Question

A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares.

A scanning code is called symmetric if its look does not change when the entire square is rotated by a multiple of $90^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides.

What is the total number of possible symmetric scanning codes?

$510$

$1022$

$8190$

$8192$

$65,534$

Answer: B

Concepts:symmetry multiplication principle

Difficulty rating: 1970

Solution:

Number rows and columns from $-3$ through $3,$ with the center at $(0,0).$ Rotations and reflections can change signs and exchange coordinates, so the orbit of $(x,y)$ is determined by the ordered pair obtained by sorting $(|x|,|y|).$ These pairs are $(u,v)$ with $0\le u\le v\le3,$ of which there are $4+3+2+1=10.$

Once one square in each orbit is colored, symmetry forces the colors of all other squares in that orbit.

There are therefore $2^{10}$ symmetric colorings before the condition about using both colors. The all-black and all-white colorings are not allowed.

The total number of valid symmetric scanning codes is $2^{10}-2=1022$ . Thus, B is the correct answer.

2018 AMC 10A Problem 20

Answer: B

Difficulty rating: 1970

Solution:

Problem 20 in Other Years