Many puzzles and games have been proposed
for different polyforms. Some use complete sets, others use
multiple copies of a single piece.
Here are some of the general types of
puzzle which have been invented. Some of these puzzles have been
studied extensively, with complete sets of solutions published
for a given set of pieces. However, the same puzzle with
different pieces may not have been investigated at all. Several
books and papers have been written about the pentominoes and
there is much information available online too, but that does
not mean that you cannot come up with something new.
The shapes which are
possible will depend upon the geometry of the pieces, so where
necessary I will give a couple of examples.
Puzzles using complete sets or copies of the same tile(s)
Puzzle Type 
Description 
Examples 

Geometric Forms 
Construction of simple
geometric shapes. 
Rectangles with polyominoes.
Hexagons with polyiamonds. 

Iconic 
Constructions which
resemble animals, trees, buildings, cars etc. 
Pentomino Zoo 

Portholes 
Constructions containing
the maximum number of unit holes. 
Tetrahex Portholes,
Pentahex Portholes 

Farms or Fences 
Constructions containing
a single hole of maximal area. 


Houses 
Constructions containing
holes of specific shapes. Often the hole shapes are
polyforms of the same set or group. 
Penthouses
with polyominoes.


Paths 
Constructions of the
longest path. Variations include mazes and paths formed from
the negative space. 
Polyomino Mazes 

Replication 
Constructions of copies
of a shape, each using a fraction of the set. 
Sextuplets
. Hexahex
Nonets 

Replication 2 
Constructions of a symmetric
shape from one or more pieces, with the remainder used to
make copies of a different shape. Butterflies, dragonflies,
propellers and snowflakes are variations on this theme. 
Tetrahex Butterflies A,
Pentomino Butterflies 

Topological Tiling 
Tiling the plane,
cylinder, torus, Mobius strip, Klein bottle and projective
plane 
Wallpaper
Patterns 

Solid Surface Tiling 
Tiling the faces of a 3
dimensional solid. Flexominoes and flexahexes are
variations. 
Tiling the faces of a cube
with pentominoes .
Flexominoes .
Flexahexes . 

Puzzles using multiple copies of the same tile(s)
Puzzle Type 
Description 
Examples 

Geometric Forms 
Construction of simple
geometric shapes. Polyominoes which can make a rectangle are
said to be rectifiable. 
Rectangles with polyominoes.
Hexagons with polyiamonds. 

Topological Tiling 
Tiling the plane,
cylinder, torus, Mobius strip, Klein bottle and projective
plane 


Compatibility 
Shapes which can be made from
copies of one tile, and duplicated using copies of another, or
the same tile in a different arrangement. 
Galvagni,
Reid & Plover Figures


Oddities 
Symmetric shapes which can be made from
an odd number of copies of one tile. 
Oddities 





