The toughest math problems that challenge the world (2023)

In 1900, German mathematician David Hilbert proposed a list of 23 math problems that would change the world. Some have been solved. Others remain. DARPA attempted to update the list a few years back. Here are the highlights.

Solving what look to be "unsolvable" math problems has been a hot topic of math and science connoisseurs for a long time. It also impacts popular movie culture—remember "Good Will Hunting" and more recently "Gifted."

The idea that amazingly difficult, conceptual, unsolvable math problems could change the world can in part be traced back to 1900 when German mathematician David Hilbert proposed his still influential 23 math problems that would change the world.

According to the School of Mathematics and Statistics at the University of St Andrews in Scotland, Hilbert began his talk with these words(translated into English), which still ring true in the world today:

“Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be towards which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?”

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Problems solved from Hilbert's original list (and who solved them)

Some of the 23 problems proposed at the time were solved, while some remain. This “Math Is Good for You” post offers a glimpse of a few of the problems and who is credited with solving them:

  • "Is there a number, which is larger than any finite number, between that of a countable set of numbers and the numbers of the continuum?" To think of a continuum, think of a number line—and all the numbers on it—without any gaps. This problem was answered by KurtGödel.
  • "Can it be proven that the axioms of logic are consistent?" Gödel also answered this problem with his "incompleteness theorem," which states that all consistent axiomatic formulations include some undecidable propositions. For more, see the short history of Euclidean and non-Euclidean geometries.
  • "Give two tetrahedra that cannot be decomposed into congruent tetrahedra directly or by adjoining congruent tetrahedra." MaxDehn showed this could be done, but he had to invent his own invariants (something that does not change under a set of transformations).

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(Video) The Simplest Math Problem No One Can Solve - Collatz Conjecture

DARPA updates Hilbert's list

A few years ago, the researchers at the U.S. government’s Defense Advanced Research Projects Agency (DARPA) began a concerted effort to update Hilbert’s list and develop a new Mathematical Challenges program for the 21st century. DARPA proposed 23 updated questions that, if answered, “would offer a high potential for major mathematical and scientific breakthroughs.”

Anthony Falcone, now president and chief technology officer atFunctor Reality, was the DARPA program manager during the last couple years of the Math Challenges program, which the agency ended in 2012.“At the time, I think the program was more a consciousness-raising effortfor mathematics. The idea was to get as many good, smart people as possible thinking about new ways to solve problems,” Falcone says. “We certainly moved the needle.”

According to DARPA, six projects were funded through 2012 to address five of the 23 challenges. According to program documents, the levels of success in the five individual challenges varied.

A simplified look at these complex findings includes:

21st Century Fluids: Developed a software implementation based on the extended von Neumann’s formula for cell growth rate from two dimensions to three dimensions.

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  • Used the software to extract statistical information from grain systems that reached the asymptotic statistical state. Since the simulation was several orders of magnitude larger than anything done before, the error bars for the extracted statistics are the smallest existing.
  • Did an analysis on two-dimensional structures, three-dimensional structures, and two-dimensional cross-sections of three-dimensional structures.
  • Studied the random topological structure of grain networks and foams, and established the existence of phase transitions based on changes in the underlying parameter.

Riemann hypothesis: Generalized the p-adic local monodromy theorem to arbitrary differential modules.

  • Developed alternate constructions for Fontaine's rings in p-adic rings.

Hodge conjecture: Proved that every projective K3 surface over an algebraically closed field contains infinitely many rational curves.

Arithmetic Langlands: Discovered the appropriate analog of complex conjugation over a finite field for “ordinary” elliptic curves and “ordinary” abelian varieties.

  • Devised and analyzed a class of pseudorandom sequences.

Algorithmic origami: Solved the “carpenter’s rule problem” regarding the straightening of polygonal chains of line segments.

  • Developed novel approaches for the accurate modeling of hydrogen bonds using the bond geometry and a well-suited mechanical model.
  • Extended proofs of Delaunay realizability (for triangulations) from maximal outerplanar graphs to arbitrary outerplanar graphs.

DARPA's math challenges still waiting to be solved

Other challenges remain unsolved. Among them are:

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  • The mathematics of the brain: Develop a mathematical theory to build a functional model of the brain that is mathematically consistent and predictive rather than merely biologically inspired.
  • The dynamics of networks: Develop the high-dimensional mathematics needed to accurately model and predict behavior in large-scale distributed networks that evolve over time, occurring in communication, biology, and the social sciences.
  • What are the symmetries and action principles for biology? Extend our understanding of symmetries and action principles in biology along the lines of classical thermodynamics, to include important biological concepts such as robustness, modularity, evolvability, and variability.
  • Geometric Langlands and quantum physics: How does the Langlands program, which originated in number theory and representation theory, explain the fundamental symmetries of physics? And vice versa?
  • Capture and harness stochasticity in nature: Address Mumford's call for new mathematics for the 21st century. Develop methods that capture persistence in stochastic environments.

Applying new and/or underutilized mathematics to real-world problems

It is this concept of bringing stochasticity, or randomness, into the fundamentals of mathematics, technology, and science that still has the opportunity to shape future developments of all kinds.

“Stochasticity should be the bedrock of math and science because it would fundamentally change the way we deal with problems in engineering, physics, and other research areas,” Falcone says. “It would have the biggest impact on technology.”

Falcone’s Functor Reality firm continues that sort of leading-edge work. The company’s website describes its role as aiming to apply new and/or underutilized mathematics to real-world problems. “It is founded on the belief that revolutionary advances will emerge from the introduction of more sophisticated mathematical techniques into technology,” Falcone says.

There continues to be an interest in solving the truly toughest math problems. Probably the most notable are the seven Millennium Prize Problems put forth by the Clay Mathematics InstituteinPeterborough, New Hampshire.“The prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude,” according to the group’s website.

It is notable that one of the seven Millennium Prize Problems—the Riemann hypothesis, formulated in 1859—also appears on DARPA's list as well as in Hilbert's address from August 1900.

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That’s one tough problem.

This article/content was written by the individual writer identified and does not necessarily reflect the view of Hewlett Packard Enterprise Company.


What is the world's hardest math problem with answer? ›

For decades, a math puzzle has stumped the smartest mathematicians in the world. x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes."

What are the 7 hardest math problems in the world? ›

Clay “to increase and disseminate mathematical knowledge.” The seven problems, which were announced in 2000, are the Riemann hypothesis, P versus NP problem, Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier-Stokes equation, Yang-Mills theory, and Poincaré conjecture.

Has 3x 1 been solved? ›

After that, the 3X + 1 problem has appeared in various forms. It is one of the most infamous unsolved puzzles in the word. Prizes have been offered for its solution for more than forty years, but no one has completely and successfully solved it [5].

What is one of the hardest math problems in the world? ›

The Riemann Hypothesis, famously called the holy grail of mathematics, is considered to be one of the toughest problems in all of mathematics.

What formula is x3 y3 z3 K? ›

The equation x3+y3+z3=k is known as the sum of cubes problem. While seemingly straightforward, the equation becomes exponentially difficult to solve when framed as a "Diophantine equation" -- a problem that stipulates that, for any value of k, the values for x, y, and z must each be whole numbers.

What is the answer for 3x 1? ›

In the 3x+1 problem, no matter what number you start with, you will always eventually reach 1. problem has been shown to be a computationally unsolvable problem.

What is the square √ 64? ›

The square root of 64 is 8, i.e. √64 = 8. The radical representation of the square root of 64 is √64. Also, we know that the square of 8 is 64, i.e. 82 = 8 × 8 = 64. Thus, the square root of 64 can also be expressed as √64 = √(8)2 = √(8 × 8) = 8.

What math problem Cannot be solved? ›

The Collatz Conjecture is the simplest math problem no one can solve — it is easy enough for almost anyone to understand but notoriously difficult to solve.

What is 1 1 in math? ›

Answer: 1 : 1 ratio means when two quantities are measured or expressed in the same proportion. The ratio a : b helps us to know how much one part of a is equivalent to one part of b. Explanation: When two quantities are taken in the same proportion, they are said to be in the ratio of 1:1.

How far has 3x 1 been tested? ›

Over the years, many problem solvers have been drawn to the beguiling simplicity of the Collatz conjecture, or the “3x + 1 problem,” as it's also known. Mathematicians have tested quintillions of examples (that's 18 zeros) without finding a single exception to Collatz's prediction.

What is the oldest unanswered math problem? ›

But he doesn't feel bad: The problem that captivated him, called the odd perfect number conjecture, has been around for more than 2,000 years, making it one of the oldest unsolved problems in mathematics.

Who made the 3x +1 problem? ›

It is named after mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate.

What is the biggest math problem ever? ›

Mathematicians worldwide hold the Riemann Hypothesis of 1859 (posed by German mathematician Bernhard Riemann (1826-1866)) as the most important outstanding maths problem. The hypothesis states that all nontrivial roots of the Zeta function are of the form (1/2 + b I).

What is the longest math proof? ›

The Stampede supercalculator used for solving the "Boolean Pythagorean triples problem." Researchers use computers to create the world's longest proof, and solve a mathematical problem that had remained open for 35 years. It would take 10 billion years for a human being to read it.

What is the longest math problem ever? ›

Andrew Wiles (UK), currently at Princeton University in New Jersey, USA, proved Fermat's Last Theorem in 1995. He showed that xn+yn=zn has no solutions in integers for n being equal to or greater than 3. The theorum was posed by Fermat in 1630, and stood for 365 years.

What is x3 y3 z3 3xyz? ›

Solution : We know that x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx).

What should be added to x3 3x2y 3xy2 y3 to get x3 y3? ›

Hence, −3x2y−3xy2 should be added to x3+3x2y+3xy2+y3 to get x3+y3.

What formula is a2 b2 c2? ›

The Pythagorean Theorem describes the relationship among the three sides of a right triangle. In any right triangle, the sum of the areas of the squares formed on the legs of the triangle equals the area of the square formed on the hypotenuse: a2 + b2 = c2.

What is 1 2 3 all the way to infinity? ›

For those of you who are unfamiliar with this series, which has come to be known as the Ramanujan Summation after a famous Indian mathematician named Srinivasa Ramanujan, it states that if you add all the natural numbers, that is 1, 2, 3, 4, and so on, all the way to infinity, you will find that it is equal to -1/12.

What is the infinite math equation? ›

x × ∞ = ∞

Why is 3x 1 impossible to solve? ›

Multiply by 3 and add 1. From the resulting even number, divide away the highest power of 2 to get a new odd number T(x). If you keep repeating this operation do you eventually hit 1, no matter what odd number you began with? Simple to state, this problem remains unsolved.

Is √ 64 a real number? ›

Yes, the square root of 64 is a real number.

How to solve √ 100? ›

The square root of 100 is 10. Therefore, 10 √100 = 10 × 10 = 100.

What's the square of 69? ›

Square Root of 69
Square Root of 69In decimal form: ± 8.30662(approx.) In radical form: ±√69
Square of 694761

How is Russian math different? ›

The main difference between Russian Math and the math being taught in schools, they say, boils down to a methodology that emphasizes derivation over memorization—of learning the reasons behind the answers—and a visual approach that helps students “see” the math, and therefore understand it better.

What is the famous math loop? ›

In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist.

What is the most famous unsolved math equation? ›

The Collatz conjecture is one of the most famous unsolved mathematical problems, because it's so simple, you can explain it to a primary-school-aged kid, and they'll probably be intrigued enough to try and find the answer for themselves.

What does [] mean in math? ›

One or both of the square bracket symbols [ and ] are used in many different contexts in mathematics. 1. Square brackets are occasionally used in especially complex expressions in place of (or in addition to) parentheses, especially as a group symbol outside an inner set of parentheses, e.g., .

Why is infinity 1 0? ›

In mathematics, expressions like 1/0 are undefined. But the limit of the expression 1/x as x tends to zero is infinity. Similarly, expressions like 0/0 are undefined. But the limit of some expressions may take such forms when the variable takes a certain value and these are called indeterminate.

What if the power is 0? ›

Any non-zero number to the zero power equals one. Zero to any positive exponent equals zero.

Is 0 an even or an odd number? ›

So what is it - odd, even or neither? For mathematicians the answer is easy: zero is an even number.

What is the 4 2 1 rule math? ›

In anesthetic practice, this formula has been further simplified, with the hourly requirement referred to as the “4-2-1 rule” (4 mL/kg/hr for the first 10 kg of weight, 2 mL/kg/hr for the next 10 kg, and 1 mL/kg/hr for each kilogram thereafter.

Why is 3n 1 a problem? ›

The 3n+1-problem is the following iterative procedure on the positive integers: the integer n maps to n/2 or 3n+1, depending on whether n is even or odd. It is conjectured that every positive integer will be eventually periodic, and the cycle it falls onto is 1 7!

Is crying over math normal? ›

Tears or anger: Tears or anger might signal anxiety, especially if they appear only during math. Students with math anxiety tend to be very hard on themselves and work under the harmful and false assumption that being good at math means getting correct answers quickly. These beliefs and thoughts are quite crippling.

Is there infinite perfect numbers? ›

It is not known whether there are infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes. As a side note, there are names for non-perfect numbers: if the sum of a number's proper factors are less than the number, it's a deficient number.

What if math never existed? ›

Now imagine how different our daily landscape would be if mathematics had never came to be. It would mean no time, no calendars, no buildings, no transportation, no recipes… the list goes on and on. Quite simply, all of the comforts which make our lives what they are today would be no more.

What is the hardest math to learn? ›

Calculus is the hardest mathematics subject and only a small percentage of students reach Calculus in high school or anywhere else. Linear algebra is a part of abstract algebra in vector space. However, it is more concrete with matrices, hence less abstract and easier to understand.

Who created Infinity math? ›

The common symbol for infinity, ∞, was invented by the English mathematician John Wallis in 1655. Three main types of infinity may be distinguished: the mathematical, the physical, and the metaphysical.

What is the most important equation in the universe? ›

E=mc^2. For our first, we'll take perhaps the most famous equation of all. Albert Einstein's 1905 equation relating mass and energy is both elegant and superficially counterintuitive. It says that energy is equal to the mass of an object in its rest frame multiplied by the speed of light squared.

Has anyone solved the Riemann Hypothesis? ›

The Riemann hypothesis, a formula related to the distribution of prime numbers, has remained unsolved for more than a century.

What is the most known math problem? ›

Dr. Wiles demonstrates to a group of stunned mathematicians that he has provided the proof of Fermat's Last Theorem (the equation x" + y" = z", where n is an integer greater than 2, has no solution in positive numbers), a problem that has confounded scholars for over 350 years.

Did Michael Atiyah solve the Riemann Hypothesis? ›

One of the most famous unsolved problems in mathematics likely remains unsolved. At a hotly-anticipated talk at the Heidelberg Laureate Forum today, retired mathematician Michael Atiyah delivered what he claimed was a proof of the Riemann hypothesis, a challenge that has eluded his peers for nearly 160 years.

What is 1 2 3 4 5 all the way to 100? ›

Therefore, the sum 1 + 2 + 3 + 4 + 5 + . . . . . . + 100 = 5050 .

Did Riemann believe in God? ›

Riemann was a dedicated Christian, the son of a Protestant minister, and saw his life as a mathematician as another way to serve God. During his life, he held closely to his Christian faith and considered it to be the most important aspect of his life.

What is the prime number paradox? ›

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.

Who accidentally solved an unsolvable math problem? ›

In statistics, Dantzig solved two open problems in statistical theory, which he had mistaken for homework after arriving late to a lecture by Jerzy Neyman.
George Dantzig
BornGeorge Bernard DantzigNovember 8, 1914 Portland, Oregon, US
DiedMay 13, 2005 (aged 90) Stanford, California, US
12 more rows

What math problem is never solved? ›

One of the greatest unsolved mysteries in math is also very easy to write. Goldbach's Conjecture is, “Every even number (greater than two) is the sum of two primes.” You check this in your head for small numbers: 18 is 13+5, and 42 is 23+19. Computers have checked the Conjecture for numbers up to some magnitude.

What is the oldest math problem? ›

Mathematicians worldwide hold the Riemann Hypothesis of 1859 (posed by German mathematician Bernhard Riemann (1826-1866)) as the most important outstanding maths problem. The hypothesis states that all nontrivial roots of the Zeta function are of the form (1/2 + b I).

What is the 1 million math problem Riemann hypothesis? ›

The Riemann hypothesis – an unsolved problem in pure mathematics, the solution of which would have major implications in number theory and encryption – is one of the seven $1 million Millennium Prize Problems. First proposed by Bernhard Riemann in 1859, the hypothesis relates to the distribution of prime numbers.

Who Solved Riemann zeta? ›

Riemann Hypothesis Solved by Sir Michael Atiyah After 160 Years, He Says.


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