum of all the rows,columns,pillars,files and poles is 2050.
4x4x4x4x4

27.Sept.2008
Awani Kumar has extended the knight's tour into fourth dimension.

He has constructed a knight's magic tour in a 4x4x4x4 hyperspace.

Sum of all the rows,columns,pillars and files is 514.
4x4x4x4

27.March 2008:
Awani Kumar's larger knight's tour magic squares:

1) 12x12
2) 16x16,diagonal
3) 14x14 semi-magic

16.December 2007:
Awani Kumar's larger knight's tour magic cubes:

1) 8x8x8
2) 8x8x8,triagonal
3) 8x8x8,triagonal
4) 12x12x12,triagonal

cube properties

**
15.October 2007: And now the 4x4x4 magic tours !
4x4x4 **

54 magic 4x4x4 tours

**
04.August 2007: Awani Kumar sent me a list of 76 16x16 magic
knight's tours , the 2 diagonals are also magic.
16x16 **

**
01.July 2007: Awani Kumar sent me a list of 1306 12x12 (semi-) magic
knight's tours !
When you have further 12x12 semi-magic knight's tours, please send them
so they can be included.
12x12 **

**
28.April 2007: Awani Kumar found a magic 3-dimensional 8*8*8 Knight's Tour !
3d-magic tour **

**
1st Nov.2005: Dan Thomasson illustrates the SMKTs
and groups them by halftours and other similarities :
Dan's pictures **

**
15th May 2004: Yann Denef redid the whole calculation with a slightly
different method in 8.22*10^15 CPU cycles, 30% faster than our program !
see his webpages at (in French) :
10th August 2004: Yann Denef improved his program, by avoiding symmetries,
now he needs 5.5*10^15 cycles = 67% of his 1st version = 46% of our program
Yann Denef's pages **

08th November 2004: I put a new c-program here to illustrate the

algorithm how to find Hamilton paths ("tours") in undirected

graphs:

20th January 2005: enumerating knight's tours on rectangular boards

**
in Deutsch :
magische Springertouren**

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.

http://www.ktn.freeuk.com/mc.htm you can download a complete list of 2240 MKTs, 140 symmetry classes

and 108 geometry-classes in computer-readable form here : (674KB)

mkts

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

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

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

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 |

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

graphics of the complexity of the ranges :
NODES2.GIF

computer-readable list of all MKTs : (674KB)
mkts.txt

picture of the 108 geometry-classes:
MKTS108B.GIF

graphics of the start and end squares (courtesy to Harold Cataquet):
reachable squares

our page on latin torus leaper tours :
latin torus-leaper-tours

our oncoming(?) project to (re)count all 8*8 knight tours :
computing knight tours

Jelliss, knight tour notes

[ most comprehensive webpage about knight tours ! ]

Velucchi, knight-links

[ special site devoted to knight tour links ]

Friedel,F.:"The Knight's Tour."

[ entertaining and curious ]

mathworld-MagicTour:

[ 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 [ article about this project with discussion ]

A.Conrad

[ how to find a tour on large boards ]

Dan Thomasson

[ the same, with another method, nice graphics ]

PDF-paper by Lee

[ about fast algorithms to count knight tours ]

Visitor number since May 1st 2003.

[return to top]

last updated: January 20th 2005
| created 2003 by Guenter Stertenbrink and Jean Charles Meyrignac mailto:sterten@posteo.de |