Evolving Polyomino Puzzles
Daniel Ashlock, Senior Member IEEE, Lauren Taylor
Abstract—A polyomino puzzle is a collection of polyominos that
can be joined to make a simple shape. The game Ten-Yen was one
of the first of these. It has ten polyomino pieces that could be used
to make a 6x6 square in a variety of ways. In this study we define
representations and fitness functions for generating polyomino
puzzles as well as developing a simple solver to compare the
evolved puzzles. The solver can be used to approximate the
number of solutions and hence the relative difficulty of the
puzzles. Two types of fitness functions are compared, the second
of which was developed to deal with scaling issues that arose with
the first. A parameter study on the algorithm is performed and it
is found that simply penalizing bad results is more effective than
parameter tuning. This study concludes by discussing potential
puzzle variants.
like all of the pieces to be distinct polyominos, as in Ten-Yen,
and that we would like the number of pieces to be relatively
large (equivalently, we would like the pieces to individually
be relatively small).
I. I NTRODUCTION
Polyomino puzzles are sets of polyominos that can be joined
to make a simple shape. Figure 1 shows a polyomino puzzle
called Ten-Yen that uses ten distinct polyominos in three
colors. There are a large number of ways to assemble the
pieces and it is possible to declare secondary challenges such
as making all the pieces that share a color form a contiguous
region, or surrounding one color with another. An example of
such a solution appears in Figure 2. If a puzzle of this sort
has one solution then it typically has many, but the number of
solutions varies substantially from puzzle to puzzle.
Fig. 1. The Ten-Yen game, patented by Multiple Products Corporation in
1950.
In this study we develop a representation and compare
several fitness functions for creating polyomino puzzles. Our
fitness criteria are based on the assumption that we would
Daniel Ashlock and Lauren Taylor are with the Department of Mathematics and Statistics at the University of Guelph,dashlock@uoguelph,
[email protected]
The authors thank the Canadian Natural Science and Engineering Research
Council of Canada (NSERC) for supporting this work.
Fig. 2. A solution to Ten-Yen with contiguous regions for each color.
The on-line encyclopedia of integer sequences (OEIS)
(https://oeis.org/) is a catalogue that lists sequences of integers
that are interesting, together with the reason or reasons they
are interesting. The sequence A000105 in the OEIS counts
the number of polyominos with n squares. The prefix of this
sequence is 1, 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, . . . telling
us there is one polyomino with no squares, one with one, one
with two, through 4655 distinct polyominos with ten squares.
The relatively small number of small polyominos constrains
the design of polyomino puzzles.
In this study we develop a simple string representation for
evolving polyomino puzzles. The challenging part of finding
a system for evolving these puzzles is the fitness function.
The representation specifies a set of polyominos that cover
a square, but we must avoid having repeated polyominos and
also trivial solutions. The initial design for the fitness function
forbids (by awarding minimal fitness) a polyomino with one
piece - a design that is clearly not a challenging puzzle. As
we will see, it is not too difficult to write fitness functions that
reward piece diversity, but neither is it trivial. Both types of
fitness functions designed in this study had large, low-quality
local optima.
The remainder of this study is structured as follows. Section II looks at other material on polyomino puzzles and
puzzle solving with evolutionary algorithms. In Section III
we explain the representation for evolving polyomino puzzles
and a shotgun solver is also outlined for sampling solutions.
Section IV gives the experimental design for testing the puzzle
evolver. Section V presents and discusses the results, including
the comparison of evolved puzzles, while Section VI draws
conclusions and outlines next steps.
II. P OLYOMINO P UZZLE AND E VOLUTIONARY P UZZLE
C ONSTRUCTION
An excellent introduction to the combinatorics of polyominoes as well as a large number of puzzles and problems can
be found in [3]. A premier source of polyomino based (and
other) puzzles is Kadon games 1 , including a modern edition
of the game Ten-Yen. Apps for puzzles like the ones evolved
in this study are also available on Google Play 2 .
A general survey on automatic content generation, which
includes the generation of puzzles, appears in [9]. Evolutionary
algorithms have been used to generate, solve, and rate the
difficulty of Sudoku puzzles [6]. A large number of publication
use evolution on Sudoku, e.g. [5], but the primary goal is
solving rather than generating the puzzles. Mazes are another
type of puzzle generated using evolution. Multiple different
sorts of mazes appear in [1]. An interesting application that
uses a reaction-diffusion representation is used to evolve mazes
from images in [10]. In general it is far more common for
evolution to be used to solve puzzles than generate them.
so that the actual chromosome for the configuration,
sans color, for the arrangement shown in Figure 1 is:
302110212302100230221002020210110110
To complete the description of the representation, the variation
operators used are two point crossover and multi-point mutation. The mutation operator chooses between 1 and M N M
positions in the chromosome and replaces them with new
values. The parameter M N M specifies the maximum number
of mutations.
A. Fitness Functions
Two fitness functions were used. The first one works well
for 6 × 6 puzzles but produces problematic results when the
puzzle size is increased to 10 × 10. The second function was
implemented to address the problems that arose for larger final
shapes. The software used to build puzzles has an equality
test for polyominos that works in spite of their flip or rotation
status. Figure 4 shows the eight forms of one of the TenYen pieces. A piece can have one, two, four, or eight forms,
depending on which rotations and flips in the dihedral group
D4 preserve or modify the squares the piece is occupying [4].
III. R EPRESENTATION AND F ITNESS
Polyomino puzzles are represented by decomposing the
final shape, e.g. a 6x6 square, into polyominos with local
connectivity specifiers. This is done by taking the individual
squares of the shape and determining which other squares they
are connected to. A loci with a value of zero generates no
connections. A one connects a square to the one to its right, a
two connects a square to the one below it, and a three connects
to the squares both to the right and below. This permits any
set of possible connections to be made.
An important semantic issue, given that some squares have
no square to their right or below them, is that a connection that
goes off the edge of the final shape is ignored. An example
of a gene that induces the polyominos in Figure 1 is given in
Figure 3. The reader is invited to check its correctness.
3
2
1
2
0
1
0
1
0
2
2
1
2
2
0
1
0
0
1
3
2
0
2
1
1
0
3
0
1
1
0
2
0
2
0
0
Fig. 3. This gene in the connectivity representation yields the polyominos in
Figure 1, in the arrangement they appear in in that figure.
The chromosome structure used in evolution is unpeeled from the template of the final shape by rows
1 http://www.gamepuzzles.com/
2 https://play.google.com/store/apps?hl=en
Fig. 4. Shown are the eight possible orientations of the L-shaped Ten-Yen
piece.
Once it is possible to test for equality of polyominos, a chromosome C is parsed to determine the connections it specifies
and then the connected regions, in the form of polyominos,
are found. If the number of times the ith polyomino occurs is
rk , the size of the ith polyomino is si , and
1 all polyominos are distinct
U=
(1)
0 otherwise
then the two fitness function are
1
f1 (C) = U + P 2
si
(2)
and
f2 (C) = U + P
s2i
1
P
+ (rk − 1)2
(3)
Note that the k index is over the polyominos for which
at least one instance occurs, while the index i is over all
polyominos no matter if they are distinct or not. We term the
first fitness function the simple fitness function and the second
the diversity penalized fitness function.
Fig. 5. An example of a 10x10 puzzle with excellent piece size and very bad
piece diversity. Note the large number of single-square polyominos.
Fig. 6. An example of a 10x10 puzzle with unique pieces but an insufficient
number of pieces.
The value U is simply the reward for having all polyominos
distinct from one another. The fractional terms require more
explanation. The term
X
s2i
At the other extreme, the solution shown in Figure 6 gains
the uniqueness penalty, placing its fitness in the range (1,2)
but has a polyomino with 99 squares, meaning that it has the
lowest possible fitness in the range (1,2), given that singlepiece solutions are not permitted. Several approaches were
used to discourage this behavior and the simplest - detecting
this local optima and forbidding it by awarding it a fitness of
zero - turned out to be the best. The other solutions for this
issue appear in the experimental design and results sections.
is the sum of the squared sizes of all polyominos that appear.
Squaring the terms causes large polyominos to contribute
disproportionately to the total; since the term is in the denominator of both fitness functions, this means that a puzzle
with smaller pieces will have a smaller denominator and hence
a higher fitness.
The term
X
(rk − 1)2
is zero if all the polyominos are distinct. Repeated polyominos
generate a positive values and, because the term is in the
denominator, it functions as a fitness penalty. The U term and
the contribution of the squared rk − 1 terms are somewhat
redundant, but serve different functions. The U term makes a
puzzle with unique pieces better than one without them. The
rk − 1-based terms create a fitness gradient that incrementally
rewards eliminating duplication.
If individual polyominos are not unique, the fitness values
are in the range (0,1), otherwise they are in the range (1,2).
Both fitness functions consider having unique polyominos to
be more desirable than keeping the size of individual polyominos small. The motivation for the second fitness function
is exemplified by the puzzle in Figure 5. This solution fails
to win the reward U for unique polyominos. If no member
of the population gains U then the most fit puzzle is one that
only uses single squares, something the solution in Figure 5 is
approximating fairly well. The penalty term involving rk − 1
in the second fitness function acts to discourage this type of
solution.
B. Exhaustive Enumeration and Sampling of Solutions
Algorithm 1 is an exhaustive enumerator. If we have a
covering of a simple shape by polyominos (a solution) then,
in any fixed scanning order of the individual squares, there is
an order of the current orientations of the polyominos in the
solution so that, when place in the first position where they fit,
they reconstruct that solution. Thus, if we exhaust all orders
of presentations of polyominos in all possible orientations, we
will enumerate all possible solutions.
Algorithm 1: Input: A set P = {p1 , p2 , . . . , pn } of polyominos and a shape S to cover.
Output: The number of possible ways to cover the shape.
Details:
For all possible orders of polyominos
Using the current order
Compute number of possible orientations for each
Initialize counter to zero
Label:Rectop
If(no polyominos)
increment counter
return
Else
For all polyominos p
For each orientation pO
Scanning by rows, if pO fits
put pO in the first place it fits
V. R ESULTS AND D ISCUSSION
Recursively call at Rectop with remaining polyominos
For purposes of discussing results, a failure is a puzzle that
Remove pO
does
not have unique pieces, a trivial solution is a puzzle with
End Scan
two pieces, and a good solution is one that has many unique
End For
End For
pieces. The case of a puzzle having a small number of unique
End For
pieces does not arise.
Algorithm 1 is O(n!) where n is the number of polyominos since the number of orientations is at most 8 it does not affect
the big-oh order of the algorithm. Even for the 6x6 puzzles
this algorithm had an unacceptable runtime so it was modified
to randomly sample polyomino orders and report the number
of solutions relative to a sample size.
In [2], the authors show that polyomino puzzles of the sort
generated in this study are NP-hard; this means that while
enumeration or sampling may work well for the relatively
small puzzles generated here, more sophisticated solvers may
be required for larger puzzles.
A. Results for the 6x6 puzzles.
IV. E XPERIMENTAL D ESIGN
The baseline evolutionary algorithm used to create puzzles
in this study is steady state [8] operating on a population of
1000 genes with M N M = 3 using single tournament selection with tournament size 7 as the model of evolution. This
model of evolution selects a set or tournament of population
members, copies the two most fit over the two least fit, and
then applies the crossover and mutation operators to the copies.
The size of the tournament controls the degree of selection
pressure, with smaller tournaments yielding lower selection
pressure. The baseline algorithm is run for 100,000 instances
of tournament selection, called mating events. A parameter
study, varying population size, the M N M parameter, and
the choice of fitness function is performed. For each set of
parameters and fitness function tested, 30 independent runs of
the algorithm are performed. An example of puzzles resulting
for the experiment with default parameters appears in Figure
7.
The experiments performed were as follows; experimental
parameters not explicitly listed are set to default values. Four
experiments using the simple fitness function on the 6x6
version of the problem with M N M set to one of 1, 3, 5,
or 7. Next a run for the 10x10 problem was performed, first
using the default parameters and simple fitness. The results
of this experiment were poor, motivating the development of
the diversity penalized fitness function. The experiment was
re-run using the new fitness function, and a large number of
2-piece solutions were located like the one shown in Figure
6. A parameter study was then run using 1,000,000 mating
events and population sizes of 3000, 5000, 7000, and 9000
with diversity penalized fitness for the 10x10 problem. Finally
experiments were run for the 10x10 problem for each fitness
function, but awarding solutions with two pieces a fitness of
zero - in effect prohibiting solutions of the type given in Figure
6.
Fig. 8. Results of the mutation parameter study for the 6x6 puzzle problem
using the first fitness function. All runs located a global optima - this figure
shows the distribution of the fraction of evolutionary time required to do so.
Four sets of runs were performed for the 6x6 puzzles with
different mutation rates. In all four experiments, every run
found a good puzzle that was globally optimal. This meant
that the parameter study on mutation, the results of which
appear in Figure 8, is somewhat moot. Rather than comparing
these runs on their fitness, we compared them on their time
to locate a global optima. The result is simple - a smaller
mutation rate located global optima sooner, but, since all the
mutation rates tested yielded 30 global optima, this is a soft
parameter choice.
We now need to explain why the optima located were
global. In the introduction the number of distinct polyominos
of small sizes were given. Every single optima located in these
experiments had polyominos of size 1, 2, 3, 4, 4, 4, 4, 4, 5,
5. Fitness rewards smallness and so this configuration uses all
five polyominos of size 4 but skips one with size three. The
constraint that there must be pieces with a total area of 36
means that, if we want to use both polyominos of size three
we get a piece size spectrum of 1, 2, 3, 3, 4, 4, 4, 5, 5, 5.
Exhaustive enumeration (calculations not shown) shows that
all other configurations are worse, so the optima found have
the best possible fitness for the 6x6 case.
This means that the algorithm presented in this study is
more than equal to the 6x6 problem. An experiment was
performed where any global optima was awarded a fitness
Fig. 7. The puzzles located using default algorithm parameters for the 6x6 puzzle problem with the simple fitness function.
of zero and the algorithm proceeded to locate 30 different
solutions exhibiting the second assortment of piece sizes given
above. This result motivated the “simply do not allow two
piece solutions” experiments for the 10x10 problem, which
had a problem with locating trivial 2-piece solutions.
The fact that the number of distinct polyominos is known
means we can also compute the number of global optima for
the 6x6 problem. There is a single polyomino of sizes 1 and
2, two of size 3, five of size 4, and twelve of size 5. This tells
us that the number of global solutions is:
2
5
12
1×1×
×
×
= 132
1
5
2
For the next most fit set of sizes this total is:
5
12
2
×
×
= 2200
1×1×
3
3
2
which suggests that the algorithm was doing well to always
locate the rarer global optima. The motivating puzzle Ten-Yen
is one of the 2200 secondary optima.
B. Relative Difficulty of 6x6 Puzzles
Table I shows the three replicates of the solution-sampler
described in Section III-B for each of the puzzles shown in
Figure 7. Three replicates were run for each puzzle to get
an idea of the sampler’s variability. The variation between
replicates is much smaller than the variation between puzzles,
which certifies that the sampler is capturing the relative
number of solutions to the puzzle well.
The evolutionary algorithm, when evolving a puzzle, certified that it found a puzzle by locating a gene that specified
one solution to the puzzle. The sampler picks 1000 different
orders of the pieces and then, for each order, exhaustively
finds all solutions which place the pieces in that order, using
all their potential orientations. The variability of this sampler
is not simple to compute due to the confounding effect of the
polyomino orientations, so three replicates were used.
The results in Table I demonstrate that there is considerable
variation in the number of solutions each puzzle has. It is
intuitive that a puzzle with more solutions is one that an
inexperienced player would have an easier time solving, so we
used the number of solutions as a surrogate for the difficulty
of the puzzles. Thus, puzzle difficulty varies considerably. A
similar study was not performed for the 10x10 puzzles because
its computational cost was quite high.
TABLE I
S HOWN ARE THE NUMBER OF SOLUTIONS FOUND IN 1000 SAMPLED
ORDERS ON 6 X 6 PUZZLES USING ALL ORIENTATIONS OF EACH
POLYOMINO - ONE ORDER MAY FIND MULTIPLE SOLUTIONS BASED ON
POLYOMINO ORIENTATION . T HE PUZZLES BEING SAMPLED ARE THOSE
SHOWN IN F IGURE 7. F OR EACH PUZZLE , THREE REPLICATES OF THE
STOCHASTIC SAMPLER WERE PERFORMED .
Puzzle
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Rep 1
14930
7882
16162
13408
4400
13482
5104
7841
9533
5729
5368
15226
8390
13931
9538
7726
27963
29083
8541
8180
3124
10579
15549
21262
7986
13377
22968
10782
25334
15292
Rep 2
15204
8231
16260
14087
3708
13576
5367
8061
9397
5255
4740
14026
7939
14715
9324
8753
27279
29240
7922
7764
3113
11413
15569
22237
8224
15268
20036
10738
27793
15154
Rep 3
14431
7773
14973
13877
3577
14321
5833
8479
9703
5287
5564
14794
8293
15202
9803
7841
28574
27555
7677
7726
3570
11685
15354
20246
7947
14485
21426
9899
27734
14974
C. Results for the 10x10 puzzles
Since the problem of locating 6x6 puzzles proved relatively
easy, 10x10 was chosen as a problem case sufficiently difficult
for meaningful experimentation. The results of experiments on
the 10x10 version of the problem are given in Table II. These
experiments attempted to address two problems, the location
of puzzles that failed to have all their pieces distinct and the
problem of finding trivial two-piece puzzles. The simple fitness
function located no good (non-trivial) puzzles and failed at
achieving piece diversity five times in thirty.
Two techniques were tried to address this problem, using
the diversity penalized fitness function and simply returning
zero fitness on trivial puzzles which Table II lists as trivial
forbidden. Changing the fitness function has a mixed outcome,
yielding 14 more good puzzles but also more than tripling
the number of puzzles that failed to achieve piece diversity.
Increasing the population size solved this problem - which is
a low cost option since the number of mating events was held
constant.
Superior to increased population size was the modification
of the fitness function to harshly penalize trivial puzzles. This
caused the simple fitness function to achieve much better
performance - something that is a bit puzzling as the number
of non-diverse puzzles dropped to 1 from 5 - but this may have
been simple luck. Implementing Div. Pen fitness in conjunction
with forbidding trivial puzzles yielded 30 good puzzles, as
did using populations of 7000 and 9000 puzzles, so two
different solutions to the performance problems of the original
algorithm both worked well.
TABLE II
R ESULTS OF THE POPULATION SIZE STUDY AND STRICT PENALTY RUNS
FOR THE 10 X 10 PROBLEM .
Fitness
Function
Simple
Div. Pen.
Div. Pen.
Div. Pen.
Div. Pen.
Div. Pen.
Simple
Div. Pen.
Population
Size
1000
1000
3000
5000
7000
9000
1000
1000
Fail
5
16
1
2
0
0
1
0
Trivial
25
0
0
0
0
0
0
0
Good
0
14
29
28
30
30
29
30
Trivial
Forbidden
no
no
no
no
no
no
yes
yes
If we look more closely at the impact of changing the
population size on the types of puzzles that are located, we see
that in at least one sense the smaller populations, excepting the
smallest, are better. Figure 9 shows the distributions of final
fitness values for the five experiments that changed population
size. The smallest population failed to find diverse puzzles far
too often, an effect that dominates the display in the left panel
of Figure 9. The zoom onto the other four runs in the right
panel shows that, while increasing population size decreased
the number of diversity-failures, it also decreased final fitness.
The good solutions in the runs with smaller population were
managing to have diverse sets of smaller pieces - one of the
goals embedded in the fitness function.
Fig. 9. Shown are box-plots of the distribution of best fitness values for the
population-size parameter study on the 10x10 problem using Div. Pen. fitness.
The top panel shows all five sets of runs while the bottom panel shows the
part of the upper range for the last four sets of runs chosen to make the
variation in fitness comparable.
VI. C ONCLUSIONS AND N EXT S TEPS
This study demonstrated a representation for evolving polyomino puzzles that had no problem creating globally optimal
6x6 puzzles, but required some tuning and a better fitness
function to create acceptable 10x10 puzzles. A sampling
algorithm demonstrated that the different evolved 6x6 puzzles
had substantial variation in their hardness. In this study each
sampling of the number of solutions of a given puzzle, for a
fixed number of greedy solution attempts, was replicated three
times because the sampling strategy was complex enough to
make it unclear how uniform the sampling was. This sampling
Fig. 10. The 10x10 puzzles located with the Dev. Pen. fitness and with 2-piece solutions forbidden.
study demonstrated acceptable replicability of the numbers of
solutions found across the three replicates to give confidence
that the relative difficulty of the puzzles was being captured.
A goal for future research is to find a more sophisticated
strategy for enumerating the solutions of these puzzles. A good
candidate is Monte-Carlo tree search (MCTS) which has been
used [6] as a solver for a game called Blokus Duo which is
not dissimilar to the polyomino puzzles proposed in this study.
Multiple replicates of the solution sampler showed that its output is reliable as a measure of relative difficulty, but complete
enumeration would yield more accurate results. Approaches
like MCTS would permit more quantified measures. Permitting
an MCTS solver a fixed amount of time and counting the
number of solutions located would be an obvious approach.
A. Solving or Designing?
In the course of performing background research for this
study, the authors found many papers on using evolution to
solve puzzles for each paper on using evolution to design
puzzles. On reflection, the authors feel this is an odd distribution of effort. There is a great deal of AI research that has
yielded subtle, powerful algorithms for tree search that can be
used, among other things, to solve puzzles. Designing puzzles,
however, is a much more difficult and more poorly specified
task. This means that it is likely to be more appropriate as a
target for evolutionary computation. Perhaps researchers are
more likely to choose easier problems, but we exhort our colleagues to put more effort into learning how to use evolution to
design puzzles. We also apologize to any evolutionary puzzle
designers whose work we missed.
B. Varying Connector Probability
A mathematical analysis of the increase in difficulty of
locating polyomino puzzles of larger sizes suggests that part
of the problem is that larger objects have more chances to
be connected or to form large, connected sub-objects. This
phenomenon is an instance of the Kolmogorov zero-one law
[7]. This, in turn, suggests a way to modify the representation
to achieve better performance.
The current representation uses four equally likely values
at each loci which form no connections, a connection to the
right, downward, or both right and downward. Modifications
to the code that generates the initial population and performs
mutation to use non-uniform probabilities, enhancing the probability of the “no connection” value for example, could be
used to tune the expected connectivity of population members,
potentially adjusting for higher connectivity in larger objects.
C. Secondary Puzzles
In the original Ten-Yen puzzle, the directions not only gave
the goal of assembling the pieces into a 6x6 square but also
gave goals such as “make the colored regions contiguous” or
“make the border of the puzzle black”. Secondary goals of
this sort make a puzzle into a collection of puzzles and hence
more interesting to potential users. The only goal of coloring
the pieces of the puzzles located in this study was to provide
relatively high-contrast drawings.
It would not be difficult to incorporate secondary puzzle
goals into the fitness function for locating polyomino puzzles.
This might require that the representation be made more
complex so that it contains information about what color each
piece is. The best way to implement such a system is another
nature next step for this research.
D. Other Shapes and Objectives
Modifying the code used in this study to make polyomino
puzzles for shapes other than squares is an almost trivial modification, so long as the size of these objects is not too large.
There is another potentially more interesting modification,
which is to evolve sets of polyominos that can fill multiple
shapes. In the simplest case, the pieces for a 6x6 puzzle could
also be asked to fill a 4x9 rectangle. The current evolutionary
setup is not one that could be modified easily to this task.
For this, a puzzle solver such as MCTS, would be needed
as the core of the fitness evaluation. The representation would
specify a collection of distinct polyominos and fitness would
be based on the ability of the solver to fill the target shapes
with those polyominos. A natural choice would be the geometric mean of the number of solutions found for the different
targets, or multi-criteria optimization could be used.
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