The figure in Panel I below is a grid of cells with four rows and four columns. The numbers on the top and on the left represent the number of cells that are to be shaded in that column and row, respectively. Which one of the options shown in Panel II below represents the grid shaded correctly?
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Count shaded cells in each row and column of every option and match them against the given totals.
Step 1: Try to Build the Grid Directly Instead of Testing Options: Rather than checking each candidate grid, we can try to work out directly which cells must be shaded, using the row total of 1 in row 2 as the strongest starting clue, since a single shaded cell in a row of 4 is the most restrictive condition to pin down. Step 2: Use the Row 2 Clue to Anchor a Column: Row 2 needs exactly 1 shaded cell. Placing it in column 2 means column 2 already has 1 of its required 2 shaded cells accounted for from row 2, and the remaining shaded cell in column 2 must come from one of rows 1, 3 or 4. Step 3: Fill Row 1, Row 3 and Row 4, and Check Consistency: Row 1 needs 3 shaded cells out of 4, meaning only one column in row 1 is left unshaded; leaving column 2 unshaded in row 1 gives row 1 shaded at columns 1, 3 and 4, matching its total of 3. Distributing the remaining shaded cells of rows 3 and 4, each needing 2, so that every column still reaches a total of exactly 2, gives row 3 shaded at columns 3 and 4, and row 4 shaded at columns 1 and 2. Adding up each column then gives column 1 equal to 2, from rows 1 and 4, column 2 equal to 2, from rows 2 and 4, column 3 equal to 2, from rows 1 and 3, and column 4 equal to 2, from rows 1 and 3, so every column comes out to exactly 2 as required. Step 4: Match to the Given Options and Final Answer: This directly constructed shading pattern, row 1 shaded at columns 1, 3, 4, row 2 shaded at column 2, row 3 shaded at columns 3, 4, row 4 shaded at columns 1, 2, is exactly the pattern shown in option (ii) of Panel II, confirming the same answer found by the direct row and column counting method. \[ oxed{ ext{Option (ii)}} \]
