The four 2s puzzle is one of those rare problems that meets you where you are. An eight-year-old can make 4 from 2+2+2−2. A college student can deploy the Gamma function to crack 7. A working mathematician can rope in complex numbers and repeating fractions. The puzzle is a perfect difficulty gradient — a staircase where every step rewards a new piece of mathematical vocabulary. What makes this blog post sing is its pacing. Eli Bendersky walks the reader up that staircase deliberately, starting with addition and subtraction, layering in exponents and factorials, then introducing the tricks — treating 22 as a valid use of two 2s, deploying the Gamma function, even pulling in complex numbers via i = √(−1). Each level feels like unlocking a new tool in a video game, and the examples are chosen to maximize that feeling of expanding capability. Then Dirac arrives and flattens the whole enterprise. His general solution — nested square roots combined with base-2 logarithms — is the kind of result that makes you laugh out loud. The core insight is almost embarrassingly elegant: applying a square root n times to 2 yields 2^(1/2^n), and then log₂ peels the exponent back out. Stack enough of these and you can extract any natural number from exactly three 2s. The fourth 2 is smuggled in via the identity 2 = 2/√(2·2), which is technically legal and aesthetically outrageous. The post correctly identifies the tension: is this cheating? Bendersky argues it's within the rules, and he's right — the puzzle specifies four instances of the digit 2 and any mathematical operations. Dirac's formula uses no other digits and deploys only standard operations (square roots, logarithms, division). The variable n is just a counter for how many times you nest the root; it doesn't appear in the final expression. The solution is adversarial in the best possible way — it honors the letter of the law while obliterating the spirit. The biographical hook — that this was a pastime for 1920s mathematicians — is a lovely touch, though Bendersky doesn't linger on it. The credit goes to Graham Farmelo's biography The Strangest Man, which has been the standard popular Dirac text since 2009. The post is essentially a well-structured retelling of one anecdote from that book, enhanced with worked examples and MathML formatting. As a piece of math exposition, this is clean, well-paced, and genuinely fun. It won't teach you new mathematics if you already know the Dirac trick, but the presentation is good enough to share with a curious teenager or a friend who thinks they hate math. The staircase structure — elementary, intermediate, advanced, then the nuclear option — is a pedagogical choice that works beautifully. What it lacks is depth: there's no discussion of whether the puzzle has interesting variants (three 3s? five 5s?), no exploration of what 'any mathematical operations' really means in formal terms, and no mention of the broader tradition of recreational mathematics from Dudeney to Gardner to Conway. But for a blog post, it does exactly what it sets out to do.
