Question

平面上的一个物体从一个格点移动到另一个格点。每一步，物体可以向右、向左、向上或向下移动一个单位。如果物体从原点出发，走一条十步路径，那么终点可能有多少个不同的位置？

An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point?

$120$

$121$

$221$

$230$

$231$

答案：B

知识点：格点奇偶性基本计数

难度评级：1820

解答：

每一步都会使坐标和改变 $1$ ，所以 $10$ 步后终点 $(a, b)$ 满足 $a + b$ 为偶数，且 $|a| + |b| \le 10$ 。任何这样的点都可到达：先走 $|a| + |b|$ 步到达它，再用剩下的偶数步往返。

可到达的点落在直线 $a + b = 2k$ 上，其中 $-5 \le k \le 5$ 。每条这样的直线与该菱形恰有 $11$ 个格点。

共有 $11$ 条直线，每条 $11$ 个点，因此共有 $121$ 个点。

因此，正确答案是 B。

Each step changes the coordinate sum by $1,$ so after $10$ steps the endpoint $(a, b)$ has $a + b$ even, and $|a| + |b| \le 10.$ Any such point is reachable: walk $|a| + |b|$ steps to it, then use the remaining even number of steps going out and back.

The reachable points lie on the lines $a + b = 2k$ for $-5 \le k \le 5.$ Each such line meets the diamond in exactly $11$ lattice points.

With $11$ lines and $11$ points each, there are $121$ points.

Thus, the correct answer is B.

2006 AMC 12B 第 18 题

答案：B

难度评级：1820

解答：

其他年份的第 18 题