题目247：双曲线下的正方形

欧拉计划

Squares under a hyperbola

Consider the region constrained by 1 ≤ x and 0 ≤ y ≤ ¹/_x.

Let S₁ be the largest square that can fit under the curve.
Let S₂ be the largest square that fits in the remaining area, and so on.
Let the index of S_n be the pair (left, below) indicating the number of squares to the left of S_n and the number of squares below S_n.

The diagram shows some such squares labelled by number.
S₂ has one square to its left and none below, so the index of S₂ is (1,0).
It can be seen that the index of S₃₂ is (1,1) as is the index of S₅₀.
50 is the largest n for which the index of S_n is (1,1).

What is the largest n for which the index of S_n is (3,3)?

题目：

考虑由 1 ≤ x 和 0 ≤ y ≤ ¹/_x 约束的区域。

设 S₁ 为曲线下能容下的最大正方形。
设 S₂ 为剩下区域能容下的最大正方形,以此类推。
设 S_n 对应的二元组 （左边，下面）表示 S_n 左边的正方形和 S_n下面的正方形。

圆形显示了一些由数字标记的正方形。
S₂ 有一个正方形在其左边，没有方块在其下面，由此 S₂ 对应的二元组为 (1,0)。
不难得到 S₃₂ 对应的二元组为 (1,1) ，与 S₅₀ 对应的二元组相同。
50 为使 S_n 对应二元组为 (1,1) 的最大整数 n。

使得 Sn 对应二元组为 (3,3) 的最大整数 n 是多少？