Question: Imagine a small box with 2x2x2 dimensions which contains 65 particles floating around. Show that at any point, there are at least 9 particles with a distance no larger than 1.7 between any 2 of them.
My attempt: I understand I must divide the box into 64 even sectors and place a particle in each. Using the pigeonhole principle,
pigeons: particles
pigeonholes: sectors
I am just struggling to prove that there are 9 particles (with distance between any two) no greater than 1.7 apart.
Any help would be appreciated.
Hint: If you take $8$ unit cubes within the cube of dimension $2 \times 2 \times 2$ and take $8$ groups of $8$ particles, the maximum distance that you can maintain between each of those groups is $\sqrt6$ and no two subgroups of those $8$ particles can be farther than $\sqrt3$ from each another without violating distance of $\sqrt3$ with another group. Now the $65$th particle will be at maximum distance of $\sqrt3$ at least from one of the groups. You still need to fill in the gaps with your working.