by **George Jelliss** and **Awani Kumar**

This new page started 4 March 2018, most recent update 22 October 2019.

Sections on this page: Evenly Even Oblongs. 8×12, 4×(4.k) 4×20, 4×24, 4×28

Oddly Even Oblongs. 12×14, 4×(4.k + 2)4×18, 4×22, 4×26, 6×(4.k) 6×12, 6×16, 6×20, 10×(4.k) 10×12, 10×16, and Unsolved Problems

It is known that magic knight's tours are possible on all (4.h)×(4.k) boards 8×8 and larger (e.g. by braid extension of 8×8 magic tours).

01 46 71 76 05 44 67 78 07 42 65 80 72 75 02 45 68 77 06 43 66 79 08 41 47 70 73 04 37 12 83 62 39 10 81 64 74 03 48 69 84 61 38 11 82 63 40 09 49 94 23 28 13 36 59 86 15 34 57 88 24 27 50 93 60 85 14 35 58 87 16 33 95 22 25 52 29 20 91 54 31 18 89 56 26 51 96 21 92 53 30 19 90 55 32 17 |

It was thought likely that there were also magic tours on 4-rank boards of this type,
but none were found until recent work by Awani Kumar (2018).

His results are published online at ArXiv 1802.09340.

He finds magic tours possible on all boards 4×(4.k) with k > 4. (None on 4×16 though there are semimagic tours).

The numbers of arithmetically distinct magic tours being:

88 on 4×20 (XL listing: 4×20) diagrams below

2076 on 4×24 (PDF listing: 4×24) one diagram below

47456 on 4×28 (too many to list here) one diagram below

All tours on 4-rank boards are asymmetric, so the 88 arithmetically distinct magic tours,
correspond to 44 geometrically distinct, numbered from either end.

Below are diagrams of all 44. The arithmetical forms 45-88 in the XL file are
the reverse numberings of these tours but not in the same sequence,

since they are arranged according to the Frénicle convention.

1-32 central link a-c, shown in groups of four

1-4 and 5-8

9-12 and 13-16

17-20 and 21-24

25-28 and 29-32

33-36 central link d-f

37-40 central link c-e

41-44 central link b-d.

One example tour

One example tour

It will be noticed that the end-points and mid-points in all the 4-rank tours (in this section and the next)

always lie within a 3×4 box, with end-points in the corners, a zebra move apart.

This apppears to be an inherent feature of magic tours on these boards, and also occurs in examples on larger boards.

In the page on General Theory of Magic Knight Tours (2003) Jelliss gave details of the theorem
concerning impossibility of magic knight tours on boards with both sides singly even,
which was first published in *The Games and Puzzles Journal* #25 (online Jan-Feb 2003).
He wrote there that: "The above theorems account for all cases except boards (4.m)×(4.n + 2).
The question remains, whether a magic knight tour on such a rectangle is possible.
I have eliminated the smallest case 4×6 by looking at all the (36) half-tours.
The next cases are: 4×10, 4×14, 8×6, 8×10, 8×14, 12×6,
12×10, 12×14, ... Is there an argument to prove the impossibility
(if so we can conclude that magic knight's tours are only possible on boards whose
sides are both a multiple of 4), or can someone come up with a counter-example?

This was followed by an Update: In 2011 Jelliss found such a counter-example, 12×14,
published on the *Jeepyjay Diary* blog, headed "A Magic Knight Rectangle":
Jeepyjay Diary 8 March 2011.
We reproduce the text here, with additions:

"Back in 2003 I was able to prove that magic knight's tours were not possible on boards (4.n + 2) by (4.m + 2), but a proof for the (4.n) by (4.m + 2) case eluded me. I now see that that is because there is no such proof! Thanks to a suggestion by John Beasley, that since there is a simple magic king tour on the 2×4 board, a magic knight tour should be possible on a sufficiently large (4.n) by (4.m + 2) board, I looked at the subject again and found two 12×14 examples last night." These are shown below.

These were constructed by the "rolling pin" method that Jelliss devised for 12×12 magic tours. He wrote that "It's surprising I hadn't thought of trying this before. It's just a matter of widening the board." The files add to 1014 = 169×6 and the ranks add to 1183 = 169×7. Each file consists of three pairs adding to 127 and three pairs adding to 211. The ranks are made up of pairs of complements adding to 169. The two tours differ only in a few moves.

141 122 143 038 139 124 127 042 045 030 131 026 047 028 144 037 140 123 128 039 044 125 130 041 046 029 132 025 121 142 035 138 119 126 129 040 043 050 031 134 027 048 036 145 120 063 034 137 014 155 032 135 106 049 024 133 011 064 061 118 013 154 033 136 015 156 051 108 105 158 146 117 012 151 062 059 016 153 110 107 018 157 052 023 065 010 115 060 149 152 111 058 017 020 109 054 159 104 116 147 150 009 114 057 094 075 112 055 160 019 022 053 091 066 007 148 093 074 113 056 095 076 021 162 103 078 006 069 092 073 008 003 082 085 168 161 096 077 100 163 067 090 071 004 083 088 167 002 081 086 165 098 079 102 070 005 068 089 072 001 084 087 166 097 080 101 164 099 |
141 122 143 038 139 124 127 042 045 030 131 026 047 028 144 037 140 123 128 039 044 125 130 041 046 029 132 025 121 142 035 138 119 126 129 040 043 050 031 134 027 048 036 145 120 149 034 137 014 155 032 135 020 049 024 133 011 150 147 118 013 154 033 136 015 156 051 022 019 158 146 117 012 151 148 059 016 153 110 021 018 157 052 023 065 010 115 060 063 152 111 058 017 106 109 054 159 104 116 061 064 009 114 057 094 075 112 055 160 105 108 053 091 066 007 062 093 074 113 056 095 076 107 162 103 078 006 069 092 073 008 003 082 085 168 161 096 077 100 163 067 090 071 004 083 088 167 002 081 086 165 098 079 102 070 005 068 089 072 001 084 087 166 097 080 101 164 099 |

a | b |

Awani Kumar has considered this case in his new ArXiv paper cited above
and finds the following numbers of arithmetically distinct magic tours,

which means half these numbers geometrically distinct, since symmetry is impossible.

16 on 4×18 See the 8 diagrams below

464 on 4×22 (XL listing: 4×22) one diagram below

9904 on 4×26 (PDF listing: 4×26) one diagram below

Awani Kumar (2018) finds 16 arithmetically distinct magic tours, numbered 1 to 16,

but there are only 8 geometrically distinct since they occur in pairs that are the reversals of each other

(1=15, 2=16, 3=9, 4=10, 5=11, 6=12, 7=13, 8=14).

In the ArXiv paper linked to above Awani Kumar reports finding eight arithmetically distinct magic tours on the 6×12 board,

and over 200 on the 6×16 board, and considers that magic tours are possible on all boards 6×(4.k) with k > 2.

**6×12**

Here are diagrams of the four geometrically distinct magic tours on the 6×12 board found by Kumar.

The first two are also shown in numerical form. Sum of 6-cell lines 219. Sum of 12-cell lines 438.

The righhand 6×8 part of the first shows exact axial symmetry.

001 036 067 042 017 020 065 044 015 024 061 046 068 041 034 003 066 043 016 019 064 045 014 025 035 002 037 072 021 018 051 010 023 062 047 060 040 069 004 033 052 055 022 063 050 011 026 013 005 032 071 038 007 030 057 054 009 028 059 048 070 039 006 031 056 053 008 029 058 049 012 027 |
001 036 071 042 063 044 011 028 013 050 059 020 070 041 034 003 010 029 062 045 060 019 014 051 035 002 037 072 043 064 027 012 049 058 021 018 040 069 004 033 030 009 046 061 024 015 052 055 005 032 067 038 007 026 065 048 057 054 017 022 068 039 006 031 066 047 008 025 016 023 056 053 |

a | b |

The other two tours differ from each other only in the orientation of the 3×4 area that includes the end-points.

c d

**6×16**

Here are diagrams of the two magic tours on this board that are shown in numerical form in Kumar (2018).

These also have sections 6×12 and 6×8 that show exact axial symmetry.

a b

**Update 22 October 2019:** Awani Kumar sends the following further two 6×16 tours.

The first is reentrant and the second has end-points on adjacent cells.

The left 6×8 half of this tour also has exact axial symmetry.

43 88 53 06 59 86 51 02 61 84 23 28 63 82 17 30 54 07 44 87 52 01 60 85 24 27 62 83 16 29 80 65 89 42 05 58 45 50 03 96 71 22 25 76 81 64 31 18 08 55 92 39 04 95 46 49 26 75 72 21 34 15 66 79 41 90 57 10 93 48 37 12 73 70 35 14 77 68 19 32 56 09 40 91 38 11 94 47 36 13 74 69 20 33 78 67 |
17 10 91 76 15 12 93 74 35 40 61 58 33 42 63 56 78 89 16 11 92 75 02 13 72 59 34 41 62 57 32 43 09 18 77 90 01 14 73 94 39 36 71 60 31 44 55 64 88 79 20 07 96 83 24 03 70 49 26 37 66 53 30 45 19 08 81 86 05 22 95 84 25 38 51 68 47 28 65 54 80 87 06 21 82 85 04 23 50 69 48 27 52 67 46 29 |

c | d |

**6×20**

**Update 22 October 2019:** Awani Kumar sends the following two 6×20 tours.

The first is reentrant and has the same pattern in the middle 4 files as tour 6×12b above.

This suggests that it could be expanded by braids to 10×16 or 10×20 in the same manner as shown below.

The second has end-points on opposite sides of the board, a {0,5} move apart.

The left 6×12 part of this tour has exact axial symmetry,
and the right 6×8 part is identical to that of tour 6×16d above.

a b

So far we only have the following two examples. These suggest that solutions exist for all cases k > 2:

**10×12 and 10×16**

These are magic tours formed by extending one of Awani Kumar's 6×12 magic tours with a braid.

The first of these was reported, with diagram, in *Jeepyjay Diary* 24 September 2019.
The other was constructed while updating this web-page.

The file and rank sums of the first are 605 and 726 (multiples of 121), and of the second 805 and 1288 (multiples of 161).

These tours can easily be expanded onto larger boards by extending the braids, e.g. to 12×14, 12×18, 14×16, 16×18, etc.

107 072 015 048 105 074 017 046 103 076 019 044 014 049 106 073 016 047 104 075 018 045 102 077 071 108 051 012 069 110 041 022 079 100 043 020 050 013 070 109 052 011 080 099 042 021 078 101 001 060 119 066 111 068 023 040 025 086 095 032 118 065 058 003 010 053 098 081 096 031 026 087 059 002 061 120 067 112 039 024 085 094 033 030 064 117 004 057 054 009 082 097 036 027 088 091 005 056 115 062 007 038 113 084 093 090 029 034 116 063 006 055 114 083 008 037 028 035 092 089 |
069 014 091 148 071 012 089 150 051 032 109 130 053 030 107 132 092 147 070 013 090 149 072 011 110 129 052 031 108 131 054 029 015 068 001 080 159 086 151 088 033 050 035 116 125 042 133 106 146 093 158 085 078 003 010 073 128 111 126 041 036 117 028 055 067 016 079 002 081 160 087 152 049 034 115 124 043 040 105 134 094 145 084 157 004 077 074 009 112 127 046 037 118 121 056 027 017 066 005 076 155 082 007 048 153 114 123 120 039 044 135 104 144 095 156 083 006 075 154 113 008 047 038 045 122 119 026 057 065 018 097 142 063 020 099 140 061 022 101 138 059 024 103 136 096 143 064 019 098 141 062 021 100 139 060 023 102 137 058 025 |

The case of 8-rank boards **8×(4.k + 2)** remains to be solved.

Adding simple braids to the 4×(4.k + 2) solutions doesn't seem to work,
since the accessible braids are not of the correct "0" type.

The "rolling pin" method used for the 12×14 has also been tried but without success.

The question of whether reentrant tours or open tours with different end-point separation than {2,3} are possible

has been answered (22 Oct 2019) by the four new tours by Kumar shown above.