# 16 by 16 Magic Knight's Tours

Sections on this page: Braid MethodSurround MethodExtended Quartes MethodSplit and Fix Method

As with the 12 by 12 tours we show most of the tours graphically, leaving the reader to put in the numbers and check the magic totals if desired. The numbering starts and ends at the cells marked with a black blob. The tours are oriented as they were first published. The magic constant for the 16×16 board is 2056.

## éBraid Method

The earliest known example of a magic knight's tour on the 16 by 16 board was by Maxwell Wihnyk, published in Schachzeitung 1885, and shown in the first diagram here. The tour is constructed by extending the braid of the Beverley tour 27a. For the numerical version see the page on History of Magic Tours. Its diagonal sums are: odd = 1898, even =2090, when numbered from file 12, rank 8.

The second braid tour shown here is by Murray, applying the method of E. Lange of extending an 8×8 tour onto larger boards by adding successive braids in the form of a border round the tour. The one shown here (using the 8×8 rhombic tour 14b) has the even diagonal adding to the magic constant 2056, though Murray did not draw attention to this property. (The odd diagonal adds to 2104, when numbering is from cell i12.)

Many other tours of braid type can be formed, in which the arrangement of the braids can be much more free-form, but none of them as far as I know show both diagonals magic. Surprisingly Murray seemed more interested in trying to work out how many tours of these types could be constructed than in finding special cases with interesting properties, though he did place emphasis on the property of each 4×4 block itself being magic, as in Beverley's original tour.

An irregular tour based on the Beverley structure and braids, and achieving the magic sum in both diagonals, was sent to me in August 2006 by Mr S. K. Singh of Raebareli, India and published in the last online issue 45 of The Games and Puzzles Journal which I reproduce here, together with a geometrical diagram:

 1 126 191 196 5 122 187 200 9 120 183 202 11 118 181 204 192 195 2 125 188 199 6 121 184 201 10 119 182 203 12 117 127 190 193 4 197 186 123 8 211 18 111 176 113 180 205 14 194 3 128 189 124 7 198 185 110 175 210 17 208 13 116 179 129 254 63 68 171 252 21 70 173 212 19 112 177 114 15 206 64 67 130 253 22 69 172 251 20 109 174 209 16 207 178 115 255 132 65 62 249 170 71 24 213 168 107 26 215 166 105 28 66 61 256 131 72 23 250 169 108 25 214 167 106 27 216 165 133 244 47 76 147 248 45 88 149 232 43 92 151 228 29 104 60 77 146 245 46 73 148 233 44 89 150 229 42 93 164 217 243 134 75 48 247 144 87 50 231 154 91 40 227 152 103 30 78 59 246 145 74 49 254 143 90 39 230 153 94 41 218 163 135 242 55 82 139 238 51 86 155 224 35 96 159 226 31 102 58 79 138 239 54 83 142 235 38 97 158 225 34 95 162 219 241 136 81 56 237 140 85 52 223 156 99 36 221 160 101 32 80 57 240 137 84 53 236 141 98 37 222 157 100 33 220 161

## éSurround Method

E. Lange also devised another type of surround method, in which a central 8×8 tour is joined to six blocks each 4×8. The patterns of moves within these added panels take the forms typified by the alternative right-hand sides that can be substituted in Beverley's 27a to form the related tours 27b, c, d, e, f. (See the history section for illustration). The tour shown here is Lange's original, published in Lehmann 1932. (The diagonal sums are: odd = 2214, even = 1898, when the origin is h5.)

## éExtended Quartes Method

This method, was introduced by H. J. R. Murray in Fairy Chess Review August 1942, where he wrote; “The principle of construction ... is really an extension of ... squares and diamonds on the 8 by 8. Take any magic tour composed of squares and diamonds entirely, and insert the moves connecting their terminals on the central 8 by 8 of a 16 by 16 board. Then replace the squares and diamonds by circuits of squares and diamonds round the quarterboard of the 16 by 16 in which the terminals happen to lie. One pair of these is taken clockwise and the other anticlockwise.” The parent tour used is 12b. In his 1951 manuscript he explained that the method could also be applied to tours containing beverley or irregular quartes, provided that certain elaborate conditions are satisfied. The main ones being that the parent tour must have X = 60 and N = 20 for every line, and the quartes must all be broken in the same way, either 2 and 2 or 3 and 1. Diagonals; odd = 2432, even = 1680, numbered from i12.

H. J. R. Murray died in 1955 and a year later his method was applied by Helga Em. de Vasa and by T. H. Willcocks, who corresponded on the subject, to produce the first diagonally magic knight's tours, which we show in the other three diagrams. The two de Vasa solutions, very similar, based on 00a, were published separately in 1962, one in Les Secrets du Cavalier by G. D'Hooghe and the other in Recreational Mathematics Magazine in an article by Willcocks, who later published his own tour, based on 34d, in Journal of Recreational Mathematics in 1968. Manuscripts of correspondence between Willcocks and De Vasa on this topic were lodged with the Murray collection in the Bodleian Library, Oxford, in 1991.

## éSplit and Fix Method

The three tours shown here were used by the editor (George Jelliss) as the illustrations on a Greetings card sent out at Christmas 1991 and New Year 1992. These tours are all diagonally magic and formed by a new method that I called ‘Split and Fix’ since it involved cutting an existing 8×8 magic tour in half, placing the halves at opposite edges of the 16×16 and joining up the loose ends in a biaxially symmetric pattern, which also meets certain other criteria. The tours used are 16a, 27i, 03b. The biaxial property means that the numbers in the files add in complementary pairs to 257 while in the ranks half add to 193 and half to 321 (adding to twice 257). In the tours derived from 27i and 03b the pairs adding to 257 are symmetric with respect to the horizontal axis.

However, in the tour derived from 16a the pairs adding to 257 occupy pairs of adjacent squares. In this tour each connecting strand has a strange type of symmetry, in that if the rth move takes the knight to the mth file then the rth move from the other end of the strand takes the knight to the (16 – m)th file. Thus the middle move crosses the vertical median. This ensures that each file that contains k also contains 257 – k (in fact they form domino pairs).

In these tours the fact that the sections of the split tour do not overlap the diagonals makes it easier to ensure that the diagonals add to the magic constant. Others that I tried with the components differently placed could not be made magic in the diagonals. It may be noted that the Wihnyk tour can be regarded as formed by splitting the Beverley tour and placing the two parts back to back in the centre of the larger board.

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