Interior Hole Problems

There are numerous problems in which a set of polyforms is used to enclose a hole or holes of a particular type.  You can use any polyforms, but the 8 pieces of the EasterEgg puzzle are used for the following four examples:

 

1. Largest Interior Hole.  Try to enclose the largest single hole possible which can be of any shape.  In the example below, the hole is 58 hexagons big.  Can you find a bigger hole?

farm

 

2. Farms.  Try to enclose the largest single hole of a specified shape (inside farm), or use a frame of a specified shape (outside farm).  The following example is an inside farm, enclosing a hexagon:

hexagonal inside farm

 

3. Swiss Problem.  Try to enclose the largest number of single unit holes (like a piece of Swiss cheese).  These can be any shape, but note that the following example is symmetrical.  Can you find a solution with more than seven holes?

swiss cheese problem

 

4. Narrow Passage.  Try to enclose the largest hole which is no wider than one unit at any point (proposed by Anneke Treep).   This example has a passage 16 units long, but it is possible to make passages at least 18 units long with these pieces.

narrow passage

 

For further reading about Interior Hole problems, see the World Game Review, issues 9 and 11.