Given a container which width is W and height is H, I'd like to fit N squares of maximal size S in it.
Example 1
W = 240
H = 210
N = 7
S = 70
Example 2
W = 200
H = 230
N = 23
S = 40
How would you calculate S from W, H, and N in O(1)?
Assume $W \ge H$ (otherwise swap them). If you could pack them perfectly the area would say $S=\sqrt{\frac {WH}N}$. If you use that side you get $m=\lfloor \frac HS \rfloor$ across the height.
Start with $m+3$ (just a guess for safety) across the height, compute
Side that fits in height $\frac H{m+3}$
Number required in width $n=\lceil\frac N{m+3}\rceil$
Side that fits in width $\frac Wn$ Now decrease the number across the height and recompute the sides It should increase and then start to decrease. Report the maximum.