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Problem 247

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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)?

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双曲线下的正方形

考虑由1 ≤ x和0 ≤ y ≤ ¹/_x所表示的区域。

记S₁是能够放进曲线下区域的最大正方形。
记S₂是能够放进剩余区域的最大正方形，并依此类推。
S_n的索引是一个数对(left, below)，分别表示S_n左侧的正方形数目和S_n下放的正方形数目。

上图展示了一些这样的正方形，并标上了相应的数字顺序。
S₂的左侧有一个正方形，下方没有正方形，因此S₂的索引为(1,0)。
可以看出S₃₂的索引和S₅₀的索引同为(1,1)。
使得S_n索引为(1,1)的数n最大是50。

使得S_n索引为(3,3)的数n最大是多少？

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