## Tuesday, 8 March 2011

### A Magic Knight Rectangle

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 knight+wazir tour on the 2x4 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 12x14 examples last night, of which this is the first:

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

It is constructed by the "rolling pin" method that I devised for 12x12 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 = 169x6 and the ranks add to 1183 = 169x7. 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.

#### 1 comment:

1. Great work George!
CONGRATULATIONS!!
Please also publish more such solutions.
By the way, can we have a closed magic tour or on a smaller rectangle?

Regards,
Awani Kumar
INDIA