Repository of problems worth solving
World Solve is a public institutional repository for identifying real unresolved problems across science, society, health, governance, economics, and local systems. Each entry is intended to be citable, inspectable, and actionable.
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4194Total
4194Open
0Solved
9Humans
open
Global / Unspecified, Global
Mathematicians have many tools for distinguishing knots, but no single method works perfectly and efficiently for every possible case. This unsolved problem sits at the heart of the mathematical field of knot theory.
mathematics-logic
global-unspecified
every
knot
don
know
whether
told
apart
simple
efficient
resolved
philosophy-ethics
haven
possible
mathematical
open
Global / Unspecified, Global
Starting from any positive number and repeatedly applying a basic rule always seems to eventually reach the number one, but no proof confirms this holds for every possible starting point. Its simplicity contrasted with its difficulty has made it famous among mathematicians and hobbyists alike.
mathematics-logic
global-unspecified
rule
seems
always
haven
proven
collatz
conjecture
strikingly
simple
lead
whether
know
possible
every
mathematical
resolved
open
Global / Unspecified, Global
Certain simple-sounding tiling and packing puzzles remain surprisingly resistant to a complete general solution. These problems test the limits of combinatorics and geometric reasoning.
mathematics-logic
global-unspecified
haven
resolved
unit
squares
needed
minimum
tile
square
given
odd
whether
possible
know
every
mathematical
philosophy-ethics
open
Global / Unspecified, Global
This question, closely related to the once-open Poincaré Conjecture, has been resolved in three dimensions but opens further mysteries in higher-dimensional analogues. Exploring these generalizations remains an active area of topology.
mathematics-logic
global-unspecified
don
know
whether
every
simply
connected
three-dimensional
shape
holes
always
haven
resolved
philosophy-ethics
possible
open
Global / Unspecified, Global
Some conjectures suggest that most true statements in sufficiently complex systems may have no short proof at all, only extremely long or even infinite ones. This unresolved question challenges our basic assumptions about what makes mathematics knowable.
mathematics-logic
global-unspecified
all
long
haven
proven
whether
almost
mathematical
truths
require
infinitely
philosophy-ethics
resolved
know
possible
every
open
Global / Unspecified, Global
The famous four-color theorem was resolved for flat maps, but generalizing the problem to more complex or higher-dimensional surfaces remains unsettled in several cases. This continues to be explored in graph theory and topology.
mathematics-logic
global-unspecified
color
flat
haven
determined
exact
minimum
number
colors
needed
any
whether
philosophy-ethics
know
resolved
possible
every
open
Global / Unspecified, Global
Some sequences and structures appear chaotic without ever settling into repetition, and proving whether repetition is truly avoidable in specific systems remains unresolved. This connects deeply to dynamical systems and number theory.
mathematics-logic
global-unspecified
whether
don
know
every
sufficiently
well-behaved
mathematical
pattern
must
eventually
philosophy-ethics
resolved
haven
possible
systems
open
Global / Unspecified, Global
Simple geometric fairness questions like this remain surprisingly open for many classes of shapes, especially in three or more dimensions. Solving them has practical relevance to resource division and computational geometry.
mathematics-logic
global-unspecified
haven
proven
whether
possible
always
divide
any
shape
equal-area
pieces
know
resolved
philosophy-ethics
every
mathematical
open
Global / Unspecified, Global
Primes appear scattered with no obvious repeating pattern, and finding a true predictive formula, rather than just an approximation, remains unsolved. This is closely tied to the deeper mysteries of the Riemann Hypothesis.
mathematics-logic
global-unspecified
formula
haven
proven
whether
there
always
predicts
next
prime
number
know
possible
every
philosophy-ethics
mathematical
resolved
open
Global / Unspecified, Global
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.
mathematics-logic
global-unspecified
every
groups
finite
don
know
whether
group
understood
piece
some
philosophy-ethics
resolved
haven
possible
mathematical
open
Global / Unspecified, Global
Quantum computers show promise for specific problems like factoring, but whether they offer a genuine speedup across all hard computational problems remains unproven. This question sits at the intersection of mathematics, physics, and computer science.
mathematics-logic
global-unspecified
computers
whether
quantum
haven
resolved
efficiently
solve
every
currently
considered
know
philosophy-ethics
possible
mathematical
open
Global / Unspecified, Global
True randomness is difficult to define and even harder to prove, since any formula-based sequence is, by definition, predictable if you know the formula. This unresolved tension underlies debates in both mathematics and philosophy of probability.
mathematics-logic
global-unspecified
sequence
haven
determined
whether
possible
construct
truly
random
numbers
only
know
resolved
philosophy-ethics
every
mathematical
open
Global / Unspecified, Global
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.
mathematics-logic
global-unspecified
dimensions
don
know
exact
largest
number
moves
required
solve
any
whether
haven
resolved
possible
every
philosophy-ethics
open
Global / Unspecified, Global
This boolean satisfiability problem is central to computer science, and no efficient general method is known for solving every possible case quickly. It remains a benchmark problem for understanding computational complexity.
mathematics-logic
global-unspecified
haven
resolved
efficiently
determine
any
given
set
logical
statements
whether
know
philosophy-ethics
possible
every
mathematical
open
Global / Unspecified, Global
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.
mathematics-logic
global-unspecified
don
know
whether
set
numbers
never
precisely
constructed
only
compass
haven
possible
resolved
philosophy-ethics
every
mathematical
open
Global / Unspecified, Global
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.
mathematics-logic
global-unspecified
proven
game
draw
haven
whether
every
possible
strategy
complete
information
resolved
philosophy-ethics
know
mathematical
open
Global / Unspecified, Global
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.
technology-computing
global-unspecified
general
haven
determined
whether
there
exists
efficient
algorithm
solving
all
mathematics-logic
philosophy-ethics
know
possible
resolved
every
open
Global / Unspecified, Global
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.
technology-computing
global-unspecified
true
given
don
know
whether
every
statement
whole
numbers
eventually
philosophy-ethics
haven
resolved
mathematics-logic
possible
open
Global / Unspecified, Global
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.
mathematics-logic
global-unspecified
haven
proven
whether
possible
always
tile
rectangle
perfectly
only
squares
know
resolved
philosophy-ethics
every
mathematical
open
Global / Unspecified, Global
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.
technology-computing
global-unspecified
limits
haven
resolved
whether
number
steps
needed
sort
any
list
philosophy-ethics
know
mathematics-logic
possible
every
mathematical
