2018 AMC 10A Problems/Problem 20

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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 [i]symmetric[/i] 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?

$\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}$

Solution

Draw a $7 \times 7$ square.

$\begin{tabular}{|c|c|c|c|c|c|c|} \hline a & b & c & d & c & b & a \\ \hline b & e & f & g & f & e & b \\ \hline c & f & h & i & h & f & c \\ \hline d & g & i & j & i & g & d \\ \hline c & f & h & i & h & f & c \\ \hline b & e & f & g & f & e & b \\ \hline a & b & c & d & c & b & a \\ \hline \end{tabular}$

There are two choices for each letter, for a total of $2^{10} = 1024$ codes. Two codes must be subtracted for an answer of $\fbox{\textbf{(B)} \text{ 1022}}$

     ~Nosysnow

See Also

2018 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 19
Followed by
Problem 21
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions
2018 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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