World Solve

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.

Some games are proven to always end in a draw or a specific winner under perfect play, but for many complex games, this remains unresolved due to sheer computational scale. This unsolved area touches game theory, logic, and computer science.

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.

While specific solutions exist for certain rectangles, a full general theory predicting exactly which rectangles allow this kind of tiling remains incomplete. This puzzle blends recreational mathematics with serious combinatorics.

Sorting algorithms have been optimized extensively, but proving a true absolute lower bound across all possible sorting methods remains an unresolved question in computational theory. This affects the fundamental efficiency limits of computing.

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.

Physical laws impose some theoretical limits on computation speed and energy use, but the exact universal boundary, if one truly exists, remains unresolved. This connects computer science directly to fundamental physics.

Irrational numbers have infinite, non-repeating digits, and while we can calculate them to enormous precision, no simple predictive pattern for their exact sequence has been found. This touches on deep unresolved questions in number theory.

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.