A classic recreational math puzzle challenges people to create any target number using exactly four instances of the digit 2 and any mathematical operations—no other digits allowed.
According to the source, the puzzle appeals to mathematicians across skill levels. Elementary school students can solve simple cases using basic arithmetic. Middle school students equipped with knowledge of exponents and factorials can reach a wider range of numbers. More advanced mathematicians employ techniques like the Gamma function, integrals, repeating fractions, and combinatorial operators to tackle tougher targets.
The number 7 is particularly notorious for its difficulty. While challenging with elementary methods, it becomes manageable with tools like the Gamma function or complex numbers.
This puzzle appears to have been popular among mathematicians in the 1920s—until Paul Dirac discovered a general solution that works for any natural number. Dirac’s approach centers on nested square roots combined with base-2 logarithms.
The general formula uses three instances of the digit 2, creating a small technical issue. However, since 2 equals 2^1, any single 2 can be replaced with 2^(2/2), introducing a fourth 2 while preserving the solution’s validity. This modification aligns with the puzzle’s rules.
The elegance of Dirac’s solution lies in its universality: it generates any target number through a systematic application of repeated square roots and logarithmic operations. The only practical challenge is accurately drawing all the nested square root symbols required for larger numbers.
According to the source, this historical account comes from Graham Farmelo’s biography of Dirac, “The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius.”
Key facts
- The four-2s puzzle asks mathematicians to create any target number using exactly four instances of the digit 2 and mathematical operations
- Paul Dirac discovered a general solution in the 1920s using nested square roots and base-2 logarithms
- Dirac’s formula technically uses three 2s, but a fourth can be added by expressing 2 as 2^(2/2) without breaking the puzzle rules
- The puzzle appeals to mathematicians at different levels, from elementary school students to university graduates