This new page started 4 March 2018.

To show the new magic tours 4×18 and 4×20 constructed by Awani Kumar.

The 16 arithmetically distinct solutions 4×18 tours appear in his paper
"Studies in Tours of Knight on 4 by n Boards"

Published online at ArXiv 1802.09340

He has sent the 88 arithmetically distinct solutions 4×20 as an XL file.

On this page I show diagrams of the 8 and 44 geometrically distinct solutions.

Plus examples sent subsequently on longer boards.

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).

Awani Kumar (2018) finds 88 arithmetically distinct magic tours,

which means 44 geometrically distinct since symmetry is impossible. Here are diagrams of all 44.

The arithmetical forms numbered 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 Frenicle 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.

Awani Kumar (email 8 Mar 2018) says there are 464 arithmetically distinct magic tours on the 4×22 board.

which means 232 geometrically distinct since symmetry is impossible.

He has sent these 4×22 as an XL file.

Here are diagrams of one magic tour for each of the sizes 4×22 to 4×28 that he provided in an earlier email (7 Mar 2018)

He states that magic tours exist on all 4×2k boards for k>8.

Although it is known that magic knight's tours are possible on all 4h×4k boards 8×8 and larger
(e.g. by braid extension of 8×8 magic tours), the only non-square example that has actually been published
as far as I am aware is this one, a simple braid extension of Beverley's tour,
that I gave in *Variant Chess* 1992 (vol.1, issue 8, p.105) just to show it is possible.

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 |

In the page on General Theory of Magic Knight Tours I give details of the theorem concerning impossibility of magic knight tours on boards with both sides singly even. I wrote there that: "The above theorems account for all cases except boards 4m × (4n + 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 I found such a counter-example, 12×14, published on my Diary blog, headed A Magic Knight Rectangle: Jeepyjay Diary 8 March 2011. I reproduce the text here, with additions:

Back in 2003 I was able to prove that magic knight's tours were not possible on boards 4n+2 by 4m+2, but a proof for the 4n by 4m+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 4n by 4m+2 board, I looked at the subject again and found two 12×14 examples last night, shown below.

They were constructed by the "rolling pin" method that I devised for 12×12 magic tours. 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. They 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 |