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Problem 280

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Ant and seeds

A laborious ant walks randomly on a 5x5 grid. The walk starts from the central square. At each step, the ant moves to an adjacent square at random, without leaving the grid; thus there are 2, 3 or 4 possible moves at each step depending on the ant’s position.

At the start of the walk, a seed is placed on each square of the lower row. When the ant isn’t carrying a seed and reaches a square of the lower row containing a seed, it will start to carry the seed. The ant will drop the seed on the first empty square of the upper row it eventually reaches.

What’s the expected number of steps until all seeds have been dropped in the top row?
Give your answer rounded to 6 decimal places.

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蚂蚁和种子

一只勤劳的蚂蚁在5x5的网格上从中心方格开始随机行走。每一步它都随机移动到相邻的一格，但不会离开网格；因此，根据它所在的位置，它每一步有2、3或4种不同的选择。

在开始前，网格的最下一行的每一个格子上都摆有一粒种子。如果蚂蚁到达最下一行的格子，此时格子上有种子而蚂蚁并没有携带种子的话，蚂蚁将会拿起种子。它将在之后第一次到达最上一行没有种子的格子时放下种子。

所有的种子都被放到最上一行所需的期望步数是多少？
将你的答案四舍五入到六位小数。

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