Hilbert's problems facts for kids

In 1900, a famous mathematician named David Hilbert shared a list of 23 big math puzzles. These problems were very important and helped guide what mathematicians worked on for many years. After Hilbert passed away, another problem was found in his notes. Sometimes, people call this Hilbert's 24th problem. It was about finding ways to show if a solution to a problem was the simplest possible.

Out of the 23 problems, three were still unsolved in 2012. Three others were too unclear to ever be fully solved. Six problems were partly solved. Because Hilbert's problems were so important, the Clay Mathematics Institute made their own list of similar puzzles in 2000. They called these the Millennium Prize Problems.

What Were Hilbert's Problems?

Hilbert's problems were a collection of big challenges in mathematics. They covered many different areas of math. Solving them pushed the boundaries of what mathematicians knew.

Why Were They Important?

These problems were very influential. They inspired mathematicians to explore new ideas and develop new tools. Even if a problem wasn't fully solved, trying to solve it often led to important discoveries.

What Happened to the Problems?

Many of Hilbert's problems have been solved over the years. Some have clear answers that most mathematicians agree on. For example, problems like the 3rd, 7th, 10th, and 14th have been fully resolved.

However, some problems have solutions that are still debated. For instance, the solution for problem 18, known as the Kepler conjecture, used a computer-assisted proof. This means a computer helped check the proof. Some people find this controversial because it's very hard for a human to check every step.

A few problems are still unsolved today. These include problem 8, which is the famous Riemann hypothesis, and problem 16. Other problems, like 4 and 23, were described as too vague to ever be fully "solved." Problem 6 is seen more as a physics problem than a math problem.

Here are a few examples of the problems and their status:

• Problem 1: The continuum hypothesis. This asks if there's a "size" of set between integers and real numbers. It was shown to be impossible to prove or disprove using standard math rules.
• Problem 2: Proving that the basic rules of arithmetic are consistent (meaning they don't lead to contradictions). Mathematicians like Kurt Gödel and Gerhard Gentzen made big steps here, but there's still debate if it's fully solved.
• Problem 3: Can any two polyhedra (3D shapes with flat faces) of the same volume be cut into pieces and reassembled into each other? The answer is no, which was proven in 1900.
• Problem 8: The Riemann hypothesis. This is about the Riemann zeta function and prime numbers. It's one of the most famous unsolved problems in math.
• Problem 10: Finding a way to tell if a certain type of equation (a Diophantine equation) has whole number solutions. It was proven in 1970 that no such method exists.

Images for kids

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  David Hilbert, the mathematician who created the list of problems.

See also

In Spanish: Problemas de Hilbert para niños