Making any integer with four 2s (2025)

A popular mathematics challenge invites participants to create any given integer using precisely four instances of the digit 2 and a variety of mathematical operations. The puzzle remains engaging across various skill levels, from elementary to advanced. Middle school students, for instance, learn to employ exponentiation and factorials to expand the range of numbers they can generate.

More sophisticated techniques can also be used to achieve the desired outcome. One intriguing example involves the Gamma function, which makes producing the number 7 much simpler. Advanced mathematical skills enable individuals to create an increasing number of integers. Discussion forums even showcase complex number manipulations and repeating fractions as effective methods for achieving this task.

Mathematicians in the 1920s found this pastime particularly entertaining. However, the late physicist Paul Dirac eventually solved the puzzle by developing a general solution applicable to every number. Consequently, the task now requires employing base-2 logarithms. To satisfy the puzzle's requirement of four 2s, one can express 2 as the square root of (2+2).

Although some may view this modification as cheating, it adheres to the puzzle's rules. For example, the number 7 can be represented using four 2s and elementary mathematical operations. This demonstrates that any integer can be expressed in this manner, albeit the challenge lies in accurately executing the square roots. The story originates from Graham Farmelo's book The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius.