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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.

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

Unsolved problems are easy to understand the statement of:

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  • $\begingroup$ Additionally: I recently had a big success with using Polya's Conjecture and the number 906,150,257 from the YouTube hashtag #MeagFavNumbers. The YouTube video by SparksMaths was very good. It shows how a really well respected mathematician had a simple, easy to understand conjecture, but turned out to be incorrect but only proven so about 40 years later. $\endgroup$ – ruferd Nov 10 '20 at 13:14
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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 $10$.

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