Today I’ll wrap up the current discussion of Target Tiling with a look at how to achieve symmetry. Also, I’ll describe something I call pseudo-symmetry, which is similar to and behaves somewhat like symmetry.
For Target Tiling on a board of width = 5, there are a total of 32 different row patterns (5-cell patterns of filled and empty cells). Of these, there are 8 that are left-right mirror symmetric (they are the same when order of cells is reversed). Last week I illustrated the 6 “non-trivial” symmetric patterns (omitting both the empty pattern and the completely filled pattern). That leaves 24 row patterns that are non-symmetric. These non-symmetric patterns match up into groups of 4 related patterns: each group containing the 2 mirror reflections of one non-symmetric pattern, each occurring twice — once with the center cell filled and once with the center cell empty. So fundamentally there are 6 different asymmetric patterns (each generating a related group of 4 out of the 24 patterns).
The next image illustrates all 6 possible non-symmetric row patterns (with center-cell empty, and omitting the mirror reflections):
Each of these asymmetric patterns (and the variants with reflection and center filled) can be first transformed into a symmetric pattern, and then cleared by a symmetric wipeout using the symmetric macro-operators described in my blog post last week.
I will present an illustration using a single 1-cell pattern, starting with just a single cell in the 2nd row (1 cell left of center). Here is the starting board position:
This non-symmetric pattern can be transformed into a symmetric pattern with just 2 T-piece placements. The first T piece is placed to the right with its stem pointing right, which yields this position:
Now place the second T piece to the left with stem pointing up, yielding this symmetric position (before the filled row clears):
After the filled row clears, the pattern is still symmetric:
This symmetric pattern can be wiped out with the application of symmetric macro-operators as discussed last week.
The macro-level description could be:
Repeat 2 times: DO [ Clear-Center-Hole + Clear-3 ]
This takes a total of 14 moves.
Transforming Pseudo-symmetry into Symmetry
Here is a second example, of clearing a non-symmetric pattern by first transforming it into a pseudo-symmetric pattern (symmetric if rows were superimposed on one another).
To illustrate this, consider the challenge of clearing a T-shaped start pattern (where the T is off-center to begin with):
This position is non-symmetric. There is an easy pattern I discovered for clearing it, though. The first move is placing a T on the left with stem down, achieving a nearly symmetric position (what I call pseudo-symmetry):
The only feature that breaks symmetry here is that the single filled cells in the left and right columns are in different rows. Ignoring this difference, the position is symmetric. It turns out that this can be transformed into an actually symmetric position with 2 moves. The first move is to place a T at far right with stem in so as to fill the hole at the right of the 2nd row.
When the row clears we arrive at this pattern:
Now a 2nd T piece placed at the left (with stem pointing right) yields a true symmetric position:
This symmetric position can be wiped out with just 1 more piece filling the T-shaped hole.
There are many more patterns to learn, but with practice, one builds up skills for clearing all the non-symmetric patterns.
My experience is that symmetric patterns are significantly easier to clear, so it is worthwhile to learn how to achieve symmetry from non-symmetric patterns.
Check out Target Tiling — it’s free, it’s fun, and it’s fantastically challenging!
Instructions on how to download and run my game are on my web page:
Next Week: The Power of Collaboration