of course it's a matter of semantics whether such chains are "guessing". personally, i would not consider them to be guessing, rather more advanced rules than "this column must contain a 5, which must be in this square". how about something like "if a=1, then b=2, and then c=3 ... which forces a contradiction. so a =/= 1." is that "guessing"? i would say "no".
be y" really doesn't change its nature to my view. And so I'm still not convinced that that particular grid can be solved without guessing..
the response from the automated system(?) seems cleaner than the chains i found ... but requires translation into humanese, or better yet, mathematics. ) I tossed it at Sudoku Susser, which is excellent at defining ) logical steps. Turns out this is an XY-wing example. ) ) * Squares R4C1 (XY), R6C3 (XZ) and R4C9 (YZ) form an XY-Wing pattern on <9>. ) All squares that are buddies of both the XZ and YZ squares cannot be <9>. ) ) R4C2 - removing <9> from <39> leaving <3>. ) R6C9 - removing <9> from <49> leaving <4>. i'm not familiar with "wings", "buddies", etc. but can surmise their meaning by examining the grid: +-------+-------+-------+ | 3 6 7 | . 4 . | 5 9 2 | | 9 5 2 | 3 7 6 | 4 1 8 | | 8 1 4 | 2 5 9 | 6 3 7 | +-------+-------+-------+ | e h 8 | 6 2 . | 7 5 g | | 5 . 6 | 4 9 . | 2 8 1 | | . 2 f | . 8 5 | 3 6 i | +-------+-------+-------+ | 6 4 3 | 9 1 2 | 8 7 5 | | . . . | . 3 4 | 9 2 6 | | 2 . . | . 6 . | 1 4 3 | +-------+-------+-------+ we have e in {1, 4} , f in {1, 9} and g in {4, 9} . since f =/= e =/= g , we have 9 in {f, g} . (note the duality between squares and values.) then from f =/= h =/= g , we have h =/= 9 , and similarly i =/= 9 . that determines the values h = 3 and i = 4 , and the remaining values are easy to fill in from here. is that "guessing"? my opinion is "no", but noone has ever cared about my opinion anyway! mike