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? (The figure is representative)

• 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.
\[ \boxed{\text{Colour exactly } c, d, \text{ and } i \text{ to achieve two lines of symmetry (minimum = 3).}} \]