Question

下图是一排 $10$ 个正方形组成的网格。每个正方形都有一条从左下顶点连到右上顶点的对角线。一只虫子沿线段从顶点走到顶点，永不经过同一条线段两次，并且在水平线段或对角线段上绝不从右向左走。设 $N$ 为虫子从左下角 $(A)$ 走到右上角 $(B)$ 的路径数。图中粗线段展示了一条从 $A$ 到 $B$ 的路径。求 $\sqrt{N}$ 。

The figure below shows a grid of $10$ squares in a row. Each square has a diagonal connecting its lower left vertex to its upper right vertex. A bug moves along the line segments from vertex to vertex, never traversing the same segment twice and never moving from right to left along a horizontal or diagonal segment. Let $N$ be the number of paths the bug can take from the lower left corner $(A)$ to the upper right corner $(B).$ One such path from $A$ to $B$ is shown by the thick line segments in the figure. Find $\sqrt{N}.$

答案

答案：243

知识点：格路乘法原理

难度评级：2230

解答：

令 $A = (0, 0)$ ， $B = (10, 1)$ 。每次水平或对角移动都向右，所以虫子的 $x$ 坐标从不减小；它会恰好一次穿过每个 $10$ 个竖条，并在相应正方形的三条向右线段中选一条：底边、顶边、或对角线。

这十个选择决定整条路径。每次穿越会以确定的高度到达（底边到低处，顶边或对角线到高处），也以确定的高度离开（底边或对角线从低处，顶边从高处），所以在每条竖直线上，虫子恰好在到达高度和离开高度不同时走那条竖直线段；每条竖直线段至多需要一次，因此不会重复线段。两端同理：虫子从 $A$ 的低处开始，到 $B$ 的高处结束，必要时使用端点处的竖直线段。反过来，每串选择都给出一条合法路径。

因此 $N = 3^{10}$ ，且 $\sqrt{N} = 3^5 = 243$ 。

Put $A = (0, 0)$ and $B = (10, 1).$ Every horizontal and diagonal move goes rightward, so the bug's $x$ -coordinate never decreases, and it crosses each of the $10$ vertical strips exactly once, using exactly one of that square's three rightward segments: the bottom edge, the top edge, or the diagonal.

These ten choices determine the whole path. Each crossing arrives at a definite height (bottom edge: low; top edge or diagonal: high) and departs at a definite height (bottom edge or diagonal: low; top edge: high), so at each vertical line the bug traverses the vertical segment exactly when the arrival and departure heights differ — and each vertical segment is needed at most once, so no segment repeats. The same applies at the ends: the bug starts low at $A$ and finishes high at $B,$ using the end verticals if necessary. Conversely, every sequence of choices yields a valid path.

Therefore $N = 3^{10},$ and $\sqrt{N} = 3^5 = 243.$

2026 AIME II 第 2 题

答案：243

难度评级：2230

解答：

其他年份的第 2 题