hi Bob,

Recently I saw a post at rec.puzzles regarding a permutation puzzle, which
the poster had done some analysis using magma. I wanted to do the same sort
of analysis with GAP.

Here's my take on it:

gap> f := FreeGroup(["a","b","c","d"]);;
gap> a := f.1;; b:= f.2;; c:= f.3;; d := f.4;;

gap> pgens := [(1,2,3,7,11,10,9,5),               
(2,3,4,8,12,11,10,6), 
(5,6,7,11,15,14,13,9), 
(6,7,8,12,16,15,14,10)];;

gap> puzzle := Group(pgens);;

gap> h := GroupHomomorphismByImages(f, puzzle, [a,b,c,d], pgens);
[ a, b, c, d ] -> [ (1,2,3,7,11,10,9,5), (2,3,4,8,12,11,10,6), 
  (5,6,7,11,15,14,13,9), (6,7,8,12,16,15,14,10) ]
gap> gword := PreImagesRepresentative(h, (15,16));;

But I'm not absolutely certain this is the shortest word:)

gap> Length(gword);
796257

Perhaps someone else knows how to find a shorter one.

Andrew

p.s I'm very curious to know what the puzzle is.

p.p.s The generators you give for the puzzle generate the symmetric group
on the 16 points.