Completing a symmetric figure on a grid

The grid shows some shaded squares and a dashed line. The dashed line is a line of symmetry of the finished figure. Shade the least number of squares so that the figure is symmetric about the dashed line. Try it first. Pause if you like. There are shaded squares on both sides. Take them one at a time. Shade only for squares that have no partner yet. Start with the shaded square in the first row, third column. The dashed diagonal crosses the square just to its left. Move one square left along the first row, to the square crossed by the dashed line. From that square, move one square straight down. We reach the matching shaded square in the second row. The square in the first row and its partner in the second row are both shaded already. We need no extra shading for this pair. The path is one square left, then one square down. The row count and column count match. Now fold the purple tracing square across the dashed diagonal. It fits exactly onto the partner we found. In the second row, move one square left from the other shaded square to the dashed line. Then move one square down and shade its partner. From the left shaded square in the third row, move two squares up to the dashed line. Then move two squares right and shade its partner. The first part of this path goes up the column. The second goes right along the row. Both parts are two squares long. From the shaded square on the right in the third row, move one square left to the dashed line. Then move one square down and shade its partner. Every shaded square now has a partner the same distance on the other side of the line. We shaded no square that did not have to be shaded. The least number of squares to shade is one plus one plus one, which equals three. The completed grid is the answer. Checking the existing pair first stopped us shading more squares than necessary.

The grid shows half of a symmetric figure. The dashed line is its line of symmetry. Shade six more squares to make it symmetric about the dashed line. Try it first. Pause if you like. This line runs across the grid. Will the square at the very top match a square nearest the line, or at the very bottom? First column: the shaded square is three squares above the line. Count three squares below in the same column and shade that square. Second column: the higher shaded square is three squares above the line. Shade the square three below in that column. Second column: the shaded square under it is two squares above the line. Shade the square two below in that column. Third column: the higher shaded square is two squares above the line. Shade the square two below in that column. Third column: the shaded square under it is one square above the line. Shade the square one below in that column. Fourth column: the shaded square is one square above the line. Shade the square one below in that column. The number of squares to shade is one plus one plus one plus one plus one plus one, which equals six. Count to the line, then match on the other side. A fold checks that the squares fit exactly.