Dlanod is building huts (the end)

The plan is to place the huts in a "honeycomb pattern," in $p$ rows with $q$ cottages each, as shown in the attached figure, where $p=6$ and $q=7$ . We can place the huts onto a rectangular plot with Length $L=6q+3$ and width $W=6+3\sqrt{3}(p-1)$ . For our construction, we will choose the largest $p$ such that $W<L$ , namely, $p=\lceil {\frac{2q-1}{\sqrt 3}}\rceil$ , so that the huts will fit on a square plot with side length $L=6q+3$ , with a little extra yard space left for some "luxury" huts in the top and/or bottom rows. The amount of land we need per cottage is $\frac{(6q+3)^2}{pq}=\frac{(6q+3)^2}{q\lceil {\frac{2q-1}{\sqrt 3}\rceil}}$ The limit of this expression, as we let $q$ go to infinity, is $18\sqrt 3<31.2<10\pi$ , so that the amount of land we need per cottage will be $<10\pi$ for sufficiently large $q$ , namely, for all $q\geq 200$ and for some smaller ones.

Once again, $\boxed{\text{Dlanod is right}}$ . The resort will be huge, of truly Dlanodian proportions.

Regardless of the details of the arrangement, Dlanod will need to buy at least $18 \sqrt 3 > 31.17$ square meters per hut, though.

Dlanod's architect is saying that the density of all circle packings in a square have density that never exceeds $90$ %.

Our friendly website tells us that the first circle packing with a density of greater than $90$ % involves $8281$ circles, with the next involving $9503$ . As might be expected, this packing is almost hexagonal.