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A 2025 post by Eli Bendersky explains how a puzzle requiring exactly four instances of the digit 2 can be extended to any natural number. The method uses repeated square roots and base-2 logarithms, with a small expression supplying the fourth 2; the post credits the general approach to Paul Dirac.
A 2025 post by Eli Bendersky explains a way to represent any natural number using exactly four instances of the digit 2, extending a recreational math puzzle that asks solvers to build target numbers from four 2s. The method uses repeated square roots and base-2 logarithms, and the post credits the underlying general solution to physicist Paul Dirac.
The puzzle’s rules allow mathematical operations but restrict the digits: the expression must use exactly four 2s and no other digits. What counts as an allowed operation can change how difficult a target is. Basic arithmetic handles some numbers, while exponents, factorials, and joining two digits to make 22 expand the possibilities. More advanced functions can make targets that are otherwise hard to express accessible.
Bendersky describes a general construction based on nested square roots. Repeatedly taking the square root of a quantity and then applying base-2 logarithms can be arranged to recover a target natural number. The count of repeated roots is a helper in the explanation, not a digit that appears in the final expression; the source says the construction works by choosing the appropriate number of repetitions.
As presented in the post, the general formula initially uses three 2s, rather than the four required by the puzzle. Bendersky says the gap can be closed with the identity 2 − 2 = 0: adding that zero to a suitable expression supplies the fourth instance without changing its value. The author acknowledges that some solvers might regard this as bending the rules, but argues it fits the stated constraint.
A General Recipe for the Puzzle
The appeal of the construction is not that it gives a short or practical way to calculate numbers. Instead, it turns a collection of individual challenges into a general recipe: rather than finding a separate clever expression for each target, the solver can use the target to determine how many square-root steps are needed. That shifts the puzzle from isolated ingenuity toward a proof that the permitted notation can reach every natural number.
It also shows how puzzle rules shape what counts as a solution. If only familiar arithmetic operations are permitted, some targets may be difficult or unreachable under a solver’s chosen toolkit. Allowing factorials, special functions, or digit concatenation changes the landscape. The four-2s problem is therefore both a number puzzle and a small lesson in how mathematical definitions and operations affect what can be expressed.
For readers, the key distinction is between an elegant expression for a particular number and a method that applies to an entire class. The post’s claim is the latter: with the specified operations and the convention that the fourth 2 can be introduced as zero, the construction can represent any natural number. The result is a recreational demonstration, not a claim about a new mathematical discovery in 2025.
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From Four Digits to Nested Roots
The four-2s puzzle is a familiar style of recreational mathematics: a fixed set of digits must be combined to make a chosen target. The exact range of answers depends on the rules. Bendersky notes that students can start with elementary operations and expand their options as they learn exponents and factorials. Treating two adjacent 2s as 22 adds another convention, while more advanced tools—including the Gamma function—can produce further expressions.
The post also mentions examples using integrals, repeating fractions, and combinatorial operators, as well as an example involving complex numbers. These references illustrate how far solvers can take the puzzle, but the central point of the 2025 report is the general construction credited to Dirac. Bendersky says the approach was a pastime among mathematicians in the 1920s and cites Graham Farmelo’s biography, The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius, as the source through which the author encountered the story.
Rules and Formula Details
The source presents the method in prose and refers to equations that are not reproduced in the supplied material. That means the exact layout of the general formula, including the precise placement of each logarithm and nested root, cannot be independently checked from this text alone. The article should therefore describe the approach at the level supported by the report rather than reconstructing missing notation.
There is also a rules question: the post treats adding 2 − 2 as a valid way to use the fourth digit, but not every version of the puzzle necessarily accepts a zero-valued addition or the same set of functions. The source does not specify a formal list of permitted operations. It also does not define whether “natural number” includes zero; its claim is framed around target natural numbers, and the article does not extend it beyond that stated scope.
Checking the Construction
The next step for readers who want to verify the claim is to consult Bendersky’s original post and inspect the full equations, then test the construction on individual targets. The source gives 7 as an example of expressing a number with four 2s, but the supplied text omits the displayed expression, so it cannot be quoted or checked here. A complete verification would require the original notation and a clear statement of the allowed operations.
No follow-up announcement or new mathematical result is described in the source material. The development is an explanatory account of a recreational-math result and its attribution to Dirac, rather than a new research paper. The open issue for any particular version of the puzzle remains whether its rules permit the functions and the zero-valued term used in the general recipe.
Key Questions
What is the four-2s puzzle?
It asks solvers to make a chosen target number using exactly four digit 2s and no other digits. The permitted mathematical operations determine which expressions count.
How does the general method work?
Bendersky describes a construction using repeated square roots and base-2 logarithms. The number of repeated roots acts as a helper that can be adjusted to the target natural number.
How does the formula use exactly four 2s?
The post says the general expression first uses three 2s, then adds 2 − 2 = 0. Because that term has value zero, it adds a fourth 2 without changing the result.
Who is credited with the general solution?
Bendersky credits the approach to Paul Dirac and says the puzzle was a pastime among mathematicians in the 1920s. The post cites Graham Farmelo’s biography of Dirac as the author’s source.
Is the method accepted under every version of the puzzle?
Not necessarily. Some versions may restrict operations or reject adding a zero-valued term solely to meet the digit count. The source does not set out a universal rulebook, so acceptance depends on the version being played.
Source: eli.thegreenplace.net
