2018 AMC 10A Problem 20

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

Classify grid squares by symmetry orbits

Choose one color per orbit, excluding the two constant colorings

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.

Problem 20 in other years