By large boards here we mean having side greater than 10. For 12×12 examples see the magic tours section.
14×14 board = 196 cells This board, like the 10×10 and 6×6, admits complete tours with quaternary symmetry, but the earliest printed example I have come across is the first diagram below by Archibald Sharp from his book Linaludo (1925). The second example is from Kraitchik (1927).
The next two examples, from Kraitchik (1927) and Murray (1942) are simply formed of four 7×7 open tours joined together, but this board offers scope for much more interesting constructions.
The next six are my own, The first, from 1986, places my 10×10 tour incorporating a Maltese cross within a complete border braid as frame. Then comes another Maltese cross design.
The next two attempt tartan plaid effects, based on the '35' and '45' arrangements of nightrider lines (where every 3rd and 5th, or every 4th and 5th line in a set of parallels is turned at right angles).
The next has a central mosaic pattern. The last example includes sequences of seven successive three-knight-move triangles in the central region.
16×16 board = 256 cells For tours of this size see also the pages on magic tours.
The example shown here is by Pierre Dehornoy (2003) and shows how to construct a tour with most of the moves in one direction on a square board of side 4n (n > 3). The tour is composed of 4n 9 pieces of 7 types (the 16×16 example uses one of each type). The four types of piece that impinge on the corners are each used once (extended where necessary to fit the new board width), the three types of piece that fit between these are repeated (with lengthening straight pieces) an appropriate number of times.
18×18 board = 324 cells This tour (by G.P.Jelliss 1999) was used as the cover illustration on issue 16 of The Games and Puzzles Journal (1999). It shows oblique quaternary symmetry (i.e. unchanged by 90 degree rotation), with the central area using a repeating (wall-paper type) pattern borrowed from Archibald Sharp's Linaludo (1925).