Panel I below shows a grid of cells with four rows and four columns. The numbers on the top of each column and on the left of each row tell us how many cells in that column or row must be shaded. The column totals, left to right, are 2, 2, 2, 2, and the row totals, top to bottom, are 3, 1, 2, 2. Panel II below shows four candidate grids, labelled (i) to (iv), with some cells shaded grey.
Which one of the options shown in Panel II represents the grid shaded correctly?
Show Hint
Add up the shaded cells in each row and each column separately, and match both sets of totals against the numbers given in Panel I.
Panel I fixes how many cells must be shaded in each column, 2, 2, 2, 2 from left to right, and each row, 3, 1, 2, 2 from top to bottom. Since every column needs the same total of 2, checking columns first is a quick filter, and then the row totals confirm which single grid is the real answer.
(i): Every column in this grid has exactly 2 shaded cells, so it passes the column check. But its row totals come out as 3, 2, 1, 2, and the puzzle needs 3, 1, 2, 2. Rows 2 and 3 have swapped their shading counts, so this grid is wrong.
(ii): Every column again has exactly 2 shaded cells. Checking the rows gives 3, 1, 2, 2, in that exact order, which matches Panel I perfectly.
(iii): The columns here do not all come out to 2, one column ends up with 3 and another with only 1, so this grid already fails the column check before the rows are even considered.
(iv): The row totals for this grid are 3, 1, 1, 3, which does not match the required 3, 1, 2, 2, so it fails on the rows even though some columns may look right.
Only option (ii) keeps every column at 2 shaded cells and lines up the rows as 3, 1, 2, 2 in the correct order.
Let's summarize:
A quick column check, all columns must equal 2, rules out (iii) immediately.
A row check, 3, 1, 2, 2 in order, then separates the correct grid from (i) and (iv), which get the right total shaded cells but in the wrong rows.
