# Smallest region that can contain all free $n$-ominoes.

A nine-cell region is the smallest subset of the plane that can contain all twelve free pentominoes, as illustrated below. (A free polyomino is one that can be rotated and flipped.)

A twelve-cell region is the smallest subset of the plane the can contain all $35$ free hexominoes. What is the smallest region of the plane that can contain all $108$ free heptominoes (shown below)? All $369$ free octominoes? Also, is there an existing OEIS sequence for this? If not, I'll add one once there is a bit more data. (The sequence begins $1, 2, 4, 6, 9, 12, \cdots$.)

• If $a(n)$ is the minimum size region covering all $n$-ominoes, then I've constructed examples to show $a(7) \leq 17$, $a(8) \leq 22$ and $a(9) \leq 26$. – Peter Kagey Jun 26 '18 at 20:50
• You can make the bound $a(n) \leq n \cdot \left\lceil \dfrac{n}{2} \right\rceil$. It's not very good though. – Jeffery Opoku-Mensah Jun 26 '18 at 21:19
• For the heptomino case, a search over all shapes containable in a $4 \times 7$ bbox only return regions of size $17$. I don't know how to extend the emulation to $5 \times 7$ bbox but this is a strong indication that $a(7) = 17$. – achille hui Jun 26 '18 at 21:40
• if I didn' make any mistake, $a(8) \le 20$. example container 00000111-00000111-00111111-11111111 – achille hui Jun 26 '18 at 21:49
• Some upper bounds$$\begin{array}{rcl} n & a(n)\le & \text{example container}\\ \hline 7 & 17 & \small\verb/11-111-11111-1111111/\\ 8 & 20 & \small\verb/111-111-111111-11111111/\\ 9 & 27 & \small\verb/111-111-11111-1111111-111111111/\\ 10 & 31 & \small\verb/111-1111-111111-11111111-1111111111/\\ 11 & 38 & \small\verb/111-1111-1111-1111111-111111111-11111111111/\\ 12 & 43 & \small\verb/1111-1111-11111-11111111-1111111111-111111111111/ \end{array}$$ – achille hui Jun 28 '18 at 20:33