## Squares under a hyperbola

### Problem 247

Published on Friday, 29th May 2009, 09:00 pm; Solved by 945Consider the region constrained by 1 ≤ `x` and 0 ≤ `y` ≤ ^{1}/_{x}.

Let S_{1} be the largest square that can fit under the curve.

Let S_{2} 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.

S_{2} has one square to its left and none below, so the index of S_{2} is (1,0).

It can be seen that the index of S_{32} is (1,1) as is the index of S_{50}.

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)?