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 basic counting

Difficulty rating: 1800

Solution:

Under the symmetry group of the square, the $49$ cells break into orbits, and every cell in an orbit must have the same color. Give a cell coordinates $(i,j)$ relative to the center, where $-3\le i,j\le3.$ Rotations and reflections can change signs and interchange the coordinates, so each orbit has one representative with $0\le i\le j\le3.$ There are $1+2+3+4=10$ such pairs. Each orbit is black or white, giving $2^{10}$ colorings, but the all-black and all-white grids are excluded. So there are $2^{10} - 2 = 1022$ symmetric scanning codes.

Thus, the correct answer is B.

2018 AMC 12A Problem 15

Answer: B

Difficulty rating: 1800

Solution:

Problem 15 in Other Years