Queen's Tours

Closed Bimagic Queen's Tours on an 8x8 Board

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A closed bimagic* queen's tour which is pandiagonally magic.

Tour c-08

This is one of the 8 closed bimagic queen's tours which start from d1, the setup square of the white queen.
c-08 is dedicated to my sister Gisela. She left her tour on earth before her 62^th birthday at exit e8.

Introduction

* Magic tours are already called bimagic if the entries satisfy the conditions of a semi-bimagic square.
William Walkington constructed the first bimagic queen's tour on an 8x8 board in July 2020.
In August 2020 I searched for further bimagic queen's tours in the data set of semi-bimagic 8x8 squares.
Including William Walkington's tour there exist 62 open bimagic queen's tours up to symmetry.
A tour is deemed closed if a queen's move is possible from the square with entry 64 to the square with entry 1.
Up to symmetry there are 44 known closed bimagic queen's tours on a 8x8 board.

More details can be found in our Paper on Bimagic Queen's Tours

(PDF)

See also William Walkington's blog about bimagic queen's tours (external link)

Classification of the 44 closed bimagic queen's tours

These tours are different representatives of only 4 classes of semi-bimagic squares.
Attention, none of the normalized representatives shown below is a magic tour.

semi-bimagic
class a

01 08 27 30 42 47 52 53
16 09 22 19 39 34 61 60
20 21 10 15 59 62 33 40
29 28 07 02 54 51 48 41
38 35 64 57 13 12 23 18
43 46 49 56 04 05 26 31
55 50 45 44 32 25 06 03
58 63 36 37 17 24 11 14

semi-bimagic
class b

01 08 27 30 44 45 50 55
16 09 22 19 37 36 63 58
20 21 10 15 57 64 35 38
29 28 07 02 56 49 46 43
39 34 61 60 14 11 24 17
42 47 52 53 03 06 25 32
54 51 48 41 31 26 05 04
59 62 33 40 18 23 12 13

semi-bimagic
class c

01 08 27 30 42 47 52 53
16 09 58 63 20 21 38 35
23 18 33 40 11 14 61 60
26 31 04 05 49 56 43 46
36 37 22 19 64 57 10 15
45 44 55 50 06 03 32 25
54 51 48 41 29 28 07 02
59 62 13 12 39 34 17 24

semi-bimagic
class d

01 08 28 29 43 46 50 55
16 09 34 39 61 60 19 22
23 18 57 64 38 35 12 13
26 31 03 06 52 53 41 48
36 37 14 11 17 24 63 58
45 44 56 49 07 02 30 27
54 51 47 42 32 25 05 04
59 62 21 20 10 15 40 33

The complementary pairs (two entries with sum 65) have three different structures.


Structure X semi-pandiagonal complementary entries have the distance dx=1, dy=1.	Structure Y pandiagonal complementary entries have the distance dx=4, dy=4. (To avoid chaos only 6 pairs are shown.)	Structure Z complementary entries have different distances.

The 6 different kinds of closed queen's tours.

Structure X Class a	Structure X Class b
Structure Y Class a	Structure Y Class b
Structure Z Class c and d

Lists of all bimagic queen's tours on an 8x8 board

The 62 open bimagic queen's tours as semi-bimagic squares

The 44 closed bimagic queen's tours as semi-bimagic squares

All 62·8 = 496 open tour matrices

All 44·8 = 352 closed tour matrices

The 62 open bimagic queen's tours in chess notation

The 44 closed bimagic queen's tours in chess notation

Back to Bimagic Squares

All parts of the website about bimagic queen's tours are published under a Creative Commons licence CC BY-NC-SA 4.0:
https://creativecommons.org/licenses/by-nc-sa/4.0/