Variations on Difficult Sudoku-by5 Logic Puzzles (Vol 1) Step-by-Step Instructions for Puzzle 33   (265/369) Diff: 10/99.9999.9994 Printable puzzle or puzzle with step-by-step solutions 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 'HyperSudoku-Xtreme' puzzles.

Step 1. There are 36 independent moves.

• The only spot left for 4 in the 4th row in the upper left puzzle is at (4, 4).
• The only spot left for 3 in the 5th row in the upper left puzzle is at (9, 5).
• The only spot left for 5 in the 7th column in the upper left puzzle is at (7, 3).
• The only spot left for 4 in the 4th 3x3 square in the upper left puzzle is at (3, 5).
• The only spot left for 6 in the 4th 3x3 square in the upper left puzzle is at (1, 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 8 in the 6th 3x3 square in the upper left puzzle is at (7, 4).
• The only spot left for 8 in the 7th 3x3 square in the upper left puzzle is at (2, 7).
• The only spot left for 5 in the 1st diagonal in the upper left puzzle is at (8, 8).
• The only spot left for 3 in the 1st diagonal in the upper left puzzle is at (2, 2).
• The only spot left for 2 in the 6th 3x3 square in the upper right puzzle is at (20, 5).
• The only spot left for 8 in the 8th 3x3 square in the upper right puzzle is at (16, 8).
• The only spot left for 6 in the 2nd diagonal in the upper right puzzle is at (15, 7).
• The only spot left for 7 in the 3rd row in the center puzzle is at (7, 9).
• The only spot left for 1 in the 8th row in the center puzzle is at (10, 14).
• The only spot left for 4 in the 8th column in the center puzzle is at (14, 15).
• The only spot left for 5 in the 5th 3x3 square in the center puzzle is at (10, 11).
• The only spot left for 8 in the 7th 3x3 square in the center puzzle is at (8, 15).
• The only spot left for 9 in the 7th 3x3 square in the center puzzle is at (8, 13).
• 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 9th 3x3 square in the center puzzle is at (15, 15).
• The only spot left for 8 in the 2nd 3x3 inner square in the center puzzle is at (13, 10).
• The only spot left for 9 in the 2nd 3x3 inner square in the center puzzle is at (14, 9).
• The only spot left for 9 in the 3rd row in the lower left puzzle is at (2, 15).
• The only spot left for 3 in the 4th 3x3 square in the lower left puzzle is at (2, 18).
• The only spot left for 4 in the 5th 3x3 square in the lower left puzzle is at (6, 18).
• The only spot left for 4 in the 7th 3x3 square in the lower left puzzle is at (3, 20).
• The only spot left for 1 in the 2nd diagonal in the lower left puzzle is at (2, 20).
• The only spot left for 3 in the 2nd diagonal in the lower left puzzle is at (6, 16).
• The only spot left for 3 in the 2nd row in the lower right puzzle is at (20, 14).
• The only spot left for 3 in the 5th row in the lower right puzzle is at (19, 17).
• The only spot left for 2 in the 7th column in the lower right puzzle is at (19, 16).
• The only spot left for 2 in the 8th column in the lower right puzzle is at (20, 19).
• The only spot left for 2 in the 3rd 3x3 square in the lower right puzzle is at (21, 14).
• The only spot left for 4 in the 5th 3x3 square in the lower right puzzle is at (18, 16).
• The only spot left for 3 in the 7th 3x3 square in the lower right puzzle is at (13, 19).

Step 2. There are 33 independent moves.

• The only spot left for 5 in the 5th row in the upper left puzzle is at (1, 5).
• The only spot left for 4 in the 8th row in the upper left puzzle is at (5, 8).
• The only spot left for 9 in the 9th row in the upper left puzzle is at (3, 9).
• The only spot left for 6 in the 9th column in the upper left puzzle is at (9, 2).
• The only spot left for 4 in the 1st 3x3 square in the upper left puzzle is at (2, 3).
• The only spot left for 1 in the 5th 3x3 square in the upper left puzzle is at (4, 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 9 in the 5th 3x3 square in the upper left puzzle is at (6, 4).
• The only spot left for 5 in the 6th 3x3 square in the upper left puzzle is at (9, 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 4 in the 9th 3x3 square in the upper left puzzle is at (8, 9).
• The only spot left for 6 in the 9th 3x3 square in the upper left puzzle is at (7, 8).
• The only spot left for 3 in the 9th 3x3 square in the upper left puzzle is at (8, 7).
• The only spot left for 8 in the 1st diagonal in the upper left puzzle is at (1, 1).
• The only spot left for 5 in the 7th 3x3 square in the upper right puzzle is at (13, 7).
• The only spot left for 2 in the 1st diagonal in the upper right puzzle is at (21, 9).
• The only spot left for 3 in the 2nd diagonal in the upper right puzzle is at (13, 9).
• The only spot left for 5 in the 3rd row in the center puzzle is at (11, 9).
• The only spot left for 5 in the 9th column in the center puzzle is at (15, 14).
• The only spot left for 8 in the 4th 3x3 square in the center puzzle is at (7, 12).
• The only spot left for 3 in the 9th 3x3 square in the center puzzle is at (14, 13).
• The only spot left for 8 in the 3rd 3x3 inner square in the center puzzle is at (10, 13).
• The only spot left for 9 in the 6th column in the lower left puzzle is at (6, 21).
• The only spot left for 1 in the 8th 3x3 square in the lower left puzzle is at (4, 19).
• The only spot left for 4 in the 9th 3x3 square in the lower left puzzle is at (7, 21).
• The only spot left for 9 in the 2nd diagonal in the lower left puzzle is at (5, 17).
• The only spot left for 1 in the 9th row in the lower right puzzle is at (17, 21).
• The only spot left for 4 in the 1st column in the lower right puzzle is at (13, 17).
• The only spot left for 1 in the 9th column in the lower right puzzle is at (21, 13).
• 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 1 in the 9th 3x3 square in the lower right puzzle is at (19, 20).
• The only spot left for 4 in the 1st diagonal in the lower right puzzle is at (20, 20).
• The only spot left for 3 in the 1st diagonal in the lower right puzzle is at (18, 18).

Step 3. There are 35 independent moves.

• The only spot left for 6 in the 9th row in the upper left puzzle is at (4, 9).
• The only spot left for 6 in the 2nd column in the upper left puzzle is at (2, 1).
• The only spot left for 4 in the 2nd 3x3 square in the upper left puzzle is at (6, 2).
• The only spot left for 8 in the 4th 3x3 square in the upper left puzzle is at (3, 6).
• The only spot left for 9 in the 6th 3x3 square in the upper left puzzle is at (8, 6).
• The only spot left for 1 in the 7th 3x3 square in the upper left puzzle is at (1, 8).
• The only spot left for 9 in the 1st diagonal in the upper left puzzle is at (7, 7).
• The only spot left for 6 in the 2nd diagonal in the upper left puzzle is at (3, 7).
• The only spot left for 2 in the 2nd diagonal in the upper left puzzle is at (5, 5).
• The only spot left for 2 in the 8th row in the upper right puzzle is at (14, 8).
• 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 3 in the 9th 3x3 square in the upper right puzzle is at (19, 8).
• The only spot left for 4 in the 2nd diagonal in the upper right puzzle is at (16, 6).
• The only spot left for 1 in the 3rd row in the center puzzle is at (9, 9).
• The only spot left for 1 in the 2nd column in the center puzzle is at (8, 11).
• The only spot left for 4 in the 4th column in the center puzzle is at (10, 7).
• The only spot left for 3 in the 6th column in the center puzzle is at (12, 15).
• The only spot left for 6 in the 8th column in the center puzzle is at (14, 11).
• The only spot left for 3 in the 2nd 3x3 square in the center puzzle is at (11, 8).
• The only spot left for 8 in the 2nd 3x3 square in the center puzzle is at (11, 7).
• The only spot left for 7 in the 3rd 3x3 inner square in the center puzzle is at (9, 14).
• The only spot left for 3 in the 3rd 3x3 inner square in the center puzzle is at (10, 12).
• The only spot left for 5 in the 4th 3x3 inner square in the center puzzle is at (12, 13).
• The only spot left for 4 in the 4th column in the lower left puzzle is at (4, 15).
• The only spot left for 3 in the 4th column in the lower left puzzle is at (4, 13).
• The only spot left for 2 in the 5th 3x3 square in the lower left puzzle is at (5, 16).
• The only spot left for 8 in the 8th 3x3 square in the lower left puzzle is at (4, 20).
• The only spot left for 4 in the 8th 3x3 square in the lower left puzzle is at (5, 19).
• The only spot left for 3 in the 2nd 3x3 square in the lower right puzzle is at (17, 15).
• The only spot left for 7 in the 3rd 3x3 square in the lower right puzzle is at (21, 15).
• The only spot left for 1 in the 4th 3x3 square in the lower right puzzle is at (13, 18).
• The only spot left for 1 in the 5th 3x3 square in the lower right puzzle is at (16, 17).
• The only spot left for 4 in the 7th 3x3 square in the lower right puzzle is at (15, 21).
• The only spot left for 8 in the 9th 3x3 square in the lower right puzzle is at (20, 21).
• The only spot left for 9 in the 2nd diagonal in the lower right puzzle is at (15, 19).

Step 4. There are 43 independent moves.

• The only spot left for 9 in the 1st row in the upper left puzzle is at (5, 1).
• The only spot left for 1 in the 2nd row in the upper left puzzle is at (5, 2).
• The only spot left for 1 in the 3rd row in the upper left puzzle is at (8, 3).
• The only spot left for 2 in the 9th row in the upper left puzzle is at (6, 9).
• The only spot left for 2 in the 2nd column in the upper left puzzle is at (2, 6).
• The only spot left for 2 in the 7th column in the upper left puzzle is at (7, 2).
• The only spot left for 1 in the 1st 3x3 square in the upper left puzzle is at (3, 1).
• The only spot left for 9 in the 3rd 3x3 square in the upper left puzzle is at (9, 3).
• The only spot left for 7 in the 5th 3x3 square in the upper left puzzle is at (6, 5).
• The only spot left for 2 in the 9th 3x3 square in the upper left puzzle is at (9, 7).
• The only spot left for 7 in the 1st diagonal in the upper left puzzle is at (3, 3).
• The only spot left for 7 in the 2nd diagonal in the upper left puzzle is at (2, 8).
• The only spot left for 1 in the 5th row in the upper right puzzle is at (15, 5).
• The only spot left for 6 in the 8th row in the upper right puzzle is at (20, 8).
• The only spot left for 2 in the 3rd 3x3 square in the upper right puzzle is at (19, 2).
• The only spot left for 4 in the 7th 3x3 square in the upper right puzzle is at (15, 8).
• The only spot left for 4 in the 1st diagonal in the upper right puzzle is at (13, 1).
• The only spot left for 9 in the 2nd diagonal in the upper right puzzle is at (21, 1).
• The only spot left for 2 in the 3rd row in the center puzzle is at (10, 9).
• The only spot left for 6 in the 8th row in the center puzzle is at (13, 14).
• The only spot left for 2 in the 1st column in the center puzzle is at (7, 10).
• The only spot left for 4 in the 7th column in the center puzzle is at (13, 12).
• The only spot left for 7 in the 8th column in the center puzzle is at (14, 7).
• The only spot left for 1 in the 2nd 3x3 square in the center puzzle is at (12, 7).
• The only spot left for 9 in the 4th 3x3 square in the center puzzle is at (9, 11).
• The only spot left for 1 in the 5th 3x3 square in the center puzzle is at (11, 10).
• The only spot left for 6 in the 7th 3x3 square in the center puzzle is at (9, 15).
• The only spot left for 2 in the 7th 3x3 square in the center puzzle is at (8, 14).
• The only spot left for 7 in the 8th 3x3 square in the center puzzle is at (10, 15).
• The only spot left for 7 in the 9th 3x3 square in the center puzzle is at (13, 13).
• The only spot left for 3 in the 1st 3x3 inner square in the center puzzle is at (9, 10).
• The only spot left for 4 in the 2nd 3x3 inner square in the center puzzle is at (12, 10).
• The only spot left for 7 in the 2nd 3x3 inner square in the center puzzle is at (12, 8).
• The only spot left for 6 in the 3rd 3x3 inner square in the center puzzle is at (8, 12).
• The only spot left for 3 in the 7th row in the lower left puzzle is at (8, 19).
• The only spot left for 4 in the 1st 3x3 square in the lower left puzzle is at (1, 14).
• The only spot left for 3 in the 8th 3x3 square in the lower left puzzle is at (5, 21).
• The only spot left for 7 in the 2nd row in the lower right puzzle is at (16, 14).
• The only spot left for 5 in the 5th row in the lower right puzzle is at (14, 17).
• The only spot left for 8 in the 4th 3x3 square in the lower right puzzle is at (13, 16).
• The only spot left for 8 in the 7th 3x3 square in the lower right puzzle is at (15, 20).
• The only spot left for 9 in the 8th 3x3 square in the lower right puzzle is at (16, 20).
• The only spot left for 9 in the 1st diagonal in the lower right puzzle is at (21, 21).

Step 5. There are 30 independent moves.

• The only spot left for 7 in the 2nd row in the upper left puzzle is at (4, 2).
• The only spot left for 7 in the 5th column in the upper left puzzle is at (5, 7).
• The only spot left for 7 in the 9th column in the upper left puzzle is at (9, 4).
• The only spot left for 2 in the 1st 3x3 square in the upper left puzzle is at (1, 3).
• The only spot left for 9 in the 1st 3x3 square in the upper left puzzle is at (1, 2).
• The only spot left for 2 in the 2nd 3x3 square in the upper left puzzle is at (4, 1).
• The only spot left for 7 in the 3rd 3x3 square in the upper left puzzle is at (8, 1).
• The only spot left for 7 in the 4th 3x3 square in the upper left puzzle is at (1, 6).
• The only spot left for 2 in the 6th 3x3 square in the upper left puzzle is at (8, 4).
• The only spot left for 1 in the 8th 3x3 square in the upper left puzzle is at (6, 7).
• The only spot left for 5 in the 5th row in the upper right puzzle is at (17, 5).
• The only spot left for 7 in the 9th 3x3 square in the upper right puzzle is at (21, 8).
• The only spot left for 8 in the 2nd diagonal in the upper right puzzle is at (20, 2).
• The only spot left for 9 in the 4th row in the center puzzle is at (15, 10).
• The only spot left for 9 in the 5th 3x3 square in the center puzzle is at (11, 12).
• The only spot left for 7 in the 6th 3x3 square in the center puzzle is at (15, 12).
• The only spot left for 3 in the 6th 3x3 square in the center puzzle is at (15, 11).
• The only spot left for 2 in the 6th 3x3 square in the center puzzle is at (13, 11).
• The only spot left for 6 in the 8th 3x3 square in the center puzzle is at (11, 13).
• The only spot left for 6 in the 1st 3x3 inner square in the center puzzle is at (10, 10).
• The only spot left for 2 in the 4th 3x3 inner square in the center puzzle is at (12, 12).
• The only spot left for 2 in the 8th row in the lower left puzzle is at (9, 20).
• The only spot left for 8 in the 1st 3x3 square in the lower left puzzle is at (2, 14).
• The only spot left for 2 in the 2nd 3x3 square in the lower left puzzle is at (6, 15).
• The only spot left for 2 in the 7th 3x3 square in the lower left puzzle is at (2, 19).
• The only spot left for 6 in the 4th 3x3 square in the lower right puzzle is at (15, 17).
• The only spot left for 7 in the 5th 3x3 square in the lower right puzzle is at (17, 16).
• The only spot left for 6 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 7 in the 8th 3x3 square in the lower right puzzle is at (18, 19).

Step 6. There are 18 independent moves.

• The only spot left for 5 in the 2nd 3x3 square in the upper left puzzle is at (6, 1).
• The only spot left for 5 in the 8th 3x3 square in the upper left puzzle is at (4, 7).
• The only spot left for 8 in the 6th row in the upper right puzzle is at (13, 6).
• The only spot left for 5 in the 8th column in the upper right puzzle is at (20, 9).
• The only spot left for 5 in the 3rd 3x3 square in the upper right puzzle is at (19, 1).
• The only spot left for 8 in the 6th 3x3 square in the upper right puzzle is at (21, 4).
• The only spot left for 7 in the 2nd diagonal in the upper right puzzle is at (19, 3).
• The only spot left for 5 in the 2nd row in the lower left puzzle is at (5, 14).
• The only spot left for 7 in the 3rd row in the lower left puzzle is at (1, 15).
• The only spot left for 6 in the 7th row in the lower left puzzle is at (7, 19).
• The only spot left for 8 in the 9th row in the lower left puzzle is at (1, 21).
• The only spot left for 7 in the 6th column in the lower left puzzle is at (6, 20).
• The only spot left for 7 in the 2nd 3x3 square in the lower left puzzle is at (5, 13).
• The only spot left for 6 in the 7th 3x3 square in the lower left puzzle is at (2, 21).
• The only spot left for 5 in the 9th 3x3 square in the lower left puzzle is at (8, 20).
• The only spot left for 8 in the 9th 3x3 square in the lower left puzzle is at (9, 19).
• The only spot left for 2 in the 1st diagonal in the lower left puzzle is at (1, 13).
• The only spot left for 8 in the 8th 3x3 square in the lower right puzzle is at (16, 19).

Step 7. There are 26 independent moves.

• The only spot left for 1 in the 1st row in the upper right puzzle is at (16, 1).
• The only spot left for 6 in the 6th row in the upper right puzzle is at (19, 6).
• The only spot left for 6 in the 1st column in the upper right puzzle is at (13, 3).
• The only spot left for 8 in the 5th column in the upper right puzzle is at (17, 3).
• The only spot left for 1 in the 8th column in the upper right puzzle is at (20, 6).
• The only spot left for 4 in the 9th column in the upper right puzzle is at (21, 7).
• The only spot left for 1 in the 3rd 3x3 square in the upper right puzzle is at (21, 3).
• The only spot left for 6 in the 4th 3x3 square in the upper right puzzle is at (14, 4).
• The only spot left for 8 in the 5th 3x3 square in the upper right puzzle is at (18, 5).
• The only spot left for 4 in the 6th 3x3 square in the upper right puzzle is at (19, 4).
• The only spot left for 7 in the 1st diagonal in the upper right puzzle is at (18, 6).
• The only spot left for 5 in the 4th row in the lower left puzzle is at (9, 16).
• The only spot left for 6 in the 1st column in the lower left puzzle is at (1, 17).
• The only spot left for 7 in the 2nd column in the lower left puzzle is at (2, 17).
• The only spot left for 6 in the 1st 3x3 square in the lower left puzzle is at (3, 13).
• The only spot left for 5 in the 1st 3x3 square in the lower left puzzle is at (2, 13).
• 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 5 in the 5th 3x3 square in the lower left puzzle is at (4, 17).
• The only spot left for 6 in the 6th 3x3 square in the lower left puzzle is at (8, 16).
• The only spot left for 6 in the 8th 3x3 square in the lower left puzzle is at (5, 20).
• The only spot left for 7 in the 9th 3x3 square in the lower left puzzle is at (8, 21).
• The only spot left for 7 in the 1st diagonal in the lower left puzzle is at (4, 16).
• The only spot left for 7 in the 2nd diagonal in the lower left puzzle is at (3, 19).
• The only spot left for 5 in the 4th column in the lower right puzzle is at (16, 16).
• The only spot left for 8 in the 5th 3x3 square in the lower right puzzle is at (17, 18).
• The only spot left for 8 in the 2nd diagonal in the lower right puzzle is at (19, 15).

Step 8. There are 23 independent moves.

• The only spot left for 7 in the 5th row in the upper right puzzle is at (13, 5).
• The only spot left for 9 in the 6th row in the upper right puzzle is at (15, 6).
• The only spot left for 1 in the 7th row in the upper right puzzle is at (17, 7).
• The only spot left for 9 in the 1st column in the upper right puzzle is at (13, 2).
• The only spot left for 5 in the 2nd column in the upper right puzzle is at (14, 3).
• The only spot left for 6 in the 5th column in the upper right puzzle is at (17, 9).
• The only spot left for 8 in the 1st 3x3 square in the upper right puzzle is at (14, 1).
• The only spot left for 6 in the 2nd 3x3 square in the upper right puzzle is at (18, 1).
• The only spot left for 5 in the 4th 3x3 square in the upper right puzzle is at (15, 4).
• The only spot left for 9 in the 5th 3x3 square in the upper right puzzle is at (16, 4).
• The only spot left for 7 in the 6th 3x3 square in the upper right puzzle is at (20, 4).
• The only spot left for 1 in the 9th 3x3 square in the upper right puzzle is at (19, 9).
• The only spot left for 9 in the 4th row in the lower left puzzle is at (3, 16).
• The only spot left for 5 in the 4th 3x3 square in the lower left puzzle is at (1, 18).
• The only spot left for 9 in the 6th 3x3 square in the lower left puzzle is at (9, 18).
• The only spot left for 7 in the 6th 3x3 square in the lower left puzzle is at (7, 18).
• The only spot left for 6 in the 3rd row in the lower right puzzle is at (18, 15).
• The only spot left for 6 in the 4th row in the lower right puzzle is at (21, 16).
• The only spot left for 8 in the 2nd 3x3 square in the lower right puzzle is at (18, 13).
• 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 5 in the 9th 3x3 square in the lower right puzzle is at (21, 19).
• The only spot left for 6 in the 1st diagonal in the lower right puzzle is at (19, 19).
• The only spot left for 6 in the 2nd diagonal in the lower right puzzle is at (16, 18).

Step 9. There are 17 independent moves.

• The only spot left for 4 in the 5th column in the upper right puzzle is at (17, 2).
• The only spot left for 7 in the 1st 3x3 square in the upper right puzzle is at (15, 2).
• The only spot left for 9 in the 2nd 3x3 square in the upper right puzzle is at (18, 3).
• The only spot left for 5 in the 2nd 3x3 square in the upper right puzzle is at (16, 2).
• The only spot left for 3 in the 5th 3x3 square in the upper right puzzle is at (17, 4).
• The only spot left for 4 in the 8th 3x3 square in the upper right puzzle is at (18, 9).
• The only spot left for 3 in the 1st diagonal in the upper right puzzle is at (15, 3).
• The only spot left for 8 in the 4th row in the lower left puzzle is at (7, 16).
• The only spot left for 2 in the 6th row in the lower left puzzle is at (3, 18).
• The only spot left for 8 in the 4th 3x3 square in the lower left puzzle is at (3, 17).
• The only spot left for 2 in the 6th 3x3 square in the lower left puzzle is at (7, 17).
• The only spot left for 5 in the 6th column in the lower right puzzle is at (18, 20).
• The only spot left for 9 in the 7th column in the lower right puzzle is at (19, 13).
• The only spot left for 5 in the 2nd 3x3 square in the lower right puzzle is at (17, 13).
• The only spot left for 6 in the 3rd 3x3 square in the lower right puzzle is at (20, 13).
• The only spot left for 9 in the 6th 3x3 square in the lower right puzzle is at (20, 18).
• The only spot left for 6 in the 8th 3x3 square in the lower right puzzle is at (17, 20).

Step 10. There are 4 independent moves.

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