Cognition Lab
Lab 0: Problem Solving
Problem 1. Draw four straights lines that pass through each of the nine dots without removing
your pencil from the paper.
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Problem 2. Arrange six matches so that they form four triangles with all sides equal to the length
of one match.
Problem 3. In each of the problems below you have 3 empty jars (A, B, & C) with the capacities
listed. Your job is to measure out a volume of water (listed as “desired amount”) by using the
three jars.
Problem
Capacity of jar A
Capacity of jar B
Capacity of jar C
Desired amount
a.
21
127
3
100
b.
14
163
25
99
c.
18
43
10
5
d.
9
42
6
21
e.
20
59
4
31
f.
23
49
3
20
g.
18
48
4
22
h.
14
36
8
6
Problem 4. Think of as many uses as you can for a flathead screwdriver.
Problem 5. For each triplet below, think of a word that goes with each of the three words listed
to form a familiar phrase or compound word.
Example: snow, down, out (Answer: fall. snow fall, downfall, fall out)
a. ache, sweet, burn
b. off, top, tail
c. top, pin Panama
d. law, case, dress
Problem 6. Assume that a steel pipe is embedded in the concrete floor of a bare room, as shown in
the Figure. The inside diameter is 0.06 inches larger than the diameter of a ping-pong ball (1.5
inches) that is resting gently at the bottom of the pipe. You are one of six people in the room,
along with the following objects:
100 feet of clothesline
A carpenter’s hammer
A chisel
A box of Wheaties
A file
A wire coat hanger
A monkey wrench
A light bulb
List as many ways as you can think of to get the ball out of the pipe without damaging the ball,
tube, or floor.
Problem 7. You are given 4 separate pieces of chain that are each 3 links in length (see left
diagram). It costs $100 to open a link and $150 to close a link. All links are closed at the
beginning of the problem. Your goal is to join all 12 links of chain into a single circle (see right
diagram). Your total budget for forming the circle is $750.
Problem 8. A hobo can make 1 whole cigar from every 5 butts he finds. How many cigars can he
make if he finds 25 cigar butts?
Problem 9. Ten male senators are on their way to the Inaugural Ball. A crowd of disgruntled
taxpayers attacks them with a volley of snowballs, knocking each senator’s top hat to the ground.
A helpful page retrieves the hats and hands one to each senator— but without checking to see
who owns which one. What is the exact probability that exactly 9 senators will receive their own
hats?
Problem 10. A conversation took place between two friends, a philosopher and a mathematician,
who had not seen or heard from each other in years. The mathematician, who had an exceedingly
good memory, asked the philosopher how many children he had. The philosopher replied that he
had three. The mathematician then asked how old the children were. Her friend, who knew how
much most mathematicians enjoy puzzles, said that he would give her a number of clues to the
children’s ages.
The philosopher’s first clue: “the product of the children’s ages is 36.” The mathematician
immediately replied that this was insufficient information.
The philosopher’s second clue: “All of the children’s ages are integers; none are fractional
ages such as 1½ years old.” Still, the mathematician could not deduce the correct answer.
The philosopher’s third clue: “The sum of the three children’s ages is identical to the
address of the house where we played chess together often, years ago.” The mathematician still
required more information.
The philosopher then gave his fourth clue: “the oldest child looks like me.” At this point,
the mathematician was able to determine the ages of the three children. Here is your problem:
What were the ages of the three children?
Instructor’s guide to Lab 0: Problem Solving
Procedure: Divide the class up into 3 teams. Give them the first page of problems (#1 - 3) and
have them “race” to see who can solve the problems first. Provide actual matches (6 per group)
to the teams for problem #2. After the winning team finishes and the other teams are mostly
done, go over the answers by asking different teams to explain their answers. As you go through
the problems explain what psychological points they are meant to illustrate.
Then give each team the second page of problems (#4 - 6) and give them a specified
amount of time (say, 10 min) to work out their answers. For problem #5 there are “right”
answers (although it’s possible that there are additional answers besides the ones I’ve listed
below). For Problems #4 and #6 there are no right answers. Have each team tell the rest of the
class what solutions they came up with while you compile a “Master list” of all solutions on the
blackboard.
Finally, give each team the last page of problems (#7 - 10) and given them a specified
amount of time (say, 10 min) to work on them. Then go through the problems and have team
share their solutions.
Problem solutions and what they illustrate:
Problem solving is a classic way to study cognitive psychology, because so many real life
cognitive operations involve solving problems. A “problem,” according to this research tradition,
involves a starting state, a goal, and a series of actions that one could take to reach the goal (this
is the state-action formulation of problem space).
Functional fixedness. Problems #1 - 3 illustrate functional fixedness (a.k.a. mental set).
Functional fixedness occurs when the problem solver imposes constraints that don’t really exist.
An action or move is seen to have a particular function, and problem solvers don’t think to use
that action for a different function.
In problem #1, the nine dots problem, people often think that their lines need to stay
within the box formed by the dots. To solve the problem they need to “think outside the box” and
draw lines forming a triangle that is larger than the box.
In problem #2, the match stick problem, people often think that the matches must be
arranged in two dimensions— say on the flat surface of a table. The solution to this problem is to
form a 3-dimensional pyramid where each of the 4 sides is a an equilateral triangle.
In problem #3 (Luchin’s water jug problem), the first six items are solved by using the
formula B - A - 2C. This formula will also work for the last two items. However, these last two
items can also be solved with a simpler operation: A + C for item g and A - C for item h. See if
anyone fell for the trick and used the more complex formula. Here, the first 6 items caused
problem solvers to get into a “mental set” and stick to it even when a faster and better solution
was possible. As the students whether this is good or bad. On the one hand, they missed a
quicker solution. On the other hand, it may be more efficient to stick with the old formula if it
works, because it is easier than trying simpler formula on every problems (which would have been
a waste of time in the first 6 items).
Creative Problem Solving. One question in problem solving research is what makes for a
creative solution to a problem. The usual definition of creativity is that a creative solution is both
novel and functional; that is, it has to work to solve the problem, but also be a solution no one
else has thought of before. Problems #4 - 6 represent measures of creative problem solving.
Problem #4 is the “Alternative Uses Test”. The idea is that if you can think of a lot of
uses for a screwdriver, some of them will be novel and useful. Note that answering this question
requires that one not be functionally fixated. If one is fixated on the screwdriver’s function of
turning screws, then one will not think of alternative uses such as lever, paper weight, weapon, or
coffee stirrer.
Problem #5 is the remote association test. The answers are heart, spin, hat suit. Here the
idea is that to be novel one has to think of remote associations— associations that are related to
the given words, but not closely related. You have to be able to think of a lot of remote
associates to be able to find one that is related to all three words. Ask students how they solved
this problem? Did they try out a lot of associates? Or did the solution just come to them?
Problem #6 is like the Alternative Uses Test because the goal is to think of as many
solutions as possible, some of which will be novel. Some solutions students might come up with
are to chew the Wheaties to make a paste which is then used to glue the clothesline to the ping
pong ball. Or to use the lightbulb filament as a very thin noose to go around the ball. They
probably won’t think of solutions such as urinating into the tube so that the ball floats to the top.
This is an another example of functional fixedness— they don’t view urine as a tool, and they
impose constraints on the situation (like not urinating in public) which are not stated in the
problem description. After revealing this bizarre solution, see how many additional solutions
students can think of. Encourage them to view the other 5 people as tools, and see if that
prompts additional solutions.
Process analysis. Process analysis is a technique for problem solving in which one thinks through
each step of the solution process and imagines what the problem solver would or would not know
at each step.
Problem #7 is actually just another example of functional fixedness— I just put it in this
group because it’s so hard. Usually people open up a link at the end of chain A and attach it to
chain B. They then open up a link at the end of chain AB and attach it to C, which they then
attach to D. At this point they have used up their budget and have a snake rather than a necklace.
The solution is to open up all 3 links of chain A. Use one link to combine B & C, another to
combine C & D, and the third to combine D & B.
Problem #8 is an example of process analysis because one is supposed to think about what
the hobo would do. He uses his 25 butts to make 5 cigars. Then what does he do? He smokes
them, of course. And what does he have left? 5 butts, from which he makes a 6th cigar. So the
answer is 6.
Problem #9 might look like a complicated probability problem, but it is actually a process
analysis problem. Picture the page holding the last hat. Nine of the senators have their own hats.
To whom will the page give the 10th hat? To the 10th senator, of course. And if all the other
senators have their own hats, then the 10th hat must belong to the 10th senator. So, the probability
that exactly 9 senators will get their own hats is 0%. Another way to think about the problem is
that, for hats to be mixed up, at least two senators must get the wrong hats.
Problem #10 is the hardest problem of all, and it’s possible that no one will solve it. The
last clue seems like no clue at all, and yet the mathematician uses it to solve the puzzle. To get
the solution, you have to think through what the mathematician knew at each step.
Step 1. Knowing the 3 ages have a product of 36 is no help, because an infinite set meets
this criterion.
Step 2. Knowing that all 3 ages are integers means the mathematician has these
possibilities:
sum
1
1
36
38
1
2
18
21
1
3
12
16
1
4
9
14
1
6
6
13
2
2
9
13
2
3
6
11
3
3
4
10
Step 3. Next the philosopher says that the sum of the ages is the same as the address
where they use to play chess. Presumably the mathematician with the good memory would
remember this address. So why doesn’t she know the solution at this point? It must be because
this clue doesn’t give a unique solution. Note that there are two set of ages that add to the same
total (13). The mathematician must be unsure which of these sets is right. So, we are down to
two sets:
1
6
6
2
2
9
Step 4. The final clue is that “the oldest looks like me.” This tells the mathematician that
there is an oldest. One set of ages has twins as oldest. So the correct answer must be the other
set: 2, 2, 9.
Closing. In closing, you might explain that it’s important for cognitive psychologists to find out
what problems are hard (and why) and what mistakes people make (and why), because patterns of
error offer clues as to the psychological processes that people go through in solving problems.
For example, we could look at all the unsuccessful attempts to solve the 9-dot problem and see
how often the lines went outside the box. If they rarely did, that would suggest that people were
constraining themselves to stay inside the box (explaining why they failed to solve the problem).
If they often went outside the box, that would suggest that people see the necessity of going
outside the box but fail to identify exactly how. Finally, students shouldn’t feel bad if they got
some problems wrong, as these are designed to be hard precisely because of psychological
properties such as functional fixedness.