Question

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 figure below consists of 20 unit squares arranged as shown. In addition to the given black squares, up to 5 more may be coloured black. Which one among the following options depicts the minimum number of boxes that must be coloured black to achieve two lines of symmetry?

• A. d
• B. c, d, i
• C. c, i
• D. c, d, i, f, g

Hint:

When asked for two lines of symmetry on a rectangular grid, first target the natural vertical (between middle columns) and horizontal (through the middle row) axes, then fix only the mismatched mirror pairs.

Answer 1

The Correct Option is B

Step 1: Identify the intended symmetry axes.
For a $4 \times 5$ grid, the two natural symmetry lines are: (i) a vertical line between columns 2 and 3, and (ii) a horizontal line through the middle row (row 3).

Step 2: Check which existing blacks already satisfy symmetry.
- Horizontal symmetry pairs: $(r1,c2)\leftrightarrow(r5,c2)$, $(r1,c3)\leftrightarrow(r5,c3)$, $(r2,c1)\leftrightarrow(r4,c1)$, $(r2,c4)\leftrightarrow(r4,c4)$ are already black–black and hence fine. - The mismatch for horizontal symmetry occurs at $(r4,c3)$ which is black; its mirror across row 3 is $(r2,c3)=d$ (currently white). Thus d must be coloured black.

Step 3: Enforce vertical symmetry (mirror across columns 2 and 3).
- Cell $(r4,c3)$ is black; its vertical mirror is $(r4,c2)=i$ (white). Hence i must be coloured black. - With d black (from Step 2), its vertical mirror is $(r2,c2)=c$; therefore c must also be black for vertical symmetry.

Step 4: Minimality check.
Colouring c, d, and i makes all horizontal and vertical mirror pairs match. No other cells are required; choosing fewer (e.g., only c,i) fails the horizontal pair $(r2,c3)\leftrightarrow(r4,c3)$, and any extra cells (e.g., f,g) are unnecessary.
\[ \boxed{\text{Colour exactly } c, d, \text{ and } i \text{ to achieve two lines of symmetry (minimum = 3).}} \]