25 darts land inside a regular-hexagonal dart board with edge length the square root of 3. Show that at least five of the darts land inside the same unit circle.

Solution

Proof by Snail_Erato: Pigeon hole principle. The circle is precisely the circumcircle of each of the 6 triangles in which we can dived the hexagon. if none had more than four darts it could not make more than 24 darts: Q.E.D!

25 darts land inside a regular-hexagonal dart board with edge length the square root of 3. Show that at least five of the darts land inside the same unit circle. https://t.co/T2iegveADP By @MarshallWBuck pic.twitter.com/45UVMTvEYi

— Mirangu (@Mirangu1) July 7, 2023