2018 AMC 10A Problems/Problem 20: Difference between revisions
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<math>\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}</math> | <math>\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}</math> | ||
==Solution== | |||
Draw a <math>7 \times 7</math> square. | |||
<math> \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} </math> | |||
There are two choices for each letter, for a total of <math>2^{10} = 1024</math> codes. Two codes must be subtracted for an answer of <math>\fbox{\textbf{(B)} \text{ 1022}}</math> | |||
~Nosysnow | |||
== See Also == | |||
{{AMC10 box|year=2018|ab=A|num-b=2|num-a=4}} | |||
{{MAA Notice}} | |||
Revision as of 15:44, 8 February 2018
A scanning code consists of a 
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 
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 
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?
Solution
Draw a square.
There are two choices for each letter, for a total of 
codes. Two codes must be subtracted for an answer of 
~Nosysnow
See Also
| 2018 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 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 | ||
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