2020 AMC 12A Problem 6

Question

In the plane figure shown below,

3

of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?

4

5

6

7

8

Answer: D

Concepts:symmetry

Difficulty rating: 1270

Small hint:

The two symmetry lines must be the central vertical and central horizontal lines of the rectangle

Big hint:

Each shaded square forces its reflections across both lines to be shaded too

Solution:

For both symmetries, the lines must be the vertical and horizontal center lines of the

5

-by-

4

grid. Every shaded square then forces the squares obtained by reflecting it across each line.

The top square lies off-center, so its reflection group has

4

squares, requiring

3

more. The middle square sits on the central column, so its group has

2

squares, requiring

1

more. The bottom-right square again has a group of

4,

requiring

3

more.

The least number of additional squares is

3 + 1 + 3 = 7.

Thus, D is the correct answer.