World Solve

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.

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.

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.

Some proofs are famously long and complex, and it remains unknown whether shorter, more elegant proofs must always exist in principle, even if we haven't yet found them. This unresolved question touches on both mathematics and the philosophy of mathematical elegance.

Some infinite series have been solved exactly, revealing elegant connections to constants like pi, while many similar-looking series remain stubbornly unresolved. This ongoing challenge continues to drive research in mathematical analysis.

Gödel's theorems already limit what's possible here for sufficiently powerful systems, but the full landscape of alternative systems and their limitations remains an active and unresolved area of mathematical logic. This continues to shape our understanding of the foundations of mathematics itself.

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.

Combinatorial geometry problems like this remain partially solved for specific shapes and configurations, but a fully general guarantee across all possible line arrangements is still unproven. This connects visual intuition with rigorous mathematical guarantees.

Certain classes of expressions can be checked for equivalence efficiently, but a fully general and guaranteed method covering every possible mathematical expression remains unresolved. This unresolved question has real consequences for automated theorem proving and symbolic computation.

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.

The Prime Number Theorem offers an excellent approximation, but the finer details of exactly how actual prime counts deviate from this smooth prediction remain tied to the still-unproven Riemann Hypothesis. Fully resolving this would refine one of number theory's most central results.

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.