The original Kakeya problem is a really nice one for students, in that it's easy to understand and visualise, but has an unintuitive answer.

In this sense it's like Borel-Kolmogorov and imho much better than Banach-Tarski which is trotted out frequently, but actually very difficult to understand (both relatively and absolutely).

https://en.wikipedia.org/wiki/Kakeya_set#Kakeya_conjecture

https://en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_parad...