Discussion
A line of symmetry is defined as a line that divides a figure into two parts in a way such that each part is a mirror image of the other part about that line.
The given figure consists of 16 unit squares arranged as shown. In addition to the three black squares, what is the minimum number of squares that must be coloured black, such that both PQ and MN form lines of symmetry? (The figure is representative)
Answer: C
The minimum number of extra black squares is 5.
Number columns 1 to 4 from the left and rows 1 to 4 from the top. PQ is the vertical centre line, so it maps column c to column 5 − c. MN is the diagonal from the bottom-left corner to the top-right corner, so it swaps a square’s column with its height above the bottom edge. For both lines to be lines of symmetry, the black set must be unchanged by both reflections, and therefore by every symmetry of the square they generate.
The given black squares are (row 1, column 3), (row 2, column 1) and (row 4, column 3). Applying the two reflections repeatedly to any one of them gives the same family of eight squares: the two middle squares of the top row, the two middle squares of the bottom row, and the two middle squares of the left and right columns. All three given squares lie in this family, so the black set must be all eight of them, and 8 − 3 = 5 more squares are needed.
Three or four squares make PQ a symmetry line but leave some square without its mirror image in MN; six is more than necessary.
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