The answer is Ï€^2/6: What’s the question?

Posted by: Gary Ernest Davis on: May 11, 2011

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The Basel problem

Pietro Mengoli

Leonhard Euler

One question that has $\frac{\pi^2}{6}$ as an answer was posed by Pietro Mengoli in 1644:

“What is the value of $1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\ldots$ ?”

This became known as the Basel Problem, and Leonhard Euler solved it and announced his solution in 1735 when he as 28 years old.

Euler showed – at first not entirely rigorously – that

$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\ldots =\frac{\pi^2}{6}$

Visible lattice points

A lattice point in the plane is a point with integer coordinates:

We say a lattice point $(m,n)$ is visible from the origin $(0,0)$ when the line segment joining $(0,0)$ to $(m,n)$ contains no other lattice points.

A question that has $\frac{\pi^2}{6}$ as an answer is:” What is the number of times we expect to choose a random lattice point until we get a point that is visible from the origin ?”

The reason for this is that the probability that a randomly chosen point is visible from the origin is $\frac{6}{\pi^2}$ , and so choosing a random lattice point is like flipping a weighted coin that comes up “VISIBLE” $\frac{6}{\pi^2}$ of the flips and “NOT VISIBLE $1-\frac{6}{\pi^2}$ of the flips.Â The waiting time until a visible point is therefore given by a geometric distribution and is just the reciprocal of the probability of choosing a visible point.

Co-prime integers

The lattice points $(m,n)$ that are visible from the origin are exactly those with $m \textrm{ and } n$ having no common factors (that is, being co-prime).

So $\frac{\pi^2}{6}$ is also the waiting time until two integers chosen at random are co-prime.

Why this type of question?

The reason for asking what is the question, given a result, is that such questions stimulate one to find creative answers, an answer being a question that leads to the desired result.

A teacher, for example, can see relatively easily how much relevant mathematics a student knows by asking a question of this type.

Here’s a simple yet revealing question to ask people at all levels of mathematical attainment: “The answer is 10. What is the question?”

Try it on a few people, preferably in groups: the answers may amaze you.

I can guarantee people will try, and you and they will be amused by the different answers given.

Tags: Basel problem, Euler, Mengoli

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