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Making any integer with four 2s (2025)
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- DeepHorus
A recreational math puzzle asks participants to express any natural number using exactly four 2s and any operations. The puzzle has challenged people from elementary school to university. Physicist Paul Dirac found a general solution that works for every number, using nested square roots and logarithms.
In the 1920s, mathematicians enjoyed this puzzle as a pastime. Dirac's general solution, however, effectively ended the challenge. The method is described in Graham Farmelo's book "The Strangest Man." This book details Dirac's life and work.
Dirac's approach uses repeated square roots and base-2 logarithms. For a number n, it takes the square root of 2 n times, then applies log base 2. The original formula uses three 2s. By replacing a 2 with β(2Γ2), which equals 2 using two 2s, the expression uses exactly four 2s.
The number 7 is notoriously difficult to express with four 2s using basic operations. Dirac's method makes it easy and uses only elementary operations. This shows any natural number can be generated with four 2s. Dirac's formula provides a universal recipe for any natural number.
Some may claim Dirac's solution is cheating, but it relies on standard mathematical tools. Others admire the elegance of the general formula. The solution has sparked debate about what counts as a valid operation. The puzzle still holds educational value for teaching logarithms and exponents.
The four-2s puzzle remains a fun way to engage students with mathematical creativity. Dirac's solution, though a spoiler, highlights the power of mathematical thinking. It continues to appear in recreational math circles and classrooms. It remains a popular challenge.
