Variations on Sudoku-by5 Logic Puzzles (Vol 1) Step-by-Step Instructions for Puzzle 21   (275/369) Diff: 58/99.9442.AA34.3465.7944.2765.444D.6378.4599.9753.7366.2B32.1111.1223.2444 The numbers in the puzzle below refer to the step number at which the value of the location can be deduced. Locations without step numbers contain the initial values found in corresponding puzzle in your edition of the book.

Rows are numbered top to bottom, columns from left to right, and rectangles from left to right and then top to bottom. Coordinates are (col, row). Precise details on numbering within each group are specified in the on-line instructions for 'Sudoku-by5' puzzles.

Step 1. There are 19 independent moves.

• The only spot left for 8 in the 1st 3x3 square in the upper left puzzle is at (3, 1).
• The only spot left for 5 in the 8th 3x3 square in the upper left puzzle is at (6, 8).
• The only spot left for 3 in the 9th 3x3 square in the upper left puzzle is at (9, 8).
• The only spot left for 5 in the 7th row in the upper right puzzle is at (16, 7).
• The only spot left for 2 in the 9th row in the upper right puzzle is at (16, 9).
• The only spot left for 2 in the 4th 3x3 square in the upper right puzzle is at (13, 4).
• The only spot left for 9 in the 5th 3x3 square in the upper right puzzle is at (18, 5).
• The only spot left for 8 in the 6th 3x3 square in the upper right puzzle is at (20, 6).
• The only spot left for 2 in the 1st row in the center puzzle is at (7, 7).
• The only spot left for 1 in the 3rd column in the center puzzle is at (9, 9).
• The only spot left for 8 in the 7th column in the center puzzle is at (13, 13).
• The only spot left for 9 in the 4th 3x3 square in the center puzzle is at (8, 11).
• The only spot left for 2 in the 7th 3x3 square in the center puzzle is at (8, 13).
• The only spot left for 6 in the 8th 3x3 square in the center puzzle is at (10, 14).
• The only spot left for 9 in the 9th 3x3 square in the center puzzle is at (13, 14).
• The only spot left for 1 in the 8th column in the lower left puzzle is at (8, 21).
• The only spot left for 3 in the 8th 3x3 square in the lower left puzzle is at (6, 20).
• The only spot left for 6 in the 8th 3x3 square in the lower right puzzle is at (16, 21).
• The only spot left for 9 in the 9th 3x3 square in the lower right puzzle is at (19, 20).

Step 2. There are 15 independent moves.

• The only spot left for 3 in the 1st row in the upper left puzzle is at (4, 1).
• The only spot left for 3 in the 3rd column in the upper left puzzle is at (3, 4).
• The only spot left for 2 in the 6th column in the upper left puzzle is at (6, 3).
• The only spot left for 6 in the 7th column in the upper left puzzle is at (7, 2).
• The only spot left for 3 in the 1st 3x3 square in the upper left puzzle is at (1, 3).
• The only spot left for 2 in the 6th 3x3 square in the upper left puzzle is at (9, 5).
• The only spot left for 3 in the 8th row in the upper right puzzle is at (21, 8).
• The only spot left for 3 in the 7th 3x3 square in the center puzzle is at (8, 15).
• The only spot left for 6 in the 9th 3x3 square in the center puzzle is at (15, 15).
• The only spot left for 3 in the 1st row in the lower left puzzle is at (2, 13).
• The only spot left for 3 in the 7th column in the lower left puzzle is at (7, 21).
• The only spot left for 2 in the 8th 3x3 square in the lower left puzzle is at (4, 20).
• The only spot left for 9 in the 9th 3x3 square in the lower left puzzle is at (8, 20).
• The only spot left for 8 in the 3rd 3x3 square in the lower right puzzle is at (21, 15).
• The only spot left for 3 in the 9th 3x3 square in the lower right puzzle is at (19, 19).

Step 3. There are 10 independent moves.

• The only spot left for 5 in the 1st row in the upper left puzzle is at (7, 1).
• The only spot left for 3 in the 5th row in the upper left puzzle is at (7, 5).
• The only spot left for 8 in the 9th column in the upper left puzzle is at (9, 4).
• The only spot left for 6 in the 4th 3x3 square in the upper left puzzle is at (2, 4).
• The only spot left for 3 in the 5th 3x3 square in the upper left puzzle is at (5, 6).
• The only spot left for 3 in the 6th 3x3 square in the upper right puzzle is at (19, 4).
• The only spot left for 3 in the 6th 3x3 square in the lower left puzzle is at (9, 17).
• The only spot left for 8 in the 7th column in the lower right puzzle is at (19, 17).
• The only spot left for 4 in the 7th column in the lower right puzzle is at (19, 14).
• The only spot left for 9 in the 3rd 3x3 square in the lower right puzzle is at (21, 13).

Step 4. There are 4 independent moves.

• The only spot left for 1 in the 1st row in the upper left puzzle is at (6, 1).
• The only spot left for 1 in the 3rd 3x3 square in the upper left puzzle is at (8, 3).
• The only spot left for 5 in the 4th row in the upper right puzzle is at (21, 4).
• The only spot left for 5 in the 7th column in the upper right puzzle is at (19, 2).

Step 5. There are 4 independent moves.

• The only spot left for 4 in the 1st row in the upper left puzzle is at (9, 1).
• The only spot left for 4 in the 2nd 3x3 square in the upper left puzzle is at (5, 2).
• The only spot left for 1 in the 5th 3x3 square in the upper left puzzle is at (5, 4).
• The only spot left for 1 in the 4th row in the upper right puzzle is at (18, 4).

Step 6. There are 2 independent moves.

• The only spot left for 1 in the 6th 3x3 square in the upper left puzzle is at (7, 6).
• The only spot left for 6 in the 4th row in the upper right puzzle is at (17, 4).

Step 7. There is 1 independent move.

• 5 must be placed at (17, 15) because it is the only value left that is not already in the 3rd row in the lower right puzzle, the 5th column in the lower right puzzle or the 2nd 3x3 square in the lower right puzzle.

Step 8. There is 1 independent move.

• 1 must be placed at (14, 15) because it is the only value left that is not already in the 9th row in the center puzzle, the 8th column in the center puzzle, the 1st 3x3 square in the lower right puzzle, the 3rd row in the lower right puzzle, the 2nd column in the lower right puzzle or the 9th 3x3 square in the center puzzle.

Step 9. There are 3 independent moves.

• The only spot left for 1 in the 8th row in the center puzzle is at (11, 14).
• The only spot left for 1 in the 7th 3x3 square in the center puzzle is at (7, 13).
• The only spot left for 1 in the 1st row in the lower right puzzle is at (16, 13).

Step 10. There are 4 independent moves.

• The only spot left for 6 in the 1st column in the center puzzle is at (7, 12).
• The only spot left for 6 in the 7th 3x3 square in the center puzzle is at (9, 13).
• The only spot left for 3 in the 8th 3x3 square in the center puzzle is at (11, 13).
• The only spot left for 6 in the 7th column in the lower left puzzle is at (7, 16).

Step 11. There are 3 independent moves.

• The only spot left for 7 in the 7th row in the center puzzle is at (12, 13).
• The only spot left for 6 in the 6th 3x3 square in the center puzzle is at (13, 10).
• The only spot left for 3 in the 1st row in the lower right puzzle is at (17, 13).

Step 12. There are 4 independent moves.

• The only spot left for 5 in the 7th row in the center puzzle is at (14, 13).
• The only spot left for 7 in the 9th row in the center puzzle is at (7, 15).
• The only spot left for 6 in the 1st row in the lower right puzzle is at (19, 13).
• The only spot left for 6 in the 2nd 3x3 square in the lower right puzzle is at (18, 14).

Step 13. There are 6 independent moves.

• The only spot left for 7 in the 7th column in the upper left puzzle is at (7, 4).
• The only spot left for 7 in the 9th 3x3 square in the lower left puzzle is at (9, 20).
• The only spot left for 6 in the 4th row in the lower right puzzle is at (13, 16).
• The only spot left for 2 in the 2nd 3x3 square in the lower right puzzle is at (16, 14).
• The only spot left for 5 in the 3rd 3x3 square in the lower right puzzle is at (20, 14).
• The only spot left for 6 in the 6th 3x3 square in the lower right puzzle is at (21, 18).

Step 14. There are 5 independent moves.

• The only spot left for 4 in the 7th column in the upper left puzzle is at (7, 8).
• The only spot left for 5 in the 9th column in the lower left puzzle is at (9, 21).
• The only spot left for 3 in the 1st column in the lower right puzzle is at (13, 20).
• The only spot left for 1 in the 3rd 3x3 square in the lower right puzzle is at (21, 14).
• The only spot left for 6 in the 5th 3x3 square in the lower right puzzle is at (17, 17).

Step 15. There are 7 independent moves.

• The only spot left for 4 in the 7th 3x3 square in the upper left puzzle is at (1, 7).
• The only spot left for 4 in the 2nd column in the center puzzle is at (8, 14).
• The only spot left for 4 in the 2nd 3x3 square in the center puzzle is at (12, 7).
• The only spot left for 5 in the 8th row in the lower left puzzle is at (3, 20).
• The only spot left for 5 in the 7th column in the lower left puzzle is at (7, 14).
• The only spot left for 4 in the 9th 3x3 square in the lower left puzzle is at (7, 19).
• The only spot left for 7 in the 5th column in the lower right puzzle is at (17, 19).

Step 16. There are 11 independent moves.

• The only spot left for 1 in the 7th 3x3 square in the upper left puzzle is at (3, 7).
• The only spot left for 1 in the 1st row in the center puzzle is at (10, 7).
• The only spot left for 8 in the 1st column in the center puzzle is at (7, 11).
• The only spot left for 5 in the 4th 3x3 square in the center puzzle is at (8, 10).
• The only spot left for 8 in the 7th 3x3 square in the center puzzle is at (9, 14).
• The only spot left for 6 in the 8th row in the lower left puzzle is at (1, 20).
• The only spot left for 5 in the 1st 3x3 square in the lower left puzzle is at (2, 15).
• The only spot left for 4 in the 6th 3x3 square in the lower left puzzle is at (9, 16).
• The only spot left for 8 in the 9th 3x3 square in the lower left puzzle is at (7, 20).
• The only spot left for 2 in the 5th column in the lower right puzzle is at (17, 18).
• The only spot left for 7 in the 9th column in the lower right puzzle is at (21, 21).

Step 17. There are 4 independent moves.

• The only spot left for 1 in the 5th 3x3 square in the center puzzle is at (12, 10).
• The only spot left for 4 in the 2nd column in the lower left puzzle is at (2, 18).
• The only spot left for 2 in the 6th 3x3 square in the lower right puzzle is at (19, 16).
• The only spot left for 2 in the 9th 3x3 square in the lower right puzzle is at (21, 19).

Step 18. There are 4 independent moves.

• The only spot left for 8 in the 5th 3x3 square in the center puzzle is at (10, 10).
• The only spot left for 4 in the 5th 3x3 square in the lower left puzzle is at (4, 17).
• The only spot left for 5 in the 6th 3x3 square in the lower right puzzle is at (19, 18).
• The only spot left for 2 in the 8th 3x3 square in the lower right puzzle is at (18, 20).

Step 19. There are 2 independent moves.

• The only spot left for 1 in the 8th row in the lower right puzzle is at (15, 20).
• The only spot left for 1 in the 8th 3x3 square in the lower right puzzle is at (18, 19).

Step 20. There are 7 independent moves.

• The only spot left for 1 in the 6th row in the lower right puzzle is at (20, 18).
• The only spot left for 8 in the 8th row in the lower right puzzle is at (16, 20).
• The only spot left for 7 in the 3rd column in the lower right puzzle is at (15, 14).
• The only spot left for 1 in the 4th 3x3 square in the lower right puzzle is at (13, 17).
• The only spot left for 8 in the 7th 3x3 square in the lower right puzzle is at (15, 21).
• The only spot left for 7 in the 7th 3x3 square in the lower right puzzle is at (14, 20).
• The only spot left for 5 in the 8th 3x3 square in the lower right puzzle is at (16, 19).

Step 21. There are 6 independent moves.

• The only spot left for 7 in the 4th row in the center puzzle is at (9, 10).
• The only spot left for 3 in the 9th 3x3 square in the center puzzle is at (14, 14).
• The only spot left for 2 in the 3rd column in the lower right puzzle is at (15, 17).
• The only spot left for 2 in the 7th 3x3 square in the lower right puzzle is at (14, 21).
• The only spot left for 5 in the 7th 3x3 square in the lower right puzzle is at (13, 21).
• The only spot left for 8 in the 9th 3x3 square in the lower right puzzle is at (20, 19).

Step 22. There are 5 independent moves.

• The only spot left for 2 in the 4th row in the center puzzle is at (15, 10).
• The only spot left for 5 in the 5th row in the lower right puzzle is at (18, 17).
• The only spot left for 5 in the 4th 3x3 square in the lower right puzzle is at (15, 16).
• The only spot left for 4 in the 7th 3x3 square in the lower right puzzle is at (13, 19).
• The only spot left for 4 in the 9th 3x3 square in the lower right puzzle is at (20, 21).

Step 23. There are 4 independent moves.

• The only spot left for 2 in the 7th 3x3 square in the upper right puzzle is at (14, 8).
• The only spot left for 3 in the 4th row in the lower right puzzle is at (16, 16).
• The only spot left for 4 in the 6th column in the lower right puzzle is at (18, 15).
• The only spot left for 3 in the 4th 3x3 square in the lower right puzzle is at (15, 18).

Step 24. There are 4 independent moves.

• The only spot left for 7 in the 4th row in the lower right puzzle is at (20, 16).
• The only spot left for 9 in the 6th column in the lower right puzzle is at (18, 16).
• The only spot left for 9 in the 2nd 3x3 square in the lower right puzzle is at (16, 15).
• The only spot left for 7 in the 5th 3x3 square in the lower right puzzle is at (16, 17).

Step 25. There are 4 independent moves.

• The only spot left for 4 in the 5th row in the lower right puzzle is at (14, 17).
• The only spot left for 9 in the 6th row in the lower right puzzle is at (14, 18).
• The only spot left for 4 in the 5th 3x3 square in the lower right puzzle is at (16, 18).
• The only spot left for 9 in the 6th 3x3 square in the lower right puzzle is at (20, 17).

Step 26. There are 4 independent moves.

• 7 must be placed at (2, 14) because it is the only value left that is not already in the 2nd row in the lower left puzzle, the 2nd column in the lower left puzzle or the 1st 3x3 square in the lower left puzzle.
• 4 must be placed at (8, 5) because it is the only value left that is not already in the 5th row in the upper left puzzle, the 8th column in the upper left puzzle or the 6th 3x3 square in the upper left puzzle.
• 5 must be placed at (1, 4) because it is the only value left that is not already in the 4th row in the upper left puzzle, the 1st column in the upper left puzzle or the 4th 3x3 square in the upper left puzzle.
• 5 must be placed at (3, 2) because it is the only value left that is not already in the 2nd row in the upper left puzzle, the 3rd column in the upper left puzzle or the 1st 3x3 square in the upper left puzzle.

Step 27. There are 6 independent moves.

• The only spot left for 4 in the 4th row in the upper left puzzle is at (6, 4).
• The only spot left for 5 in the 2nd 3x3 square in the upper left puzzle is at (4, 3).
• The only spot left for 4 in the 4th 3x3 square in the upper left puzzle is at (2, 6).
• The only spot left for 5 in the 6th 3x3 square in the upper left puzzle is at (8, 6).
• The only spot left for 5 in the 7th 3x3 square in the upper left puzzle is at (2, 9).
• The only spot left for 7 in the 7th 3x3 square in the lower left puzzle is at (3, 19).

Step 28. There are 3 independent moves.

• The only spot left for 6 in the 4th column in the upper left puzzle is at (4, 7).
• The only spot left for 6 in the 2nd 3x3 square in the upper left puzzle is at (5, 3).
• The only spot left for 9 in the 6th 3x3 square in the upper left puzzle is at (8, 4).

Step 29. There are 7 independent moves.

• The only spot left for 7 in the 3rd row in the upper left puzzle is at (9, 3).
• The only spot left for 6 in the 8th row in the upper left puzzle is at (3, 8).
• The only spot left for 9 in the 4th column in the upper left puzzle is at (4, 6).
• The only spot left for 7 in the 2nd 3x3 square in the upper left puzzle is at (4, 2).
• The only spot left for 9 in the 8th 3x3 square in the upper left puzzle is at (6, 7).
• The only spot left for 6 in the 9th 3x3 square in the upper left puzzle is at (8, 9).
• The only spot left for 6 in the 1st row in the center puzzle is at (14, 7).

Step 30. There are 8 independent moves.

• The only spot left for 9 in the 3rd row in the upper left puzzle is at (2, 3).
• The only spot left for 7 in the 5th row in the upper left puzzle is at (1, 5).
• The only spot left for 8 in the 6th row in the upper left puzzle is at (1, 6).
• The only spot left for 7 in the 3rd column in the upper left puzzle is at (3, 9).
• The only spot left for 9 in the 3rd 3x3 square in the upper left puzzle is at (9, 2).
• The only spot left for 7 in the 5th 3x3 square in the upper left puzzle is at (6, 6).
• The only spot left for 8 in the 5th 3x3 square in the upper left puzzle is at (4, 5).
• The only spot left for 6 in the 7th column in the upper right puzzle is at (19, 9).

Step 31. There are 4 independent moves.

• The only spot left for 8 in the 9th row in the upper left puzzle is at (5, 9).
• The only spot left for 1 in the 1st column in the upper left puzzle is at (1, 2).
• The only spot left for 1 in the 4th 3x3 square in the upper left puzzle is at (2, 5).
• The only spot left for 8 in the 7th 3x3 square in the upper left puzzle is at (2, 8).

Step 32. There are 5 independent moves.

• The only spot left for 2 in the 1st 3x3 square in the upper left puzzle is at (2, 2).
• The only spot left for 2 in the 7th 3x3 square in the upper left puzzle is at (1, 9).
• The only spot left for 2 in the 8th 3x3 square in the upper left puzzle is at (5, 8).
• The only spot left for 8 in the 9th 3x3 square in the upper left puzzle is at (8, 7).
• The only spot left for 8 in the 2nd row in the center puzzle is at (12, 8).

Step 33. There are 9 independent moves.

• The only spot left for 7 in the 8th 3x3 square in the upper left puzzle is at (5, 7).
• The only spot left for 7 in the 9th 3x3 square in the upper left puzzle is at (8, 8).
• The only spot left for 8 in the 7th row in the upper right puzzle is at (18, 7).
• The only spot left for 8 in the 7th 3x3 square in the upper right puzzle is at (15, 9).
• The only spot left for 9 in the 2nd row in the center puzzle is at (15, 8).
• The only spot left for 5 in the 2nd row in the center puzzle is at (10, 8).
• The only spot left for 9 in the 6th column in the center puzzle is at (12, 12).
• The only spot left for 9 in the 2nd 3x3 square in the center puzzle is at (11, 7).
• The only spot left for 8 in the 8th 3x3 square in the center puzzle is at (11, 15).

Step 34. There are 9 independent moves.

• The only spot left for 9 in the 7th row in the upper right puzzle is at (20, 7).
• The only spot left for 4 in the 7th row in the upper right puzzle is at (19, 7).
• The only spot left for 5 in the 7th 3x3 square in the upper right puzzle is at (13, 9).
• The only spot left for 1 in the 7th 3x3 square in the upper right puzzle is at (13, 8).
• The only spot left for 9 in the 8th 3x3 square in the upper right puzzle is at (17, 9).
• The only spot left for 7 in the 1st row in the center puzzle is at (13, 7).
• The only spot left for 1 in the 9th column in the center puzzle is at (15, 11).
• The only spot left for 7 in the 2nd 3x3 square in the center puzzle is at (11, 9).
• The only spot left for 5 in the 8th 3x3 square in the center puzzle is at (12, 15).

Step 35. There are 9 independent moves.

• The only spot left for 9 in the 1st row in the upper right puzzle is at (16, 1).
• The only spot left for 7 in the 7th column in the upper right puzzle is at (19, 1).
• The only spot left for 9 in the 3rd 3x3 square in the upper right puzzle is at (21, 2).
• The only spot left for 4 in the 7th 3x3 square in the upper right puzzle is at (14, 9).
• The only spot left for 3 in the 7th 3x3 square in the upper right puzzle is at (15, 7).
• The only spot left for 5 in the 5th row in the center puzzle is at (11, 11).
• The only spot left for 7 in the 6th row in the center puzzle is at (10, 12).
• The only spot left for 5 in the 6th 3x3 square in the center puzzle is at (15, 12).
• The only spot left for 7 in the 6th 3x3 square in the center puzzle is at (14, 11).

Step 36. There are 7 independent moves.

• The only spot left for 4 in the 5th row in the upper right puzzle is at (15, 5).
• The only spot left for 1 in the 2nd 3x3 square in the upper right puzzle is at (16, 2).
• The only spot left for 4 in the 3rd 3x3 square in the upper right puzzle is at (21, 3).
• The only spot left for 2 in the 5th row in the center puzzle is at (9, 11).
• The only spot left for 3 in the 6th row in the center puzzle is at (13, 12).
• The only spot left for 2 in the 5th 3x3 square in the center puzzle is at (11, 12).
• The only spot left for 3 in the 5th 3x3 square in the center puzzle is at (10, 11).

Step 37. There are 5 independent moves.

• The only spot left for 6 in the 5th row in the upper right puzzle is at (21, 5).
• The only spot left for 6 in the 4th column in the upper right puzzle is at (16, 8).
• The only spot left for 6 in the 2nd 3x3 square in the upper right puzzle is at (18, 2).
• The only spot left for 4 in the 4th 3x3 square in the center puzzle is at (9, 12).
• The only spot left for 4 in the 6th 3x3 square in the center puzzle is at (13, 11).

Step 38. There are 3 independent moves.

• The only spot left for 7 in the 5th row in the upper right puzzle is at (20, 5).
• The only spot left for 4 in the 4th column in the upper right puzzle is at (16, 6).
• The only spot left for 2 in the 2nd 3x3 square in the upper right puzzle is at (18, 3).

Step 39. There are 7 independent moves.

• The only spot left for 1 in the 5th row in the upper right puzzle is at (14, 5).
• The only spot left for 7 in the 4th column in the upper right puzzle is at (16, 3).
• The only spot left for 4 in the 6th column in the upper right puzzle is at (18, 8).
• The only spot left for 1 in the 8th column in the upper right puzzle is at (20, 9).
• The only spot left for 7 in the 5th 3x3 square in the upper right puzzle is at (18, 6).
• The only spot left for 1 in the 6th 3x3 square in the upper right puzzle is at (21, 6).
• The only spot left for 7 in the 9th 3x3 square in the upper right puzzle is at (21, 9).

Step 40. There are 3 independent moves.

• The only spot left for 7 in the 1st 3x3 square in the upper right puzzle is at (14, 2).
• The only spot left for 1 in the 1st 3x3 square in the upper right puzzle is at (15, 1).
• The only spot left for 7 in the 8th 3x3 square in the upper right puzzle is at (17, 8).

Step 41. There are 6 independent moves.

• The only spot left for 2 in the 1st row in the upper right puzzle is at (20, 1).
• The only spot left for 8 in the 2nd row in the upper right puzzle is at (17, 2).
• The only spot left for 6 in the 3rd column in the upper right puzzle is at (15, 6).
• The only spot left for 8 in the 1st 3x3 square in the upper right puzzle is at (14, 3).
• The only spot left for 2 in the 1st 3x3 square in the upper right puzzle is at (15, 2).
• The only spot left for 6 in the 1st 3x3 square in the upper right puzzle is at (13, 1).

Step 42. There are 6 independent moves.

• The only spot left for 3 in the 3rd row in the upper right puzzle is at (17, 3).
• The only spot left for 5 in the 1st 3x3 square in the upper right puzzle is at (15, 3).
• The only spot left for 4 in the 1st 3x3 square in the upper right puzzle is at (13, 2).
• The only spot left for 4 in the 2nd 3x3 square in the upper right puzzle is at (17, 1).
• The only spot left for 3 in the 3rd 3x3 square in the upper right puzzle is at (20, 2).
• The only spot left for 5 in the 4th 3x3 square in the upper right puzzle is at (14, 6).

Step 43. There are 2 independent moves.

• The only spot left for 3 in the 1st 3x3 square in the upper right puzzle is at (14, 1).
• The only spot left for 3 in the 4th 3x3 square in the upper right puzzle is at (13, 6).

Step 44. There are 2 independent moves.

• 9 must be placed at (3, 18) because it is the only value left that is not already in the 6th row in the lower left puzzle, the 3rd column in the lower left puzzle or the 4th 3x3 square in the lower left puzzle.
• 4 must be placed at (1, 13) because it is the only value left that is not already in the 1st row in the lower left puzzle, the 1st column in the lower left puzzle or the 1st 3x3 square in the lower left puzzle.

Step 45. There are 3 independent moves.

• The only spot left for 4 in the 2nd 3x3 square in the lower left puzzle is at (6, 15).
• The only spot left for 4 in the 7th 3x3 square in the lower left puzzle is at (3, 21).
• The only spot left for 9 in the 7th 3x3 square in the lower left puzzle is at (2, 19).

Step 46. There are 2 independent moves.

• The only spot left for 8 in the 2nd column in the lower left puzzle is at (2, 16).
• The only spot left for 8 in the 7th 3x3 square in the lower left puzzle is at (1, 21).

Step 47. There is 1 independent move.

• The only spot left for 8 in the 6th 3x3 square in the lower left puzzle is at (8, 18).

Step 48. There is 1 independent move.

• The only spot left for 5 in the 6th 3x3 square in the lower left puzzle is at (8, 17).

Step 49. There is 1 independent move.

• The only spot left for 7 in the 6th 3x3 square in the lower left puzzle is at (8, 16).

Step 50. There is 1 independent move.

• The only spot left for 7 in the 4th column in the lower left puzzle is at (4, 13).

Step 51. There is 1 independent move.

• The only spot left for 5 in the 4th column in the lower left puzzle is at (4, 19).

Step 52. There are 2 independent moves.

• The only spot left for 8 in the 4th column in the lower left puzzle is at (4, 15).
• The only spot left for 8 in the 8th 3x3 square in the lower left puzzle is at (5, 19).

Step 53. There are 2 independent moves.

• The only spot left for 6 in the 4th column in the lower left puzzle is at (4, 21).
• The only spot left for 1 in the 8th 3x3 square in the lower left puzzle is at (6, 19).

Step 54. There are 3 independent moves.

• The only spot left for 1 in the 4th row in the lower left puzzle is at (5, 16).
• The only spot left for 9 in the 4th column in the lower left puzzle is at (4, 16).
• The only spot left for 9 in the 8th 3x3 square in the lower left puzzle is at (5, 21).

Step 55. There are 2 independent moves.

• The only spot left for 2 in the 4th row in the lower left puzzle is at (6, 16).
• The only spot left for 9 in the 2nd 3x3 square in the lower left puzzle is at (6, 13).

Step 56. There are 4 independent moves.

• The only spot left for 2 in the 2nd row in the lower left puzzle is at (3, 14).
• The only spot left for 5 in the 6th column in the lower left puzzle is at (6, 18).
• The only spot left for 2 in the 2nd 3x3 square in the lower left puzzle is at (5, 15).
• The only spot left for 5 in the 2nd 3x3 square in the lower left puzzle is at (5, 13).

Step 57. There are 4 independent moves.

• The only spot left for 6 in the 1st 3x3 square in the lower left puzzle is at (3, 15).
• The only spot left for 6 in the 2nd 3x3 square in the lower left puzzle is at (6, 14).
• The only spot left for 2 in the 4th 3x3 square in the lower left puzzle is at (1, 17).
• The only spot left for 6 in the 5th 3x3 square in the lower left puzzle is at (5, 18).

Step 58. There are 4 independent moves.

• The only spot left for 1 in the 1st 3x3 square in the lower left puzzle is at (1, 15).
• The only spot left for 7 in the 4th 3x3 square in the lower left puzzle is at (1, 18).
• The only spot left for 1 in the 4th 3x3 square in the lower left puzzle is at (3, 17).
• The only spot left for 7 in the 5th 3x3 square in the lower left puzzle is at (5, 17).