Making any integer with four 2s

Scope and “spirit” of the game

• Many argue that once “any mathematical operation” or arbitrary functions are allowed, the puzzle becomes trivial and loses its spirit.
• Others counter that all operations (+, −, ×, ÷, powers) are functions anyway; the real issue is which functions count as acceptable, not whether “functions” are allowed.
• Several comments emphasize that this is a game, not a formal competition, so rule choices should prioritize fun rather than rigor.

Which operations are allowed?

• Debate over allowing “fancy” functions: gamma, square root, logarithms, floor/ceiling, trig, primorial, etc.
• One camp wants to restrict to integer→integer functions to stay close to the traditional “four fours” puzzle. This excludes sqrt and log but allows +, −, ×, factorial, exponentiation by integers.
• Others are fine with square roots, logs, rounding, etc., especially if they’re “common” or appear on a basic calculator.
• The gamma function and Dirac-style constructions are viewed by several as too obscure or too powerful for a kids’ puzzle.

Hidden constants and notation

• Strong criticism that sqrt hides an implicit 2 (root degree), just as log and ln hide bases like 10 or e. Some see this as using extra twos “for free.”
• Counterpoint: all notation hides structure (e.g., + hides repeated successor; factorial hides a whole product), so this line is inherently arbitrary.
• Some propose forbidding any letters/digits other than “2”; others say letters are just symbols too and the distinction is fuzzy.

Unary operators and trivialization

• Repeated unary operations (successor S(n)=n+1, increment ++, repeated sqrt) can generate all integers, making the puzzle trivial.
• Observation: if you restrict to only a finite set of n‑ary operators (n≥2) and exactly four 2s, the number of expressible values is finite, so you can’t reach all integers; at least one “useful” unary operation seems necessary.

Need for explicit rules

• Several commenters think the article’s initial rules (“any mathematical tools”) are too vague; they’d prefer a fixed, explicit operator set up front.
• Others enjoy that part of the fun is precisely pushing and renegotiating those boundaries.

Related puzzles and curiosities

• References to the classic “four fours” puzzle, Knuth’s “one 4” constructions, and mobile/online games based on similar ideas.
• Some share floating-point based constructions where expressions involving √2 and rounding artifacts yield large integers, highlighting implementation quirks rather than pure math.