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.<p>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).<p><a href="https://en.wikipedia.org/wiki/Kakeya_set#Kakeya_conjecture" rel="nofollow">https://en.wikipedia.org/wiki/Kakeya_set#Kakeya_conjecture</a><p><a href="https://en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_paradox" rel="nofollow">https://en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_parad...</a>