Heesch's problem asks how many times a shape can be completely surrounded by copies of itself without being able to tile the entire plane. Equivalently, it asks for the maximum finite Heesch number.
Heesch's Problem
See also
Heesch NumberExplore with Wolfram|Alpha
References
Eppstein, D. "Heesch's Problem." https://ics.uci.edu/~eppstein/junkyard/heesch/.Referenced on Wolfram|Alpha
Heesch's ProblemCite this as:
Weisstein, Eric W. "Heesch's Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HeeschsProblem.html