World Solve

Classifying all finite simple groups was a monumental achievement, but fully understanding how all groups relate to and build from these simple pieces remains an ongoing task. This shapes our foundational understanding of symmetry itself.

While the standard cube's maximum was determined through massive computation, generalized versions with more dimensions or larger sizes remain computationally and theoretically unresolved. This connects group theory with combinatorial optimization.

Certain constructions, like doubling a cube's volume using classical tools, were proven impossible, but a full classification of every possible constructible versus non-constructible case remains an evolving area. This connects ancient geometry with modern algebra.

Formulas exist for degrees up to four, but general algebraic solutions for higher degrees don't exist in the same closed form, leaving open questions about alternative methods. This shapes ongoing research in both pure and computational algebra.

Some mathematical truths may be true but fundamentally unprovable within any given formal system, a possibility raised by Gödel's work but not fully mapped out for all classes of statements. This unresolved boundary shapes the philosophy of computation.

Gödel's incompleteness theorem confirms this for many systems, but fully mapping which systems are exempt, if any, from this limitation remains an open and active question. This continues to shape foundational debates in mathematical logic.

Arrow's Impossibility Theorem proved certain fairness conditions can't all hold at once for common voting systems, but the full landscape of tradeoffs across all possible systems remains an active research area. This has real implications for the design of democratic institutions.

Despite extensive computation, no proof confirms whether these digits behave truly randomly forever or eventually reveal some hidden structure. This remains an open question bridging number theory and statistics.

Some modern proofs rely heavily on computer verification because they're too long or complex for any person to check by hand, raising open questions about whether shorter, purely human-verifiable proofs must always exist. This unresolved tension touches both mathematics and epistemology.

This is the famous Halting Problem, proven impossible to solve in full generality, yet many partial and practical questions around predicting program behavior remain actively studied. This shapes the theoretical limits of what computers can ever determine about themselves.

Some tiling patterns work perfectly in abstract mathematical space but haven't been proven physically constructible under all real-world constraints. This connects pure geometry with material and structural engineering.

Ramsey-type problems ask whether order must always emerge within sufficiently large or complex systems, and while some cases are proven, the general boundaries remain incompletely mapped. This connects deeply to combinatorics and the philosophy of mathematical order.

Some lower bounds on computational efficiency are proven for specific tasks, but a fully unified theory covering every possible algorithmic problem remains unresolved. This unresolved question shapes the outer limits of computer science.

Game theory has solved many two-player scenarios completely, but multiplayer games with shifting alliances and incomplete information remain far less understood in full generality. This unresolved question has real implications for economics and political strategy.

This long-standing conjecture claims there's always at least one prime number between any two consecutive perfect squares, and while verified extensively by computer, no general proof exists. Solving it would sharpen our understanding of prime distribution.

Various conjectures link the number of prime factors a number has to specific patterns of behavior, but many of these connections remain only partially proven. This continues to be explored within analytic number theory.

Modern cryptography largely depends on problems assumed, but not proven, to be computationally hard, leaving a theoretical gap in fully guaranteed security. This unresolved foundation underlies much of digital security today.

Fair division theory has strong results for two-party negotiations, but multi-party fairness guarantees become significantly harder to prove in full generality. This unresolved area connects mathematics with real-world conflict resolution.

Some deterministic functions produce output so irregular it resembles randomness, and a complete classification of exactly which rules produce this behavior remains incomplete. This unresolved boundary connects number theory, chaos theory, and computer science.

Certain recursively defined sequences appear statistically random for enormous stretches, yet whether hidden structure always eventually emerges remains an open and evolving question. This connects deeply to both number theory and the study of pseudorandomness.