AMC10 2018 A

AMC10 2018 A · Q20

AMC10 2018 A · Q20. It mainly tests Basic counting (rules of product/sum), Counting with symmetry / Burnside (rare).

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° 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?

一个扫描码由 $7 \times 7$ 的方格网格组成，其中一些方格涂黑，其余涂白。在这 49 个方格中必须至少有一种颜色的方格。扫描码被称为对称的，如果整个方形绕中心逆时针旋转 90° 的倍数时外观不变，也不改变当它反射穿过连接对角线的线或连接对边中点的线时。何种对称扫描码的总数是多少？

(A) 510 510

(B) 1022 1022

(D) 8192 8192

(E) 65,534 65534

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): None of the squares that are marked with dots in the sample scanning code shown below can be mapped to any other marked square by reflections or non-identity rotations. Therefore these 10 squares can be arbitrarily colored black or white in a symmetric scanning code, with the exception of “all black” and “all white”. On the other hand, reflections or rotations will map these squares to all the other squares in the scanning code, so once these 10 colors are specified, the symmetric scanning code is completely determined. Thus there are $2^{10}-2=1022$ symmetric scanning codes.

答案（B）：在下面给出的示例扫描码中，用点标记的那些方格在任何反射或非恒等旋转下都不会被映射到另一个被标记的方格。因此，在一个对称的扫描码中，这 10 个方格可以被任意涂成黑色或白色，但“全黑”和“全白”两种情况除外。另一方面，反射或旋转会把这些方格映射到扫描码中的所有其他方格，因此一旦这 10 个颜色被指定，对称扫描码就完全确定了。因此，对称扫描码共有 $2^{10}-2=1022$ 种。

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