Atom Frenzy

This problem makes good use of the pigeonhole principle. $\\$ Since there are $37$ atoms and the height of the cylinder is $6$ units, if we divide the cylinder into six segments of unit height, then there will always be one segment which contains at least $7$ atoms. $\\$ Now if we consider that segment and the minimum number of atoms it will have ( $7$ ) and further split the segment into six sectors, then there will always be one sector in the segment that contains at least $2$ atoms. $\\$ The maximum distance within those two atoms will then be: $\\$ (let $x$ denote the distance) $\\$ $x^2 = (radius)^2 + (height)^2$ $\\$ $x^2 = 2^2 + 1^2$ $\\$ $x^2 = 5$ $\\$ $x = \sqrt{5}$ $\\$ $x \approx 2.24$ $\\$

Note: The maximum horizontal distance is the radius because the angle measure is $60^{\circ}$ (think about it)