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

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    WORLD’S HARDEST MATH PROBLEMS

    Update: This article was last updated on 12th Oct 2023 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

    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

     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

    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

    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

    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

    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

    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

    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

    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

    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?

    Moonpreneur understands the needs and demands this rapidly changing technological world is bringing with it for our kids. Our expert-designed Advanced Math course for grades 3rd, 4th, 5th, and 6th will help your child develop math skills with hands-on lessons, excite them to learn, and help them build real-life applications. 

    Register for a free 60-minute Advanced Math Workshop today!

    Moonpreneur

    Moonpreneur

    Moonpreneur is an ed-tech company that imparts tech entrepreneurship to children aged 6 to 15. Its flagship offering, the Innovator Program, offers students a holistic learning experience that blends Technical Skills, Power Skills, and Entrepreneurial Skills with streams such as Robotics, Game Development, App Development, Advanced Math, Scratch Coding, and Book Writing & Publishing.
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    4 Comments
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    elena
    elena
    3 months ago

    does solving hard math questions even sharpen your brain? I mean I get exhausted and start panicking

    Simran Chawla
    Editor
    Simran Chawla
    2 months ago
    Reply to  elena

    Yes, solving tough math problems sharpens your brain by boosting critical thinking and problem-solving skills. While it may feel challenging, the process enhances cognitive abilities over time.

    jazzz
    jazzz
    3 months ago
    Reply to  elena

    Don’t panic, take breaks, have some time, you’ll get better eventually, not immediately.

    Jane
    Jane
    4 months ago

    Do you know there is an award price of $1 Million for solving the Riemann Hypothesis….. THIS IS SOMETHING SOO SERIOUS!!!!

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