10 Hardest Math Problems In The World With Solutions

Interested in mathematics? Do you want to know what the hardest math problem in the world is? The mysterious world of mathematics is filled with puzzling problems that can stump even the most seasoned mathematicians. WuKong Education will present the world’s 10 hardest math problems, both solved problems and unsolved problems that continue to stump the experts.

Math is more than numbers; it’s the foundation for logical thinking, creativity, and success in future studies. WuKong Math, trusted by families across 118+ countries, offers online math programs for children aged 3–18 that combine engaging lessons with training. From fundamental skills to international competition prep, our courses make math both fun and empowering for kids.

The table of the 10 hardest math problems

Mathematical problems such as the Poincaré Conjecture and Fermat’s Last Theorem took centuries to solve. However, other problems, such as the Riemann hypothesis and Goldbach’s conjecture, continue to baffle mathematicians and inspire new generations to find solutions.

It is a basic fact in number theory that there are infinitely many primes, which serves as a foundation for various conjectures related to prime numbers. One such well-known conjecture is the Twin Prime Conjecture, which questions whether there are infinitely many prime numbers that differ by 2.

Next, we take a look at some of the 10 hardest math problems. Many mathematical problems have taken mathematicians decades or even centuries to solve, while others remain unsolved.

10 Hardest Math Problems	Status
The Collatz Conjecture	Unsolved
Goldbach’s Conjecture	Unsolved
The twin prime conjecture	Unsolved
The Four Color Theorem	Solved
Riemann Hypothesis	Unsolved
The existence of odd perfect numbers	Unsolved
The Poincaré Conjecture	Solved
The solitary number problem	Unsolved
The Birch and Swinnerton-Dyer Conjecture	Unsolved
Hodge Conjecture	Unsolved

1. The Poincaré Conjecture

Problem(solved)：In 1904 the French mathematician Henri Poincaré asked if the three-dimensional sphere is characterized as a unique simply connected three manifold. This question, the Poincaré conjecture, was a special case of Thurston’s geometrization conjecture. Suppose you have a sphere. You can tie a loop of string anywhere on the string and pull the ends to close the loop. But you can’t do that on a torus (doughnut) if the string goes around through the hole.

Solution：Poincare’s conjecture says that the sphere is the only three-dimensional shape that has this loop-tightening property for every possible loop on its surface. It was proved by Grigori Perelman 20 years ago. In simple terms, he showed that every shape meeting the problem’s criteria can be stretched and shaped into a 3-sphere.

2. The Four Color Theorem

Problem(solved): Can every map on a plane be colored with only four colors such that no two adjacent regions share the same color?

Solution: The Four Color Theorem was proven with computer assistance. Mathematicians used computer programs to check a large number of map configurations and showed that four colors are always sufficient. This involved verifying that there are no maps that require more than four colors.

Example: Take a map of Europe as an example. Try coloring it using only four colors (e.g., red, blue, green, and yellow). Start with one country and color its adjacent countries with different colors. No matter how complex the map is, you’ll find that it’s always possible to color it with four colors without any two adjacent countries having the same color.

3. The Collatz Conjecture

The Collatz Conjecture is a simple but unproven mathematical conjecture that proposes a process about sequences of integers. It was proposed by German mathematician Lothar Collatz in 1937. Over the years, many mathematicians have attempted to unravel the mystery of this conjecture, but it has remained an enigma. Many mathematicians have suggested that this problem may even be out of the reach of present-day mathematics. 

The difficulty of the Collatz Conjecture lies in its unpredictable nature. The sequence generated by the simple iterative process can take various routes before eventually reaching 1, making it challenging to establish a general proof. This problem is a classic example of how a simple algorithm can lead to complex and seemingly random behavior and pose a significant challenge to mathematicians.

The function f(n) in the figure above, which cuts even numbers in half, cuts odd numbers in triples and then adds to 1, ends up with all of the numbers we examined being 1.

Problem(unsolved):The conjecture is that this is true of all natural numbers (positive integers from 1 to infinity). The conjecture deals with two simple repeating operations performed on any given positive integer and asks if it will eventually transform the given integer into 1. If the given integer is even, it will be divided by 2. If odd, it will be multiplied by 3, and have 1 added to it. Hence the iconic (3n + 1) name.

Example:the integer 16. It is even, thus once divided by 2, it becomes 8. 8 being even, will be divided again, becoming 4, then 2, and eventually 1.In theory, no matter what positive integer the operations are performed on, it will always transform into 1.

Math expert Marty Parks’ comments on the Collatz Conjecture:

“The Collatz Conjecture is a fun one because it illuminates how little we really know about math. You have a very simple recipe to follow: 1. Pick any whole number bigger than 0. 2. Divide it by 2 if its even, or times it by 3 and add 1 if its odd. So far we end up at the number 1 no matter where we start — and no one knows why! Like most of the biggest unsolved problems in mathematics, the difficulty with the Collatz Conjecture is that the tools needed to tackle the problem still need to be developed. This very simple problem is so far beyond the reach of what our current math allows to do that it will be very exciting to see how future mathematicians (or AI) will develop completely unheard-of methods to help find a proof.”

4. Goldbach’s Conjecture

The Goldbach conjecture is one of the most captivating mysteries in mathematics. It was proposed by the German mathematician Christian Goldbach in 1742. Goldbach’s Conjecture shows that every even natural number greater than 2 can be expressed as the sum of two prime numbers.

Problem(unsolved): The conjecture was first proposed by Christian Goldbach on June 7, 1742, in a letter to Leonhard Euler. In this letter, Goldbach presented the idea and conjectured that every integer greater than 2 could be expressed as the sum of two prime numbers. Euler wrote back that the first part of Goldbach’s conjecture was highly probable. He noted that “every even integer is the sum of two prime numbers”, but he was unable to provide proof.

Example: 16 = 3 + 13

Progress: Significant progress has been made in understanding this conjecture over a long period. For example, Nils Pipping verified that n = 100,000 in 1938. Later, with the advent of computers, T. Oliveira e Silva conducted distributed computer searches. By 2013 he confirmed the conjecture that “n” is less than or equal to 4×1018 (and repeated the proof for “n” up to 4 × 1017). However, a complete and rigorous proof for all even integers greater than 2 remains out of reach.

5. The twin prime conjecture

Proving the twin prime conjecture is a long outstanding problem in number theory. The twin prime conjecture was first formulated by De Polignac in 1849. De Polignac argued that for every natural number “k”, there is an infinite number of primes “p” such that “p+2k” is also prime. The case “k = 1” is the one we are interested in, the Twin Primes Conjecture.

Problem:(unsolved) It is not surprising that the twin prime conjecture revolves around twin prime numbers. These are prime numbers that are either 2 less or 2 more than another prime number, forming pairs of prime numbers such as (5, 7), (13, 15). The conjecture states that there are an infinite number of prime numbers p such that p + 2 is also prime.

Progress: In 2013, Yitang Zhang‘s (a Chinese-American mathematician primarily working on number theory) research took an important step towards proving the existence of infinitely many twin prime numbers. His research showed that there exists a finite upper bound-70 million, for which gaps between pairs of prime numbers exist infinitely often. By April 2014, this limit (the gap between two prime numbers) had shrunk to 246, indicating significant progress in understanding twin prime numbers.

6. Riemann Hypothesis

The Riemann Hypothesis, formulated by Bernhard Riemann in 1859, is a central problem in number theory that discusses the distribution of prime numbers. The hypothesis focuses on the zeros of Riemann’s zeta function(a video). Building on the work of Swiss mathematician Leonhard Euler, Riemann assumed that all non-trivial zeros of this zeta function lie on a critical line in the complex plane, the critical line Re(s) = 0.5.

Riemann hypothesis will be a landmark achievement in mathematics, especially in the field of cryptography, which is crucial for Internet security. Confirmation of the hypothesis will also greatly improve our understanding of prime numbers and will validate many mathematical papers that currently take the Riemann hypothesis as a given, thus solidifying a wide range of mathematical theories.

7. The existence of odd perfect numbers

The existence of odd perfect numbers is a deep unsolved mathematical mystery. In math, a perfect number is a positive integer “n” that equals the sum of all divisors except the number itself. In other words: n = 1 + 2 + 3 + … + (n-1). A famous example of a perfect number is 28. The divisors of 28 are 1, 2, 4, 7 and 14. However, the existence of odd perfect numbers remains uncertain.

Problem: In 1496, Jacques Lefebvre made the point that all perfect numbers can be generated according to Euclid’s law. This implied that there could be no odd perfect numbers, setting the stage for centuries of speculation. Recently, Carl Pomerance presented a heuristic argument that the existence of odd perfect numbers is highly unlikely. This argument has increased skepticism about their existence.

8. The solitary number problem

The solitary number problem delves into the field of solitary numbers, which are integers that do not have any “friends”, in the mathematical sense (e.g., they do not share a common relationship with any other number). Friendly numbers are numbers that have the same abundance index (the ratio of the sum of the number’s divisors to the number itself).

Solitary numbers include prime numbers, prime powers, and numbers for which the greatest common divisor of the number and the sum of its divisors (expressed as sigma(n)) equals 1. For example, the number 5 is a solitary number. The divisors of 5 are 1 and 5, and their sum is 6. The greatest common divisor of 5 and 6 is 1.

Problem(unsolved): While it is possible to prove the solitariness of some numbers by examining their properties, proving the solitariness of others is challenging. For example, numbers like 10, 15, and 20 are believed to be solitary numbers, but providing conclusive proof has remained elusive. The concept of solitary numbers has fascinated mathematicians for many years. While prime numbers are well-known solitary numbers, other integers also exhibit solitary properties, even if their greatest common divisor with sigma(n) is not 1.

Progress: In 2022, Sourav Mandal shed light on the potential nature of 10’s friend, proposing a specific form. It must follow if it exists, adding an intriguing layer to the problem. Furthermore, examples like 24, classified as friendly, and possessing 91,963,648 as its smallest friend, illustrate the diversity in the classification of numbers as friendly or solitary.

9. The Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer conjecture is a far-reaching and complex problem in number theory, focusing on elliptic curves. The conjecture relates the number of rational solutions (points whose two coordinates are rational numbers) on an elliptic curve to certain features of the L-function associated with that curve.

Problem(unsolved)：The resolution of this conjecture would have large implications in several areas of mathematics, particularly in number theory and algebraic geometry. It would increase our understanding of elliptic curves, which are central to many mathematical fields, including cryptography, and could lead to advancements in digital security and new encryption technologies. The conjecture is notoriously tricky to solve due to its deep connections to various complex mathematical concepts such as L-functions, elliptic curves, and modular forms.

Progress: While there has been significant progress in understanding specific cases and aspects of the conjecture, general proof or disproof remains distant. The conjecture forms part of the Langlands program, which aims to unify different areas of mathematics. Advances within this program have shed some light on the conjecture, but a complete solution is still pending.

Math expert Marty Parks’ comments on the Birch and Swinnerton-Dyer Conjecture

“The Birch and Swinnerton-Dyer Conjecture deals with elliptic curves in number theory, and these types of problems tend to require a lot of advanced ‘machinery’ to even begin approaching a basic understanding of it. In terms of difficulties faced in trying to solve this problem, the Birch and Swinnerton-Dyer Conjecture suffers from many of the same issues that the Riemann Hypothesis suffers from. It’s likely that some incredibly advanced techniques will be necessary to find a solution. If solved, there are many other math theorems in number theory that would also be solved since they use assumptions from this theorem.”

10. Hodge Conjecture

The Hodge conjecture was proposed by William Hodge in 1941 and involves algebraic geometry. It shows a fundamental relationship between collections of simple geometric pieces, called algebraic cycles, and complex shapes of certain “nice” spaces, called projective algebraic varieties. The Hodge conjecture asserts that these algebraic cycles can approximate the shapes of these varieties.

Problem(unsolved): In short, solving this problem is like putting together a very complex, abstract puzzle. The Hodge conjecture is very difficult because of its complex nature and deep connections to various areas of mathematics. It requires a deep understanding of algebraic geometry, complex geometry, and topology.

A solution to the Hodge conjecture could advance our understanding of higher-dimensional mathematical structures. It also could have applications in areas such as string theory and other parts of theoretical physics.

Math expert Marty Parks’ comments on the Hodge conjecture:

“The Hodge Conjecture is such a hard problem because many types of the structures it references don’t really exist, and it’s really hard to just create those types of structures (logically, speaking). Since the Hodge problem relates to topology (more specifically cohomotopy), its solution would provide mathematicians better tools to solve related problems in algebraic topology, helping physicists, computer scientists, and even material scientists with new methods to analyze problems related to space (in physics), to network connectivity (in computer science), and analysis of material defects (in material science).”

FAQs About Hardest Math Problems

1.What is the Hardest Math Problem?

Unsolved Math Problems

1. Riemann Hypothesis : Unproven conjecture about the distribution of prime numbers.
2. P vs NP Problem : Unknown whether P class problems equal NP class problems.
3. Navier-Stokes Problem : Unsolved issue on fluid dynamics equation solutions.
4. Hodge Conjecture : Partially proven algebraic geometry conjecture.
5. Yang-Mills Problem : Unresolved problem in quantum field theory.
6. Twin Prime Conjecture : Unproven existence of infinite primes differing by 2.
7. Goldbach Conjecture : Unproven statement about expressing even numbers as two primes.
8. Odd Perfect Number : Existence unknown.
9. Isolated Number Problem : Some cases hard to prove.
10. Birch and Swinnerton-Dyer Conjecture : Unproven link between elliptic curves and L-functions.

Solved Math Problems

1. Poincaré Conjecture : Proven that the 3-sphere is the only simply connected 3-manifold.
2. Four-Color Theorem : Proven that four colors suffice for map coloring.

2.What is the 5 Hardest Math Problems on SAT?

Temperature Equation Problem

Question: The equation C=95​(F−32) shows how temperature F, measured in degrees Fahrenheit, relates to a temperature C, measured in degrees Celsius. Which of the following must be true?

I. A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of 95​ degree Celsius.

II. A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.

III. A temperature increase of 95​ degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius.

Answer: I and II only.

Algebra Problem

Question: If 3x−y=12, what is the value of 8x⋅2y?

Answer: 212.

Geometry Problem

Question: Points A and B lie on a circle with radius 1, and arc AB has a length of 3π​. What fraction of the circumference of the circle is the length of arc AB?

Answer: 61​, 0.166, or 0.167.

Complex Numbers Problem

Question: Rewrite the expression 3−2i8−i​ in the form a+bi, where a and b are real numbers. What is the value of a?

Answer: 2.

Trigonometry Problem

Question: In triangle ABC, ∠B=90∘, BC = 16, and AC = 20. Triangle DEF is similar to triangle ABC, with each side of triangle DEF being 3 the length of the corresponding side of triangle ABC. What is the value of sinF?

Answer: 53​ or 0.6

For more detailed information, please read 5 Hardest Math Problems on SAT with Answers to explore more.

Conclusion

These are the top 10 hardest math problems in the world. Some of them have been solved perfectly, while the complexity of some still poses a challenge to the academic world. For math enthusiasts, this is an arena where they can continue to hone their problem-solving skills. WuKong provides high-quality online math classes for kids that help learners build strong foundations in problem solving and critical thinking.

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