Australian AMC 2015 - Senior Question 25

Question 25

Two spheres, one of radius 2 and the other of radius 4, have the same centre. What is the edge size of the largest cube that fits between them?

Question illustration

$\frac65$
$\frac13(\sqrt{19}+1)$
$\frac{\sqrt{21}-2}3$
$\frac23(\sqrt{22}-2)$
$\frac{12}5$