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Difficult Unformalized Group Puzzle
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The small superscript in each undetermined location is the step number at which the location must be fill before all of the locations with subsequent step numbers can be determined. A detailed description of the moves in each step is given below.

Display the puzzle with superscripts showing just the step number for the first few steps in their empty grid locations.

See the puzzle answer grid.

Deductive steps to solve the puzzle

The moves within each step may be done in any order.

All of the moves within a particular step must be done before all of the moves in the subsequent step can be guaranteed to be completed.

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 .

Step 1. There are 17 independent moves.

• The only spot left for L in the 2nd column is at (2, 6).
• The only spot left for L in the 10th column is at (10, 7).
• The only spot left for Z in the 2nd 4x3 rectangle is at (8, 2).
• The only spot left for A in the 3rd 4x3 rectangle is at (12, 2).
• The only spot left for U in the 5th 4x3 rectangle is at (7, 6).
• The only spot left for L in the 8th 4x3 rectangle is at (7, 9).
• The only spot left for A in the 8th 4x3 rectangle is at (8, 7).
• The only spot left for L in the 10th 4x3 rectangle is at (1, 10).
• The only spot left for F in the 11th 4x3 rectangle is at (7, 11).
• The only spot left for A in the 11th 4x3 rectangle is at (6, 10).
• The only spot left for D in the 1st eye is at (5, 5).
• The only spot left for M in the 1st eye is at (3, 2).
• The only spot left for Z in the 2nd eye is at (11, 5).
• The only spot left for E in the 2nd eye is at (10, 5).
• The only spot left for M in the 2nd eye is at (11, 4).
• The only spot left for L in the 2nd eye is at (9, 3).
• The only spot left for Z in the elements that make up the smile is at (9, 11).

Step 2. There are 7 independent moves.

• The only spot left for E in the 10th row is at (3, 10).
• The only spot left for M in the 4th 4x3 rectangle is at (4, 6).
• The only spot left for I in the 6th 4x3 rectangle is at (10, 6).
• The only spot left for M in the 9th 4x3 rectangle is at (9, 7).
• The only spot left for A in the 12th 4x3 rectangle is at (11, 11).
• The only spot left for I in the 2nd eye is at (8, 4).
• The only spot left for N in the elements that make up the smile is at (4, 11).

Step 3. There are 6 independent moves.

• The only spot left for E in the 11th row is at (12, 11).
• The only spot left for F in the 10th column is at (10, 8).
• The only spot left for N in the 10th column is at (10, 2).
• The only spot left for N in the 4th 4x3 rectangle is at (3, 5).
• The only spot left for N in the 12th 4x3 rectangle is at (9, 12).
• The only spot left for F in the 2nd eye is at (8, 5).

Step 4. There are 6 independent moves.

• The only spot left for N in the 5th column is at (5, 9).
• The only spot left for N in the 5th 4x3 rectangle is at (7, 4).
• The only spot left for F in the 12th 4x3 rectangle is at (12, 12).
• The only spot left for U in the 1st eye is at (2, 5).
• The only spot left for F in the 1st eye is at (5, 3).
• The only spot left for O in the 2nd eye is at (9, 2).

Step 5. There are 6 independent moves.

• The only spot left for E in the 5th column is at (5, 12).
• The only spot left for N in the 2nd 4x3 rectangle is at (6, 3).
• The only spot left for R in the 5th 4x3 rectangle is at (6, 5).
• The only spot left for O in the 6th 4x3 rectangle is at (12, 6).
• The only spot left for F in the 8th 4x3 rectangle is at (6, 9).
• The only spot left for R in the 2nd eye is at (11, 3).

Step 6. There are 7 independent moves.

• The only spot left for O in the 5th row is at (1, 5).
• The only spot left for I in the 9th row is at (2, 9).
• The only spot left for I in the 5th column is at (5, 7).
• The only spot left for O in the 2nd 4x3 rectangle is at (7, 3).
• The only spot left for E in the 2nd 4x3 rectangle is at (6, 2).
• The only spot left for O in the 5th 4x3 rectangle is at (5, 4).
• The only spot left for R in the 1st eye is at (2, 4).

Step 7. There are 13 independent moves.

• The only spot left for I in the 3rd row is at (1, 3).
• The only spot left for O in the 7th row is at (3, 7).
• The only spot left for D in the 9th row is at (4, 9).
• The only spot left for O in the 10th row is at (10, 10).
• The only spot left for A in the 2nd column is at (2, 12).
• The only spot left for E in the 2nd column is at (2, 8).
• The only spot left for I in the 6th column is at (6, 12).
• The only spot left for R in the 1st 4x3 rectangle is at (1, 2).
• The only spot left for I in the 2nd 4x3 rectangle is at (7, 2).
• The only spot left for A in the 4th 4x3 rectangle is at (4, 4).
• The only spot left for R in the 6th 4x3 rectangle is at (9, 6).
• The only spot left for I in the 9th 4x3 rectangle is at (11, 8).
• The only spot left for E in the 1st eye is at (4, 3).

Step 8. There are 16 independent moves.

• The only spot left for F in the 6th row is at (1, 6).
• The only spot left for Z in the 9th row is at (12, 9).
• The only spot left for R in the 10th row is at (7, 10).
• The only spot left for D in the 12th row is at (7, 12).
• The only spot left for D in the 1st column is at (1, 11).
• The only spot left for O in the 6th column is at (6, 8).
• The only spot left for D in the 11th column is at (11, 10).
• The only spot left for D in the 1st 4x3 rectangle is at (2, 2).
• The only spot left for E in the 4th 4x3 rectangle is at (1, 4).
• The only spot left for F in the 6th 4x3 rectangle is at (9, 4).
• The only spot left for Z in the 7th 4x3 rectangle is at (4, 7).
• The only spot left for D in the 9th 4x3 rectangle is at (9, 8).
• The only spot left for O in the 10th 4x3 rectangle is at (2, 11).
• The only spot left for R in the 12th 4x3 rectangle is at (10, 11).
• The only spot left for A in the 1st eye is at (3, 3).
• The only spot left for O in the elements that make up the smile is at (8, 12).

Step 9. There are 4 independent moves.

• The only spot left for R in the 9th 4x3 rectangle is at (12, 7).
• The only spot left for U in the 9th 4x3 rectangle is at (11, 7).
• The only spot left for U in the 12th 4x3 rectangle is at (9, 10).
• The only spot left for R in the elements that make up the smile is at (3, 9).

Step 10. There are 5 independent moves.

• The only spot left for U in the 9th row is at (8, 9).
• The only spot left for U in the 3rd column is at (3, 12).
• The only spot left for U in the 7th 4x3 rectangle is at (4, 8).
• The only spot left for R in the 8th 4x3 rectangle is at (8, 8).
• The only spot left for R in the 10th 4x3 rectangle is at (4, 12).

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