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PROBLEM SOLVING
ZYGMUNT PIZLO
University of California, Irvine
ZYGMUNT PIZLO
University of California, Irvine
ZYGMUNT PIZLO
University of California, Irvine
Modern Research on the Human Ability to Solve Problems that Have Large
Search Spaces
. Permutations and Combinations; Polynomial and Exponential Numbers
of Computations
. Nearest Neighbor Algorithm for the TSP
. Something Was In the Air: How the Cognitive Science Community Actually
Discovered the TSP
. Problems to Solve
vii
References
Index
xiv
This book is intended for undergraduate and graduate courses on Human Problem
Solving. Note that such a course has rarely been offered at American universities despite
the fact that it is as important as such traditional and widely offered courses as Sensation
and Perception and Cognitive Psychology. This book covers insight problem solving, the
role of symmetry and invariance in scientific discovery, combinatorial optimization
problems, and the contribution of gestalt psychology, especially its emphasis on mental
representations. In fact, the mental representation of problems turns out to be the
underlying theme of the entire book. The first chapter explains why mental representa-
tions are necessary in problem solving and the rest of the book describes a wide range of
possible representations and their use across all, or almost all, types of problems. The
book also includes perceptual and cognitive inferences, which are treated as solutions of
constrained optimization problems, the Theory of Mind, mathematics problems, as well
as intuitive physics and causal reasoning.
This textbook emphasizes understanding the mathematical and computational mech-
anisms underlying problem solving. I want the students to learn what is computed and
how it is computed when problems are solved. The topics, listed above, allow me to
explain the theoretical concepts inherent in solving problems. My preferred emphasis on
theory proves to be fruitful because it provides a relatively coherent, intelligible story. The
book also describes many empirical studies, but they only play a supportive role.
Structuring the class this way will prepare cognitive psychology students to explore the
related area called artificial intelligence, and it will make it easy for computer science and
engineering students to venture into the science of the mind.
Keeping this book to a manageable length required me to leave out some material,
such as the neuroscience of problem solving, reinforcement learning and decision
making. These as well as other topics can easily be added by Instructors when they use
this textbook in their classes on problem solving.
The book provides problems to solve and projects to do after each chapter. Some
problems are easy, while others are difficult or even very difficult. Each instructor may
decide which problems to use. The text throughout the entire book provides many other
problems that are solved partially or completely, as well as references to other sources that
have additional problems. The book is accompanied by a software library, written in
Python, and hosted on GitHub at the following link: https://github.com/jackvandrunen/
tsp. This software was developed by Jacob VanDrunen and it provides tools for solving
the Traveling Salesman Problem. The reader can find instructions in this link on how to
download and install the library by using Python’s package manager, as well as links to
xv
Problem Solving
Definition of the Main Concepts
(a) (b)
(c)
Figure . Inscribing a square into a triangle (from Polya, ). (a) inscribed square. (b) squares whose
sizes are not correct. (c) the dashed line intersects the right side of the triangle at the point that is the top-
right corner of the inscribed square.
Here is an example of a virtual -puzzle: www.artbylogic.com/puzzles/numSlider/numberShuffle.htm?rows=&cols=
&sqr=.
Figure . A start state (a) and a goal state (b) (from Pizlo & Li, ).
invariants and symmetries in this book). The existence of these two disjoint sets was used
years ago to popularize this puzzle. The start state was produced by swapping
(physically) tiles and . A big monetary prize was offered for a solution that would
produce the goal state shown in Figure .b by any sequence of legal moves. We now
know that this is impossible. In fact, swapping any pair of tiles physically brings the -
puzzle to the other half of the permutations. All of this will be explained in Chapter . At
this point, the -puzzle is important for us because it is a good way to illustrate the fact
that problem solving can be thought of as a goal-directed activity. Indeed, the very
definition of how the -puzzle is played includes the concept of the start state and
the goal state, and the task (problem) is to get to the goal.
The early stages of AI research did use the -puzzle, as well as other sizes of the same
type of puzzle, namely, the -, -, and -puzzle, as examples for formulating theories of
problem solving. The number of states in these puzzles is (N+)!/. Here N is the
number of tiles and (N+) is the size of the board ( , , and in the
examples just mentioned). The fact that these puzzles are members of the class of NP-
hard problems means that finding the shortest number of moves to the goal state may
require a brute-force search through most or all of the states. This kind of search is
impractical because N! is a large number and it grows very quickly with N. For the -
puzzle, the number of states that can be produced by legal moves is !/ . This is
times more than the number of neurons in your brain. Another way to illustrate how
big this number is, is to realize that if you started at the time of the Big Bang, that is,
about billion years ago, and kept producing states a day, you would have just
finished looking through all of the states in the -puzzle. And, if you produced half a
billion states per second and started at the Big Bang, you would have just finished looking
through all states of a -puzzle. Do people solving such problems actually examine a
large or a small fraction of all possible states, and if they examine only a small fraction,
how small is this fraction? You will surely be surprised by the answers when you get them.
Block
F G 6 H
H
5
D 4
E 13 2
14
15 3
C 16
1
17 2
18
1
A A
Apparatus used in preliminary training Apparatus used in the test trial
representation. But now we can see that the concept of mental representation is even
more fundamental than the gestalt psychologists claimed because mental representation
is a necessary condition for any goal-directed action, including solving problems that are
not insight problems. Without mental representations, goal-directed actions would
remain outside of modern science, and even more importantly, we humans would be
unable to plan and carry on goal-directed actions if we did not have mental representa-
tions. Without goal-directed actions, we humans could not be “intelligent.” Edward
Tolman, who worked in the first half of the twentieth century, was one of the first to use
the concept of goal-directed (purposive) behavior in his theories and experiments. Look
at Figure . taken from his paper. The rat was trained in the simple maze shown
on the left. The entrance is marked as A and the goal (food) is on top-right rendered with
an H within a circle. After the training was completed, the rat was presented with the
maze shown on the right, whose entrance was identical to the entrance of the training
maze, but the rest of the maze was changed. Faced with a blocked alley that used to go to
the goal, the rat came back to the circular chamber, and almost immediately ran along the
alley marked as which led directly to the position where the food had been located
during the training trials. The rat chose an available shortcut, when the familiar path was
blocked. This choice could not be a result of training. It was the result of the rat creating
and using an accurate spatial mental map of the maze.
Dogs and chimpanzees also can use spatial maps when they go around an obstacle.
Wolfgang Köhler used the configuration in which a dog or a chimp stands on one side of
a transparent fence, and food is placed on the other side of the fence. The animal quickly
realizes that the fence cannot be penetrated, looks around and runs around the fence.
This behavior is not trivial because the animal must, at first, face away, putting the food
out of sight as it turns and runs away from the food. But once we assume that the animal
has a spatial map of its environment, this behavior seems natural. The chicken, whose
“intelligence” I praised when I discussed its innate depth perception earlier in this
chapter, fails this “obstacle test.”
This description of two levels of analysis, one representing the physiological and the other representing the mental level,
is completely analogous to the double-aspect view in the mind/body problem, which is a characteristic of a neutral
monism. One of the best illustrations of the double-aspect view comes from William James (: ) who compared
(i) a physical description of a painting, like Raphael’s The School of Athens, where the description characterizes the
chemical composition of the paint at each point in the fresco, to (ii) a mental description, in which one would simply say
that the painting depicts a number of students, all involved in conversations. Both of these two descriptions are valid, but
one cannot be easily translated into the other.
When you stand in front of a mirror, and you move your right hand, your reflected
copy moves her left hand. How does the mirror know to reflect left and right, but not
top and bottom? (The answer is in Chapter .)
There are three rooms. One of them contains an expensive item (e.g., a car), whereas
the other two rooms contain objects with very low value. You don’t know which
room contains which item. You are asked to point to one of the rooms. After that, the
host, who knows which room contains the car, opens the door of one of the other
rooms without the car. Now you are told to make your final pick: either you can stay
with your original choice, or you can switch to the other room. Should you switch in
order to increase your chances of getting the car? (After Burkholder, .)
You start at a point and walk one mile south. Then you turn and walk one mile west.
Finally, you turn and walk one mile North. Is it possible that you ended up at your
starting point?
Glove Selection: There are gloves in a drawer: pairs of black gloves, pairs of
brown, and pairs of gray. You select the gloves in the dark and can check them only
after a selection has been made. What is the smallest number of gloves you need to
select to guarantee getting the following: (a) at least one matching pair; (b) at least
one matching pair of each color? (From Levitin & Levitin, , with permission
from Oxford Publishing Limited.)
Ferrying Soldiers: A detachment of soldiers must cross a wide and deep river with
no bridge in sight. They notice two -year-old boys playing in a rowboat by the
shore. The boat is so tiny, however, that it can only hold two boys or one soldier.
How can the soldiers get across the river and leave the boys in joint possession of the
boat? How many times does the boat pass from shore to shore in your algorithm?
(From Levitin & Levitin, , with permission from Oxford Publishing Limited.)
Inverting a Coin Triangle: Consider an equilateral triangle formed by closely packed
pennies or other identical coins like the one shown in Figure .. (The centers of the
coins are assumed to be at the points of the equilateral triangular lattice.) Flip the
triangle upside down in the minimum number of moves if on each move you can
slide one coin at a time to its new position. (From Levitin & Levitin, , with
permission from Oxford Publishing Limited.)
Sorting in : There are five items of different weights and a two-pan balance scale
with no weights. Order the items in increasing order of their weights, making no
more than seven weighings. (From Levitin & Levitin, , with permission from
Oxford Publishing Limited.)
. The Role of Brain Size: How Carnivores Solve the Puzzle Box Problem
Is a bigger brain better? It is tempting to answer “yes,” but, surprisingly, there is only very
weak experimental support for this answer. If our brain is compared to a computer’s
hardware, it is obvious that the larger the CPU and RAM, the greater the potential for
solving problems. This is true because a computer that has more elaborate hardware in
the form of the number of its transistors, is capable of handling more sophisticated and
smarter software. But, even if a computer is very elaborate, it will not be smart if it does
not have sophisticated computer programs. If sophisticated software is not installed, the
hardware, all by itself, cannot be smart. This is true with all computers. Now, what’s the
https://en.wikipedia.org/wiki/Artificial_general_intelligence.
https://en.wikipedia.org/wiki/Brain-to-body_mass_ratio.
1.0
F C
A
PM
0.8
M
F
M
0.6
V U
U
F C
0.4
F
YF
A
0.2
C
F UU
F PF
0.0
H F FCV PCF CC
MY
H U
–0.5 0.0 0.5
Figure . Left: the relation between the brain volume and the body mass. Right: the relation between
problem-solving success that mass-corrected brain volume. (From Benson-Amram et al., , with
permission from Proceedings of the National Academy of Sciences.)
in the left-hand corner of this graph, you see that shrews, which have neither language
nor abstract thinking, have a brain to body weight ratio of almost percent – five times
a human’s. So, the ratio of brain and body weight will not do, either.
The graph in Figure . (left panel) is similar to the graph just described. One
difference is that instead of brain weight it plots brain volume in milliliters (mL) on the
vertical axis. The second difference is that it shows data points for only mammalian
carnivores, rather than for all mammals. This particular study (Benson-Amram et al.,
) tried to correlate brain size with problem-solving ability. The authors tested
animals from species in families of zoo-housed animals. The axes in
Figure . (left panel) are again logarithmic, more precisely they use the natural
logarithm, whose base is e .. The weight of the smallest animal was . kg (ln
(.) = .), and the weight of the largest animal was kg (ln() = .). The brain
volume of the smallest brain was . mL (ln(.) = .), and the brain volume of the
largest brain was mL (ln() = .). The data points represent averages for indi-
vidual species. Once again, their data points are not far from the regression line which
means that the larger animals do have larger brains.
All animals were tested with a puzzle box. Their task was to move the latch and open
the door of the box. There was a favorite food bait inside. The most successful animals
were the bears, who opened the box on percent of the trials, raccoons ( percent),
and wolverines ( percent). Meerkats were least successful: they never solved the puzzle
box problem. Now look at Figure ., right panel. It shows the relationship between the
proportion correct in solving the puzzle box problem and the brain’s volume corrected
for the body mass. This corresponds to how far the brain volume is above the regression
line in the left panel of Figure .. The regression line in the right panel has a positive
http://movie-usa.glencoesoftware.com/video/./pnas./video-.
.. Why Does the Mirror Reflect Left and Right But Not Top and Bottom?
Look at Figure .a where I can be seen standing in front of a large mirror. I took this
photo of myself while I was holding a piece of paper on which I used a red marker to
draw X and Y axes, as well as a point V in the first quadrant of its coordinate system. You
can see the axes and the point in the mirror clearly because my drawing faces the mirror.
Note that you can also see what was drawn because the red marker shows through the
paper. This allows you to compare the reflected image with the drawing. First, it should
be obvious that the mirror is not reflecting left and right, or top and bottom. Both the Y-
axis on the paper and the Y-axis in the mirror image of the paper point up. So, the mirror
does not reflect the top and bottom. We have known this since philosophers looked at
mirrors in ancient Greece. More importantly, the X-axis on the paper and the X-axis in
the mirror image point in the same direction, specifically, toward the right edge of the
mirror, where “right” is defined in the coordinate system attached to my body. Clearly,
the mirror did not reflect (swap) left and right. Now, consider the third axis, the Z-axis
that represents the third dimension in this scene. Here, you have to use your imagination
because I did not draw this axis in the scene I am showing you. Let the Z-axis in the scene
be horizontal and orthogonal ( degrees) to the surface of the mirror. Furthermore, let
the Z-axis’s direction project from my torso toward the piece of paper that I am holding
in front of me. Now, imagine how the reflection of this axis would look in the mirror. It
will point from the image of my torso toward the image of the piece of paper. Clearly, the
direction of the Z-axis has been reflected (flipped) from front to back. This is what a
mirror does: it reflects the depth axis that is orthogonal to the mirror and preserves the axes
that are parallel to the mirror. If you need additional cues to this problem, consider an
explanation involving motion: imagine that I move the piece of paper to the right. The
piece of paper reflected in the mirror will move in the same direction, namely to the right
Figure .. (a) reflection in a mirror; (b) rotation around Y-axis; and (c) rotation around X-axis.
As the house door was closed and the door of the shed
stood open, Vinzi went to the hayrick. That no steps led up
to the little door was not surprising to Vinzi; he knew the
arrangement. The little shed did not rest on the ground but
stood firmly on four blocks, to keep the hay dry and
ventilated. As Vinzi knew, it was a case of clambering up to
the open door, which was so low a full-grown man had to
stoop to enter. Vinzi climbed up nimbly, and found a tall
man working inside.
"He can eat right away," said she. "Supper is just ready;
the smoke drove me from the hearth. I will serve it at once,
for we need not wait for the boys; they will soon be
coming."
Stepping into the house, she took off Vinzi's knapsack,
and the lad was soon comfortably seated at the table. As
they ate, all timidity vanished. He was ravenously hungry
for he had scarcely eaten on the journey. Somehow his
cousin must have guessed this, and long before his plate
was empty, had heaped it again. Vinzi thought he had never
eaten anything better than the steaming potatoes and the
lovely yellow cheese.
Now and again the wife would say, "Pour out some more
milk for the boy. He must be thirsty after all the wind and
dust on that long trip."
She bustled off, and her three sons soon followed, and
as Vinzi thought perhaps he could help, he would have gone
too. But his cousin beckoned him back, declaring his
knapsack had been enough of a load for that day; it was
none too light, and hanging it on his arm, they went out to
the hayrick.
When the last had been put in, his cousin said, "Now
we'll say good-night. Inside the door is a wooden bolt, just
like the one outside. With it fastened, you are sole master
of your castle."
"Yes, I'll call good and loud through the round air hole,"
promised Jos.
When they reached the house, the cousin and his wife
were still at the breakfast table. Both gave Vinzi a friendly
greeting, and Josepha set a large cup of coffee before him,
suggesting that he eat plenty of bread with it, for the fresh
mountain breeze would soon make him hungry.
Jos and Faz had all they could do to keep the cows on
the roadway and to urge them along, for fresh grass
tempted them now to one side, now to the other.
Russli ran eagerly ahead and soon turned off the road
across a pasture, until he reached a large bush whose
branches grew straight up into the air.
Vinzi looked around him. Here and there stood tall, dark
larches, through whose delicate branches one could glimpse
the blue of the heavens above. Beneath their feet stretched
the lovely green of the mountain pasture land, brightened
by the fiery red alpine roses which grew amongst the moss-
covered stones. A full mountain stream rushed along its
course, and the rocks that hindered its passage tossed it
high into snow white foam. So this was the pasture!
Vinzi now ran off to where the cows grazed and looked
about for his cousins. Across the road was a very large
treeless pasture in which browsed many cattle. A small
group of young herders were bending over a smoking spot
on the ground. Jos and Faz were among them, Vinzi saw
that. He called out to Jos with all his might, but in vain for
some time, but as soon as Jos heard him he came over to
him.
With flashing eyes the happy boy put the pipe to his
mouth and blew a high, piercing shriek. Russli himself was
frightened at it.
"Here, take it," said he. "This is yours; the others have
already taken theirs."
Taking the pipe from Faz, Vinzi said, "I will give it a trial
myself," and began to play a little tune.
"Vinzi, keep the little chap with you on the way home;
that will be helping us."
"Look there, the boys are coming with the cattle," said
the father before whose eyes Russli held the pipe. "Go and
show it to your mother. I'll soon come in."
"Are you awake again?" asked Faz who had done the
tugging, fully believing that Vinzi had fallen asleep in the
path.
And now to the house, with Faz in the lead. The mother
was waiting for them at the table, patiently enduring the
noise Russli was making with his pipe.
"Can you play 'I Sing to You with Heart and Mouth?'"
she asked.
Yes, Vinzi knew it well, and after seeking a little for the
right pitch, played with assurance. The mother sang well
and her husband joined in with a strong bass, and suddenly
Jos lifted his fine voice. Faz growled after his father, then
jumped to his mother's high notes, and Russli squeaked in
between. But the other voices were so strong, that these
false notes did not disturb the song. Mrs. Lesa was so
delighted that she begged for another song directly the first
was finished, and then another and another.