A mathematical challenge.

In 1929 in Göttingen, a challenge to express any whole number using the number 2 precisely four times, and using only well-known mathematical symbols, was introduced.

The first few numbers are easy:

1 = (2 + 2)/(2 + 2),

2 = (2/2) + (2/2),

3 = (2 x 2) - (2/2),

4 = 2 + 2 + 2 - 2.

The game became much more difficult even for Göttingen’s finest mathematical minds. Hundreds of hours were spent playing the game with higher and higher numbers - until Paul Dirac found a simple and general formula enabling *any* number to be expressed using four 2s, entirely within the rules. He had rendered the game pointless.

Dirac’s solution relies on a basic property of logarithms:

and is

where the number of radicals is exactly *n *square roots.

One may think that Dirac killed the game using only three 2s. Each symbol in the formula is very common in mathematics, so Dirac’s solution is still within the rules of the game.

Content inspired by *The Strangest Man *by Graham Farmelo