There is a nice pentomino puzzle by George Sicherman, presented on Odette de Meulemeester's site [in Dutch]: build a shape, which contains a piece surrounded by a sanitary belt (hence the name, Quarantine puzzle). Unique solutions exist for all pentominoes; i.e., there are shapes satisfying the above conditions and having a unique tiling.
The idea is simple and clear, there is no reason not to try it on hexiamonds.
For hexiamonds, we add an extra condition: the outline of the shape has to be convex. That way not all hexiamonds can be quarantined: this is impossible for three of them (1/4 of the whole set). As for unique solutions, they exist only for six pieces (1/2 of the set). The impossible cases have been proved by George Sicherman, who also is the author of most of the solutions listed below.
| Unique solution | Multiple solutions | |
|---|---|---|
| Total area of 94 cells. | ||
| Total area of 94 cells, "Egg with embryo" shape. | 11 solutions. | |
2 solutions. | ||
| Total area of 93 cells | 2 solutions. | |
| Total area of 90 cells | 7 solutions. |
George Sicherman reports at least one solution of the analogous puzzle with tetrahexes.
The puzzle is meaningless for bigger sets, as shapes are too easy to find and it is extremely unlikely that unique solutions exist. The only way the original idea could be adapted to bigger sets, is by enlarging the quarantined region or the width of the sanitary belt around it. A few examples follow with heptiamonds.
| Double shape, simple belt. | Double shape, double belt. | Simple shape, triple belt. |
When surrounded with a triple belt, 10 of the heptiamonds lead to a shape with a nice symmetry, as shown below.
Things are less beautiful with 12 other pieces.