It lately occurred to me that there are concepts that I use (and see used by others) in creating variations on polyform puzzles that I haven’t seen explained very thoroughly, and it might be helpful if I used this space for just that purpose.
Some polyomino puzzles using symmetry variations
The first of these is the use of different kinds of symmetry in defining the set of pieces used in a puzzle. (I touched on this in my post on rectangularcell pentominoes.) Normally, all combinations of rotations, translations, and reflections of a polyomino in a grid are considered to be equivalent. Leaving aside translations for the moment, the possible rotations and reflections of a polyomino are equivalent to the group of symmetries of a square. We can find variations on polyominoes by restricting the allowed symmetries to subgroups of that group. For example, the onesided polyominoes are the result of allowing only rotations, not reflections. Rhombic cell pentominoes (which Kadon sells) allow 180° rotations, plus diagonal reflections. My Agincourt puzzle allows only reflections over vertical axes, assuming that the arrows are pointing vertically. Notice that it doesn’t matter which direction the arrows point as long as they point in the same direction; this suggests that what we are interested in isn’t symmetry subgroups per se, but classes of subgroups where two subgroups that are related to each other by symmetries of the square are equivalent.
What are all of the possible variations with different allowed transformations? We can generate a representative subgroup of every class by using some combination of reflection over a particular axis parallel to the grid, a particular diagonal axis, and 90° and 180° rotations. Here’s a chart of the symmetry variations this produces.

Polyomino Type 
Reflection 
Rotation 
# of Symmetries 

Free 
Either 
90° 
8 

Parallel (a.k.a. Rectangular) 
y axis 
180° 
4 

Diagonal (a.k.a. Rhombic) 
x=y 
180° 
4 

Onesided 
None 
90° 
4 

Oriented Parallel 
y axis 
None 
2 

Oriented Diagonal 
x=y 
None 
2 

Polar Onesided 
None 
180° 
2 

Fixed 
None 
None 
1 
I chose the above terminology for the types (after keeping “free”, “onesided”, and “fixed” as established terms) in order to build in some helpful mnemonics. The types with four symmetries have short names. The types with two symmetries have longer names based on the names of the types whose symmetry groups their symmetries are subgroups of. The odd duck here is “polar onesided”, which is a subgroup of all of the larger symmetry groups, but putting “onesided” in its name makes the types with two symmetries nicely echo the names of those with four.
Here’s a chart of the number of polyominoes of each type for a given size:
Polyomino Type 
1 
2 
3 
4 
5 
6 
7 
OEIS # 
Free 
1 
1 
2 
5 
12 
35 
108 
A000105 
Parallel 
1 
2 
3 
9 
21 
68 
208 
A056780 
Diagonal 
1 
1 
3 
7 
20 
62 
204 
A056783 
Onesided 
1 
1 
2 
7 
18 
60 
196 
A000988 
Oriented Parallel 
1 
2 
4 
12 
35 
116 
392 
A151525 
Oriented Diagonal 
1 
1 
4 
10 
34 
110 
388 
A182645 
Polar Onesided 
1 
2 
4 
13 
35 
120 
392 
A151522 
Fixed 
1 
2 
6 
19 
63 
216 
760 
A001168 
(The odd entries for the polar onesided polyominoes track those for the oriented parallel polyominoes exactly for several terms, before eventually diverging. There are 4998 9ominoes for both, and 67792 polar onesided, and 67791 oriented parallel 11ominoes. It seems unlikely that this is a coincidence. Does anyone know why this occurs?)
These types can be realized geometrically by replacing square cells in a planar tiling with cells with the appropriate symmetry. Another way they can be realized is by keeping the cells square and marking them with a figure with the appropriate symmetry. This is essentially what I did by cutting arrow shaped holes in the Agincourt pieces. The latter method allows the possibility of mixing different symmetry types in the same tiling. I don’t believe I’ve seen such a problem before, so let me be the first to fill what may be a much needed gap:
Problem #28: Tile a 6×6 square with the oriented parallel, oriented diagonal, and polar onesided trominoes. No tromino should touch another of the same type.
With these symmetry subgroup based polyform variations in mind, any type of polyform on a square grid can be transformed into an entire family of polyforms. In particular, polysticks would reward exploration in this light, which does not seem to have occurred yet. A similar analysis to the one above can be made for symmetry based variations of polyiamonds and polyhexes. Bringing translation symmetry subgroups into the picture leads to things like checkered polyominoes. I may get to these in later posts; this one was getting long enough that I needed to wrap it up.
I should note that Peter Esser’s pages on polyforms cover these variations, and that his polyomino solver program can work with any of the 8 symmetry types (but not with mixed types.) (It is, sadly, a Windows binary, but I’ve been able to make it work under Wine on Linux.)