World Solve

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.

The Continuum Hypothesis asks whether there's a size of infinity strictly between the countable and uncountable, and it has been shown to be independent of standard mathematical axioms. This unresolved question touches the very foundation of set theory.

Fair division problems become extremely complex once individual preferences differ, and general efficient solutions for many people remain incompletely understood. This connects mathematics to real-world negotiation and resource allocation.

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.

This problem, part of a broader family of questions in combinatorial geometry, has known bounds but no exact general formula for every configuration. Solving it would refine our understanding of how simple geometric constraints shape complex arrangements.

Knot theory offers tools for identifying and classifying tangles, but calculating a guaranteed minimal ontangling path for every possible knot remains computationally unresolved. This connects abstract mathematics to physical intuition about rope and cords.

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.

Palindromic and reversible primes show interesting patterns, but whether infinitely many exist across all number bases remains unconfirmed. This is one of many number-theoretic curiosities still lacking rigorous proof.

While specific puzzle sizes have been fully solved computationally, generalizing the worst-case difficulty formula across all puzzle sizes remains an open combinatorial question. This links recreational mathematics with formal complexity theory.

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.

Related conjectures about how primes combine remain only partially proven for specific ranges, without a fully general confirmed pattern. This connects to broader unresolved questions about the additive structure of prime numbers.

Finding the shortest possible route through many locations becomes exponentially harder as the number of locations grows, and no efficient exact solution is known for all cases. This unresolved question sits at the heart of computational optimization theory.

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.

Chaotic systems are famously sensitive to initial conditions, and whether their long-term patterns can ever be fully captured by simpler predictive rules remains an unresolved question in dynamical systems theory. This has real implications for weather prediction and other complex natural systems.

Partition numbers grow in a complex, only partially understood way, and while excellent approximations exist, a complete exact predictive formula for all cases remains elusive. This continues to be an active area within combinatorics and number theory.

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.

Even our best physical models rely on approximations or idealized assumptions, and whether a truly complete and exact mathematical simulation is even theoretically possible remains unresolved. This touches on the deep relationship between mathematics and the physical world.

Graph theory offers many partial answers about guaranteed patterns within large networks, but a fully general and complete theory covering every possible network structure remains incomplete. This affects our understanding of everything from social networks to biological systems.

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.

While many invariants help distinguish specific knots, no single finite and fully general method has been proven to work for absolutely every possible pair of knots. This remains a central unresolved question in the mathematical study of knots.