Question

在一个符号金字塔中，如果某格下面两个格子的符号相同，则该格填 “+”；如果下面两个格子的符号不同，则该格填 “-”。下图展示了一个四层符号金字塔。有多少种方法可以填最底行的四个格子，使得金字塔顶端为 “+”？

In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?

$2$

$4$

$8$

$12$

$16$

答案：C

知识点：奇偶性乘法原理

难度评级：1310

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文字解答：

设有两个相邻的下方格子和它们上方的格子。如果已知左下格和上方格，就总能确定右下格：

如果上方格是 $+$ ，则右下格必须与左下格相同；如果上方格是 $-$ ，则右下格必须与左下格相反。

现在假设已知某一行。只要选择这一行最左格左下方的那个格子的符号，就能依次确定下一整行。

因此，顶行已知为 $+$ 时，下一行有 $2$ 种选择；同理，第三行有 $2$ 种选择，第四行也有 $2$ 种选择。底行共有 $2\cdot2\cdot2=8$ 种可能。

所以正确答案是 C。

Suppose we have two cells and the cell above them. If we are given the bottom left cell and the top cell, we can always find the bottom right cell as follows:

If the top cell is $+,$ then the bottom right cell must be the same as the bottom left cell, and if the top cell is $-,$ the bottom right cell must be the opposite of the bottom left cell.

Now, suppose we are given a row. Then, suppose we choose a value for the cell below and to the left of the leftmost cell in our given row. We then can inductively determine the entire row below our given by first finding the bottom-right cell of the leftmost cell in our row, and using that newly found cell as the bottom-left reference for the second to the left cell in the given row to find its bottom-right counterpart. The process continues on until the row below the given row is fully solved.

Therefore, since we know that the top row has a cell labelled $+,$ we have $2$ choices for the row below, depending on our choice of the bottom-left cell. Similarly, we have $2$ choices for the third row, and thus $2$ choices for the fourth row. This makes $2\cdot2\cdot2=8$ total choices for the bottom row of the sign pyramid.

Thus, the correct answer is C.

2018 AMC 8 第 19 题

答案：C

难度评级：1310

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文字解答：

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