10 World’s Hardest Math Problems With Solutions and Examples That Will Blow Your Mind

Update: This article was last updated on 7th May 2024 to reflect the accuracy and up-to-date information on the page.

The mystical world of mathematics—is home to confounding problems that can make even the most seasoned mathematicians scratch their heads. Yet, it’s also a realm where curiosity and intellect shine the brightest. 

Here are 10 of the world’s hardest math problems, with solutions and examples for those that are solved and a humble “unsolved” tag for the puzzles that continue to confound experts.

1. The Four Color Theorem

Source: Research Outreach

Problem: Can every map be colored with just four colors so that no two adjacent regions have the same color?

Status: Solved

Solution Example: The Four Color Theorem was proven with computer assistance, checking numerous configurations to show that four colors are sufficient. If you want to prove it practically, try coloring a map using only four colors; you’ll find it’s always possible without adjacent regions sharing the same color.

2. Fermat’s Last Theorem

Source: NPR

Problem: There are no three positive integers a,b,c that satisfies

an+bn=cn for n>2.

Status: Solved

Solution Example: Andrew Wiles provided a proof in 1994. To understand it, one would need a deep understanding of elliptic curves and modular forms. The proof shows that no such integers a,b,c can exist for n>2.

3. The Monty Hall Problem

Source: Towards Data Science

Problem: You’re on a game show with three doors. One hides a car, the others goats. After choosing a door, the host reveals a goat behind another door. Do you switch?

Status: Solved

Solution Example: Always switch. When you first choose, there’s a 1/3 chance of picking the car. After a goat is revealed, switching gives you a 2/3 chance of winning. If you don’t believe it, try simulating the game multiple times.

4. The Travelling Salesman Problem

Source: Brilliant

Problem: What’s the shortest possible route that visits each city exactly once and returns to the origin?

Status: Unsolved for a general algorithm

Solution Example: This is known as computer science’s most well-known optimization problems. Although there is no solution for all cases, algorithms like the Nearest Neighbor and Dynamic Programming can provide good approximations for specific instances.

5. The Twin Prime Conjecture

Source: Hugin

Problem: Are there infinitely many prime numbers that differ by 2?

Status: Unsolved

Solution Example: N/A

Recommended Reading: Pros and Cons of Math Competition

6. The Poincaré Conjecture

Problem: Can every simply connected, closed 3-manifold be homomorphic to the 3-sphere?

Status: Solved

Solution Example: Grigori Perelman proved this using Richard Hamilton’s Ricci flow program. In simple terms, he showed that every shape meeting the problem’s criteria can be stretched and shaped into a 3-sphere.

7. The Goldbach Conjecture

Source: Medium

Problem: Can every even integer greater than 2 be expressed as the sum of two prime numbers?

Status: Unsolved

Solution Example: N/A

8. The Riemann Hypothesis

Source: The Aperiodical

Problem: Do all non-trivial zeros of the Riemann zeta function have their real parts equal to 1/2?

Status: Unsolved

Solution Example: N/A

9. The Collatz Conjecture

Source: Python in Plain English

Problem: Starting with any positive integer n, the sequence n,n/2,3n+1,… eventually reaches 1.

Status: Unsolved

Solution Example: N/A

10. Navier–Stokes Existence and Smoothness

Problem: Do solutions to the Navier–Stokes equations exist, and are they smooth?

Status: Unsolved

Solution Example: N/A

Conclusion

There you have it—10 of the world’s hardest math problems. Some have been gloriously solved, giving us a brilliant glimpse into the capabilities of human intellect. Others still taunt the academic world with their complexity. For math lovers, this is the playground that never gets old, the arena where they can continually hone their problem-solving skills. So, do you feel up to the challenge?

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