Fermat's margin, and a lotus in a lake

Baudhāyana triples have a long afterlife. In the 17th century they set the French mathematician Fermat thinking.

There are infinitely many squares that are the sum of two squares. So, Fermat asked, is there a cube that is the sum of two cubes? A fourth power that is the sum of two fourth powers? In symbols: does xⁿ + yⁿ = zⁿ have a solution in natural numbers when n > 2?

In the margin of a book about number patterns, Fermat wrote that it doesn't: not one cube, not one fourth power, not ever. And then he added the most famous sentence in mathematics: 'I have found a truly marvellous proof of this statement, but the margin is too small to contain it.'

No one ever found that proof. The claim became known as Fermat's Last Theorem, and after his death the greatest mathematicians threw themselves at it. More than three hundred years of failure followed.

Then, in 1963, a ten-year-old boy named Andrew Wiles read Eric Bell's book The Last Problem, about the theorem and everyone who had failed at it. Knowing all that, he decided he would prove it. He did, in 1994.

Now a much older problem, from Bhāskarāchārya's Līlāvatī, and one you can solve today: 'In a lake surrounded by chakra and krauñcha birds, there is a lotus flower peeping out of the water, with the tip of its stem 1 unit above the water. On being swayed by a gentle breeze, the tip touches the water 3 units away from its original position. Quickly tell the depth of the lake.'

It looks as if there isn't enough information. There is. Let x be the depth, so the stem is x + 1 long. When the breeze lays it over, the stem becomes the hypotenuse of a right triangle with legs 3 and x. By the theorem, 3² + x² = (x + 1)². Expand: 9 + x² = x² + 2x + 1. The x² cancels, leaving 9 = 2x + 1, so x = 4. The lake is 4 units deep.