For example, the surround number of a cross is 2, the tiling below and its mirror image:
Which positive integers are the surround numbers of some shape? In particular, is 3 a surround number? What are the surround numbers of the small polyominoes? What is the surround number of an m x n rectangle? For what other families of polyominoes can we compute surround numbers?
Owen found a general formula for the surround number of an m×n rectangle. It's pretty complicated. An n×n square has surround number 2n^{2}1. For n ≥ 2, an n×1 rectangle has surround number (n^{4}+22n^{3}+105n^{2}56n8) / 4. For n ≥ 3, an n×2 rectangle has surround number (n^{4}+32n^{3}+344n^{2}+768n+400) / 16 if n is even and (n^{4}+32n^{3}+278n^{2}+656n+361) / 16 if n is odd.
Here are some polyominoes with surround number 3, and the people who found them:
Polyomino  Author 

Erich Friedman  
Brendan Owen  
Andrew Bayly  
Joe DeVincentis 
Here are the surround numbers of some small polyominoes:
Polyomino  Surround Number 

1  
123  
7555  
361  
7  
161  
13355  
2794  
778 
Here are the smallest polyominoes with a given surround number. These are all due to Brendan Owen.


This sequence of values of the smallest area polyomino to have surround number n: 7, 1, 5, 9, 8, 9, 9, 4, 8, 7, 9, 9, 9, 9, 8, 9, 6, 9, 8, 8, 9, 8, 8, 7, 9, 8, 7, 8, 8, 9, 8, 8, 9, 9, 8, 9, 9, 9, 9, 8, 9, 6, 8, 9, 10, 9, 9, 8, 6, 7, 8, 8, 8, 9, 8, 9, 9, 9, 10, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 7, 8, 8, 9, 9, 7, 8, 7, 9, 9, 9, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 8, 9, 9, 9, . . . is now sequence A047875 of the Encyclopedia of Integer Sequences.