What are some famous problems, which are not difficult to understand, for senior high school students

Question

I hope I am asking my question in the right forum.

I am trying to introduce some mathematical problems (Better to be famous in the math community) to a group of senior high school students with a typical background in high school mathematics like (differentiation and applications - basic probability- basic plane geometry- little combinatorics, and graph theory and basic number theory and introductory linear algebra which are common to solve a system of linear equations.)

Actually, my friends and I are trying to introduce mathematics to senior high school students; in between, we would like to present them with some famous math problems in which there were some unsuccessful attempts to solve them, but at last, they've been solved. Or famous wrong conjectures which were thought to be correct. Through this, we wanted to emphasize that even mathematicians at a high level can get wrong, and guessing wrong or failing an attempt shouldn't worry you in studying mathematics

Thank you for helping me. I am also very grateful for other suggestions for our work, besides these.

Answer 1

Some problems that are easy to understand (but not understand the solution to):

• Fermat's Last Theorem
• Four Color Theorem
• Solving the Cubic and Quartic Equations
• The impossibility of the Quintic Equation
• Archimedes' Cattle Problem
• Sudoku problems (17 being the minimum amount of moves needed in order to be solvable)
• Theorema Egregium
• Constructing a 17-gon
• Every Rubik's Cube can be solved in 20 or less moves
• 1988 IMO Problem #6

Unsolved problems are easy to understand the statement of:

• Does an odd perfect pumber exist?
• Twin Prime Conjecture
• Is there a perfect Euler Brick?
• P vs NP
• Collatz Conjecture
• Goldbach Conjecture
• Inscribed Square Problem
• Moving Sofa Problem
• Beal Conjecture
• The Congruent Number Problem

Answer 2

Euler conjectured that there are no Graeco-Latin squares of size congruent to $2$ mod $4$. He sought a proof of this for size $6$ in order to prove that the $36$ officers problem had no solution, but was unable to find one. It was later found that he was right for size $6$, but not in general. For example, there exists a Graeco-Latin square of size $10$.