I would be very interested to hear an explanation of how to solve the puzzle below without guessing. For reference, the unique solution is on the right.

  - - - O - -               O X X O X O
  X - - - - X               X O O X O X
  - - - - - O               X O X O X O                                         
  - - - X O X               O X O X O X
  - - - - - X               O X O O X X
  - - X - - O               X O X X O O

Order to fill in the board (1, 2, 3, ... 9, A, B, C, ... N):

  B C D O 2 1 
  X 3 G N 4 X 
  M N H I 5 O 
  F E F X O X 
  L K J 6 A X 
  9 8 X 7 8 O

Reasoning:

  1. finish column
  2. avoid OOO in row
  3. avoid OOO in row
  4. avoid OOO in row
  5. avoid OOO in column
  6. don't duplicate column 6
  7. avoid XXX in column
  8. wrap XX on both sides
  9. finish row
  A. finish column
  B. avoid OOO in column
  C. avoid XXX in column
  D. finish row
  E. don't duplicate column 5
  F. finish row
  G. avoid OOO in column
  H. avoid OOO in column
  I. avoid XXX in row
  J. finish column
  K. avoid OOO in row
  L. finish row
  M. finish column
  N. finish rows

Nice work! Step 6 was the one that eluded me. Using that principle I've now been able to solve several boards that I previously thought required guessing.

Yeah, finding step 6 took me much longer than any other step. (Followed by step E.)

Do you see a way to make progress on this one without guessing?

  O X - - X O
  X - - O - -
  - O X X O -
  - - O - - -
  O - X O X -
  X O - - O -