Semi-Magic Knight's Tours

Part 2: Semi-Magic Tours of Knight on 6×6 Board

By Awani Kumar, Lucknow-226010, INDIA (May 2002).


The following study finds the total number of semi-magic tours on the 6 by 6 board = 88. Of these: closed magic tours = 26; open magic tours = 62. Further notes are given at the end.

Key: D1=Diagonal sum, top left to bottom right.
D2=Diagonal sum, top right to bottom left.
Magic Constant=111. Mauve band indicates D1 + D2 = 222.
The end-cells, numbered 1 and 36, are marked in blue.


11035282512130133632813013367241301336274
3429211827141922912351419823123514192831235
3369261324238311827429231182562923118526
3033205167201522934112015229341120152293411
21431182314724173252632817325262381732256
32192215617162162510331621427103316212471033
 1  D1=84  D2=102 2  D1=98  D2=120 3  D1=98  D2=120 4  D1=118  D2=80

1369282512136132433413613243342912313423
3429211827122534314231425343122311301241332
31035261324352152253235215225328310332235
3033205167261128718212621281781129185202514
214311823142916920316291619103164716273621
3219221561710273017819202730718917281961526
 5  D1=110  D2=102 6  D1=98  D2=94 7  D1=98  D2=102 8  D1=124  D2=98

2 9 12 31 34 23 2 13 4 33 36 23 2 13 4 33 36 23 2 13 18 31 36 11
11 30 1 24 13 32 5 32 1 24 15 34 15 32 1 24 5 34 17 30 1 12 19 32
8 3 10 33 22 35 12 3 14 35 22 25 12 3 14 35 22 25 14 3 16 33 10 35
29 18 21 4 25 14 31 6 29 10 19 16 31 16 29 20 9 6 29 6 9 22 25 20
20 7 16 27 36 5 28 11 8 17 26 21 28 11 18 7 26 21 8 15 4 27 34 23
17 28 19 6 15 26 7 30 27 20 9 18 17 30 27 10 19 8 5 28 7 24 21 26
 9  D1=108  D2=114 10  D1=102  D2=120 11  D1=102  D2=120 12  D1=130  D2=92

2 13 18 31 36 11 2 15 36 19 22 17 2 17 8 35 26 23 2 25 4 33 36 11
17 30 1 12 19 32 35 28 3 16 9 20 7 34 1 24 9 36 27 32 1 12 5 34
14 3 16 33 10 35 14 1 34 21 18 23 16 3 18 27 22 25 24 3 26 35 10 13
29 24 27 4 7 20 29 4 27 10 33 8 33 6 31 12 19 10 31 28 17 8 21 6
26 15 22 9 34 5 26 13 6 31 24 11 30 15 4 21 28 13 16 23 30 19 14 9
23 28 25 6 21 8 5 30 25 12 7 32 5 32 29 14 11 20 29 18 15 22 7 20
13  D1=94  D2=128 14  D1=130  D2=92 15  D1=114  D2=110 16  D1=102  D2=120

2 25 28 9 36 11 2 25 28 21 12 23 2 25 28 21 12 23 2 27 18 31 4 29
27 8 1 12 29 34 27 10 1 24 29 20 27 20 1 24 29 10 17 12 3 28 19 32
24 3 26 35 10 13 36 3 26 11 22 13 36 3 26 11 22 13 26 1 36 13 30 5
7 16 5 20 33 30 9 6 15 32 19 30 19 16 5 32 9 30 11 16 9 22 33 20
4 23 18 31 14 21 4 35 8 17 14 33 4 35 18 7 14 33 8 25 14 35 6 23
17 6 15 22 19 32 7 16 5 34 31 18 17 6 15 34 31 8 15 10 7 24 21 34
17  D1=102  D2=120 18  D1=102  D2=120 19  D1=102  D2=120 20  D1=112  D2=110

2 33 36 19 4 17 2 35 24 13 4 33 2 35 24 13 4 33 3 16 13 30 27 22
35 10 3 16 27 20 23 12 3 34 25 14 25 12 3 34 23 14 14 29 2 23 12 31
32 1 34 21 18 5 36 1 22 15 32 5 36 1 22 15 32 5 17 4 15 28 21 26
11 22 9 28 15 26 11 16 9 20 29 26 11 26 9 30 19 16 36 7 24 1 32 11
8 31 24 13 6 29 8 21 18 27 6 31 8 21 28 17 6 31 5 18 9 34 25 20
23 12 7 30 25 14 17 10 7 30 19 28 27 10 7 20 29 18 8 35 6 19 10 33
21  D1=94  D2=128 22  D1=90  D2=120 23  D1=90  D2=128 24  D1=106  D2=112

3 16 13 30 27 22 3 28 13 36 5 26 3 30 13 36 1 28 4 1 24 13 36 33
14 31 2 23 12 29 14 19 4 27 12 35 14 19 2 29 12 35 23 12 3 34 25 14
17 4 15 28 21 26 29 2 31 18 25 6 23 4 31 18 27 8 2 5 22 15 32 35
32 7 24 1 36 11 20 15 22 9 34 11 20 15 22 9 34 11 11 8 17 28 21 26
5 18 9 34 25 20 1 30 17 32 7 24 5 24 17 32 7 26 6 29 10 19 16 31
8 33 6 19 10 35 16 21 8 23 10 33 16 21 6 25 10 33 9 18 7 30 27 20
25  D1=110  D2=112 26  D1=102  D2=124 27  D1=102  D2=120 28  D1=102  D2=128

4 1 24 13 36 33 4 1 24 13 36 33 4 1 24 13 36 33 4 1 36 13 30 27
23 12 3 34 25 14 23 14 3 34 25 12 25 12 3 34 23 14 35 12 3 28 19 14
2 5 22 15 32 35 2 5 22 15 32 35 2 5 22 15 32 35 2 5 18 31 26 29
11 16 9 20 29 26 21 18 7 28 11 26 11 26 9 30 19 16 11 34 9 22 15 20
6 21 18 27 8 31 6 29 20 9 16 31 6 21 28 17 8 31 6 23 32 17 8 25
17 10 7 30 19 28 19 8 17 30 27 10 27 10 7 20 29 18 33 10 7 24 21 16
29  D1=94  D2=120 30  D1=94  D2=128 31  D1=94  D2=128 32  D1=80  D2=142

4 13 2 33 36 23 4 27 8 35 16 21 4 27 8 35 16 21 4 27 10 33 16 21
1 32 5 24 15 34 9 34 5 20 7 36 9 36 5 20 7 34 11 32 5 20 9 34
12 3 14 35 22 25 26 3 28 15 22 17 26 3 28 15 22 17 26 3 28 15 22 17
31 6 29 10 19 16 33 10 19 6 29 14 29 10 19 6 33 14 31 12 19 6 35 8
28 11 8 17 26 21 2 25 12 31 18 23 2 25 12 31 18 23 2 25 14 29 18 23
7 30 27 20 9 18 11 32 1 24 13 30 11 30 1 24 13 32 13 30 1 24 7 36
33  D1=104  D2=120 34  D1=120  D2=98 35  D1=124  D2=98 36  D1=124  D2=102

4 27 10 33 16 21 4 27 12 31 16 21 4 27 12 31 16 21 4 27 14 29 16 21
11 34 5 20 9 32 11 30 5 20 13 32 11 32 5 20 13 30 13 30 5 20 7 36
26 3 28 15 22 17 26 3 28 15 22 17 26 3 28 15 22 17 26 3 28 15 22 17
35 12 19 6 31 8 29 10 19 6 33 14 33 10 19 6 29 14 31 12 19 6 35 8
2 25 14 29 18 23 2 25 8 35 18 23 2 25 8 35 18 23 2 25 10 33 18 23
13 36 1 24 7 30 9 36 1 24 7 34 9 34 1 24 7 36 11 32 1 24 9 34
37  D1=120  D2=102 38  D1=120  D2=102 39  D1=124  D2=102 40  D1=120  D2=98

4 27 14 29 16 21 4 27 18 31 2 29 4 27 30 13 16 21 4 27 30 13 16 21
13 36 5 20 7 30 17 12 3 28 19 32 31 12 5 20 29 14 31 14 5 20 29 12
26 3 28 15 22 17 26 5 36 13 30 1 26 3 28 15 22 17 26 3 28 15 22 17
35 12 19 6 31 8 11 16 9 22 33 20 11 32 19 6 35 8 35 32 19 6 11 8
2 25 10 33 18 23 6 25 14 35 8 23 2 25 34 9 18 23 2 25 34 9 18 23
11 34 1 24 9 32 15 10 7 24 21 34 33 10 1 24 7 36 33 36 1 24 7 10
41  D1=124  D2=98 42  D1=116  D2=110 43  D1=104  D2=142 44  D1=80  D2=142

4 33 24 13 2 35 4 33 24 13 2 35 4 33 36 19 2 17 4 35 24 13 2 33
23 12 3 34 25 14 23 14 3 34 25 12 35 10 3 16 27 20 23 12 3 34 25 14
32 5 22 15 36 1 32 5 22 15 36 1 32 5 34 21 18 1 36 5 22 15 32 1
11 8 17 28 21 26 21 18 7 28 11 26 11 22 9 28 15 26 11 8 17 28 21 26
6 31 10 19 16 29 6 31 20 9 16 29 6 31 24 13 8 29 6 29 10 19 16 31
9 18 7 30 27 20 19 8 17 30 27 10 23 12 7 30 25 14 9 18 7 30 27 20
45  D1=102  D2=132 46  D1=94  D2=132 47  D1=98  D2=128 48  D1=102  D2=128

4 35 24 13 2 33 4 35 24 13 2 33 4 35 24 13 2 33 5 18 15 22 31 20
23 12 3 34 25 14 23 14 3 34 25 12 25 12 3 34 23 14 16 23 6 19 14 33
36 5 22 15 32 1 36 5 22 15 32 1 36 5 22 15 32 1 7 4 17 32 21 30
11 16 9 20 29 26 21 18 7 28 11 26 11 26 9 30 19 16 24 27 2 11 34 13
6 21 18 27 8 31 6 29 20 9 16 31 6 21 28 17 8 31 3 8 25 36 29 10
17 10 7 30 19 28 19 8 17 30 27 10 27 10 7 20 29 18 26 1 28 9 12 35
49  D1=94  D2=120 50  D1=94  D2=128 51  D1=94  D2=128 52  D1=120  D2=102

5 18 15 22 31 20 5 18 15 22 31 20 5 18 15 34 7 32 5 24 13 36 7 26
16 33 6 19 14 23 16 33 6 19 14 23 16 35 6 31 14 9 14 19 6 25 12 35
7 4 17 32 21 30 7 4 17 32 21 30 19 4 17 8 33 30 23 4 31 18 27 8
34 1 28 11 24 13 34 27 2 11 24 13 36 27 2 23 10 13 20 15 22 9 34 11
3 8 35 26 29 10 3 8 35 26 29 10 3 20 25 12 29 22 3 30 17 32 1 28
36 27 2 9 12 25 36 1 28 9 12 25 26 1 28 21 24 11 16 21 2 29 10 33
53  D1=120  D2=138 54  D1=120  D2=112 55  D1=120  D2=102 56  D1=98  D2=124

5 28 13 36 3 26 5 30 3 22 19 32 5 30 25 12 7 32 5 32 29 20 7 18
14 19 4 27 12 35 28 23 6 31 2 21 26 13 6 31 24 11 28 21 6 17 30 9
29 6 31 18 25 2 7 4 29 20 33 18 29 4 27 10 33 8 33 4 31 8 19 16
20 15 22 9 34 11 24 27 14 35 10 1 14 19 16 3 36 23 24 27 22 15 10 13
7 30 17 32 1 24 15 8 25 12 17 34 17 28 21 34 9 2 3 34 25 12 1 36
16 21 8 23 10 33 26 13 16 9 36 11 20 15 18 1 22 35 26 23 2 35 14 11
57  D1=98  D2=124 58  D1=120  D2=102 59  D1=92  D2=130 60  D1=84  D2=138

5 36 31 12 7 20 5 36 31 12 7 20 6 19 8 29 32 17 6 19 22 15 32 17
32 13 6 19 30 11 32 13 6 19 30 11 9 30 5 18 21 28 23 30 5 18 21 14
35 4 33 10 21 8 35 4 33 10 21 8 4 7 20 31 16 33 4 7 20 31 16 33
14 25 2 23 18 29 4 25 22 3 18 29 13 10 15 22 27 24 27 24 29 8 13 10
3 34 27 16 9 22 23 34 27 16 9 2 36 3 12 25 34 1 36 3 26 11 34 1
26 15 24 1 28 17 26 15 24 1 28 17 11 14 35 2 23 26 25 28 35 2 9 12
61  D1=100  D2=122 62  D1=80  D2=142 63  D1=138  D2=98 64  D1=110  D2=126

6 19 22 29 4 31 6 33 8 15 18 31 7 24 13 36 5 26 7 30 13 36 1 24
21 2 5 32 23 28 9 14 5 32 35 16 14 19 6 25 12 35 14 19 8 23 12 35
18 7 20 3 30 33 4 7 34 17 30 19 23 8 31 18 27 4 29 6 31 18 25 2
1 10 35 14 27 24 13 10 23 2 27 36 20 15 22 9 34 11 20 15 22 9 34 11
8 17 12 25 34 15 22 3 12 25 20 29 1 30 17 32 3 28 5 28 17 32 3 26
11 36 9 16 13 26 11 24 21 28 1 26 16 21 2 29 10 33 16 21 4 27 10 33
65  D1=102  D2=120 66  D1=102  D2=120 67  D1=102  D2=124 68  D1=102  D2=120

8 19 22 15 30 17 11 14 35 2 23 26 11 14 35 2 23 26 11 16 13 30 9 32
21 14 7 18 23 28 34 3 12 25 36 1 36 3 12 25 34 1 14 3 10 33 22 29
6 9 20 29 16 31 13 10 15 22 27 24 13 10 15 22 27 24 17 12 15 28 31 8
13 34 11 2 27 24 4 33 6 29 18 21 4 21 6 17 30 33 2 27 4 21 34 23
10 5 36 25 32 3 9 16 31 20 7 28 9 16 19 32 7 28 5 18 25 36 7 20
35 12 33 4 1 26 32 5 8 17 30 19 20 5 8 29 18 31 26 1 6 19 24 35
69  D1=102  D2=120 70  D1=84  D2=138 71  D1=84  D2=124 72  D1=92  D2=130

11 22 13 36 9 20 11 22 13 36 9 20 11 22 31 18 9 20 11 22 31 18 9 20
14 3 10 21 28 35 34 3 10 21 28 15 16 1 10 21 30 33 32 1 10 21 30 17
23 12 15 34 19 8 23 12 35 14 19 8 23 12 17 32 19 8 23 12 33 16 19 8
2 33 4 27 16 29 2 33 4 27 16 29 2 15 4 27 34 29 2 15 4 27 34 29
5 24 31 18 7 26 5 24 31 18 7 26 5 24 13 36 7 26 5 24 13 36 7 26
32 1 6 25 30 17 32 1 6 25 30 17 14 3 6 25 28 35 14 3 6 25 28 35
73  D1=80  D2=142 74  D1=100  D2=122 75  D1=98  D2=124 76  D1=114  D2=108

12 9 2 35 28 25 14 1 4 33 24 35 14 1 4 33 24 35 14 1 4 35 24 33
1 34 11 26 3 36 3 22 13 36 5 32 3 32 13 36 5 22 3 22 13 32 5 36
10 13 8 29 24 27 12 15 2 23 34 25 12 15 2 23 34 25 12 15 2 23 34 25
33 30 17 6 21 4 21 18 27 8 31 6 31 28 17 8 21 6 21 18 27 8 31 6
14 7 32 19 16 23 16 11 20 29 26 9 16 11 30 19 26 9 16 11 20 29 26 9
31 18 15 22 5 20 19 28 17 10 7 30 29 18 27 10 7 20 19 28 17 10 7 30
77  D1=96  D2=112 78  D1=102  D2=120 79  D1=102  D2=120 80  D1=102  D2=118

14 9 12 31 16 29 14 11 2 29 20 35 14 25 16 9 12 35 14 25 16 9 12 35
11 22 15 28 3 32 1 28 13 36 3 30 17 8 13 36 27 10 27 8 13 36 17 10
8 13 10 33 30 17 12 15 10 19 34 21 24 15 26 11 34 1 24 15 26 11 34 1
23 34 21 4 27 2 27 18 25 6 31 4 7 18 5 22 31 28 7 28 5 32 21 18
20 7 36 25 18 5 24 9 16 33 22 7 4 23 20 29 2 33 4 23 30 19 2 33
35 24 19 6 1 26 17 26 23 8 5 32 19 6 3 32 21 30 29 6 3 22 31 20
81  D1=94  D2=128 82  D1=112  D2=108 83  D1=102  D2=120 84  D1=102  D2=120

14 25 30 7 12 23 14 25 30 7 12 23 17 10 7 30 19 28 18 29 20 7 10 27
29 6 13 24 31 8 31 6 13 24 29 8 8 21 18 27 6 31 31 6 17 28 21 8
26 15 28 9 22 11 26 15 28 9 22 11 11 16 9 20 29 26 16 19 30 9 26 11
5 36 3 16 19 32 5 32 3 16 19 36 2 35 22 15 32 5 5 32 15 22 35 2
2 27 34 21 10 17 2 27 34 21 10 17 23 12 3 34 25 14 14 23 34 3 12 25
35 4 1 18 33 20 33 4 1 18 35 20 36 1 24 13 4 33 33 4 13 24 1 36
85  D1=94  D2=128 86  D1=94  D2=124 87  D1=120  D2=124 88  D1=124  D2=128


Further Notes on the 6×6 Semi-magic Tours
added by the editor.

The tours occur in 44 pairs that are reverse numberings of each other, as follows: 1-53, 2-39, 3-36, 4-43, 5-54, 6-88, 7-87, 8-75, 9-76, 10-78, 11-79, 12-72, 13-81, 14-59, 15-82, 16-69, 17-52, 18-83, 19-84, 20-25, 21-85, 22-45, 23-46, 24-42, 26-40, 27-38, 28-29, 30-31, 32-44, 33-80, 34-67, 35-56, 37-68, 41-57, 47-86, 48-49, 50-51, 55-58, 60-70, 61-74, 62-73, 63-71, 64-77, 65-66.

The closed tour 16-69 is one of the quatersymmetric tours, and being symmetric has constant difference of 18 between numbers in diametrally opposite cells. (This semi-magic property of this quatersymmetric tour was noted by Maurice Kraitchik in his book on Le Problème du Cavalier 1927, p.36.)

The other twelve pairs of closed tours occur in three groups of four, corresponding to three geometrically congruent closed tours, numbered from different starting cells:
(a) 1-53, 10-78, 18-83, 65-66;
(b) 11-79, 17-52, 19-84, 55-58;
(c) 12-72, 13-81, 14-59, 21-85;
In case (b), where the underlying tour is piecewise symmetric, there is a constant difference of 6 or 30 between numbers in diametrally opposite cells. (Tour 19 was given by G. P. Jelliss in a note on ‘Piece-Wise Symmetric Tours’ in Journal of Recreational Mathematics Vol.28 No.1 1997 pp.63-64 and Vol.29 No.1 1998, pp.9-71; diagram on p.70)


The open tour pair 20-25 is the only one in which the diagonal sums are 111 plus or minus 1, i.e. as near as possible to the magic constant.

There are 50 tours in which D1 + D2 = 222, consisting of 24 closed 36 open. This group includes all the closed tours except the pair 1-53.


Further Notes on larger 6×N Semi-magic Tours
(added by the editor August 2003).

6×8 Board

Addendum: Jean-Charles Meyrignac (26 August 2003) reports the following results: On the 6×8 board, my program found the 5 following tours (it only searched for tours beginning in a corner): [the ranks all add to 196]

  1  36  23  38   7  30  47  14 
 22  39   6  31  46  15   8  29 
 35   2  37  24   9  28  13  48 
 40  21  32   5  16  45  10  27 
  3  34  19  42  25  12  17  44 
 20  41   4  33  18  43  26  11 
-------------------------------
121 173 121 173 121 173 121 173 

  1  36  23  38  25  12  47  14 
 22  39  34   3  46  15  10  27 
 35   2  37  24  11  26  13  48 
 40  21   4  33  16  45  28   9 
  5  32  19  42   7  30  17  44 
 20  41   6  31  18  43   8  29 
-------------------------------
123 171 123 171 123 171 123 171 

  1  20  43  30  11  18  41  32 
 44  29   2  19  42  31  12  17 
 21   4  27  48  13  10  33  40 
 28  45  24   3  34  37  16   9 
  5  22  47  26   7  14  39  36 
 46  25   6  23  38  35   8  15 
-------------------------------
145 145 149 149 145 145 149 149 

  1  22  47  30  11  14  39  32 
 46  29   2  23  38  31  12  15 
 21   4  27  48  13  10  33  40 
 28  45  24   3  34  37  16   9 
  5  20  43  26   7  18  41  36 
 44  25   6  19  42  35   8  17
------------------------------- 
145 145 149 149 145 145 149 149 

  1   8  45  38  17  10  47  30 
 44  39   2   9  46  29  16  11 
  3  24   7  18  37  48  31  28 
 40  43  36  25   6  19  12  15 
 23   4  41  34  21  14  27  32 
 42  35  22   5  26  33  20  13 
-------------------------------
153 153 153 129 153 153 153 129 

(total running time: 6 minutes)

6×10 Board No results to hand.

6×12 Board.

Here are two 6×12 examples constructed by joining together two of Awani Kumar's 6×6 semimagic tours, suitably chosen. The 6-cell lines add to 219 and the 12-cell lines to 510 and 366 (the magic constant would be 438).

26 39 24 31 28 53  8 57  6 13 10 71
37 32 27 52 23 30 55 14  9 70  5 12
40 25 38 29 54 51 58  7 56 11 72 69
33 36 47 20 43 22 15 18 65  2 61  4
48 41 34 45 50 19 66 59 16 63 68  1
35 46 49 42 21 44 17 64 67 60  3 62
26 39 24 43 52 41 56 69 66 13 10 71
37 44 27 40 23 54 67 14 57 70 65 12
28 25 38 53 42 51 58 55 68 11 72  9
45 36 47 32 19 22 15  6 17  2 61 64
48 29 34 21 50 31 18 59  4 63  8  1
35 46 49 30 33 20  5 16  7 60  3 62

Similar tours can be constructed by joining 6×6 semi-magic tours end-to-end to any length 6k.


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