VII. GAME TYPE 4: NETWORKING GAMES
are network games? These games involve the spatial connections
between elements. These problems could involve any elements that
form a network, or a series of pathways. Like computers linked
in a network, the elements of network games are connected. For
example, you could imagine networks that represent various roads
connecting several cities, or perhaps the flow of information
between a group of office workers, or a series of electric bulbs
linked in a circuit. Your task in solving these games is to determine
the nature of these connections between elements.
of Network Games
Like the other games we have looked at, the elements will usually
be presented in the initial premise, or set-up of the game. The
conditions will then describe the connections between the elements.
It is these connections, or the relationships between the elements
that are important for network games. These conditions might
tell you that the connection between two elements is bi-directional
(two-way) or unidirectional (one-way). A condition might tell
you that two elements are not connected directly, in which case
a pathway might require multiple steps to get from one element
to another. Other elements could be dead ends, that is, once
you have connected to them, you cannot move through the network
to reach another element. We will see examples of these different
kinds of conditions as we work through the sample games.
How to Solve
one advantage of network games is that once you have created
the network flow diagram describing the elements and the relationships
between the elements, the questions themselves are usually quite
straightforward. Usually all the answers can be quickly determined
by simply reading what you have drawn. The network flow diagram
is therefore critical for these games. However, you must learn
to create effective flow diagrams, which is what we will show
you here. These games commonly have many more conditions than
the other types of games, and unless you are able to do some
preliminary sorting of information, it is easy to create a network
diagram that is too complicated and impossible to use. Again,
by learning how to translate the conditions of the game accurately
and represent them graphically in a precise way, you will become
well-equipped in the skills required to solve these problems.
with other games, start with a roster of the elements. If the
elements are simply letters, such as X, Y, and Z, then you can
just use those letters as the elements. If the premise gives
longer names, such as four towns called Fayestown, Gainesville,
Huron, and Indica, simply abbreviate the names to the first letter
of each word (F, G, H, I). Creating the roster is especially
critical if some of the elements are not specifically mentioned
in the conditions. By writing out the roster for each game, you
won't accidentally forget one or more of the elements when trying
to answer the questions.
the roster has been created, create a diagram in which the elements
are arranged circularly. Next, read through the conditions and
represent them symbolically on this diagram. Let's start with
some basic guidelines for how to effectively represent this information.
Since we are dealing with connections, we will rely heavily on
the use of arrows. Pay close attention to the directionality
of connections. Is the connection one-way? Use a line with an
arrow pointing in the direction of the flow. Is the connection
two-way? Have arrows pointing in both directions between the
two elements. How will we represent the elements themselves?
A simple strategy is just to use the letters you have assigned
in the roster. For example, let's look at a sample game with
a simple premise and only two sentences describing the conditions.
Sample Game One
Five cities- A, B, C, D, and
E - are connected by a series of roads. At most, one road connects
any two cities. All cities are connected to at least one other
city. All of the roads connecting these five cities are either
one-way or two-way roads as follows (assume that the roads described
are the only roads connecting these cities):
roads between A and C, A and D, and B and D are all two-way roads.
The roads from A to B, C to E,
and E to A are all one-way roads.
can we represent this information? First of all, we will need
to represent the cities. In this case, the roster consists of
A, B, C, D, and E, and we will use these letters. How do we know
how to arrange the cities? We don't know anything about the positioning
of the cities with relation to each other, but we don't need
to. For this problem, we are only interested in the connections
between the cities, not their actual locations. Therefore, we
can arrange them any way we want. What you will find, however,
is that the best way to arrange the elements in a network game
is in a circular form, with the elements spaced evenly. The reason
for this is so we have room in the center of the diagram, or
the space between the elements, in which we will put the information
about the connections. For this game, let's use lines to represent
the roads that connect the elements (in this case, the five cities)
and use arrows to represent the directionality. Your flow chart
might look something like this.
Once you have summarized all
of the information from the conditions, review the initial premise
to make sure that your diagram does not violate any of the conditions
outlined there. Also, be sure that any additional information
provided within the premise has been included in your diagram.
In this case, we can double-check that in our diagram, each city
is connected to at least one other city, and that there is at
most one road between any two cities. This diagram meets all
of those criteria.
Now, let's try to answer some
questions for this sample game.
(1) Which city has the greatest
number of direct road connections to the other cities?
(A) City A
(B) City B
(C) City C
(D) City D
(E) City E
This question asks about direct
connections. Another way to think about this question would be,
which city has the greatest number of roads connecting it to
other cities? Let's look at our diagram, and start with city
A. From city A, roads lead to B, C, D, and a road enters from
E. Since we have only five cities, and city A has four connections,
this is likely to be the right answer! But let's first go through
the other answer choices first so we can be sure. City B can
be reached from A, but from B, one must travel to D. These are
fewer connections than A, so we can eliminate this possibility.
City C is connected to A and D by a two-way road, and a one-way
road that leads to E, for three direct connections. This is still
fewer than the four connections of A. City D also has three connections,
and city E has only two, and both one-way. Therefore, A is the
Let's try another question.
(2) What is a complete list of
the cities from which a traveler could not drive directly to
(A) City C only
(B) City D only
(C) Cities C and D only
(D) Cities C and E only
(E) Cities C, D, and E only
Again, this question is straightforward
with the help of our network diagram. From city C, city B cannot
be reached directly. This is definitely one of the cities, but
we do not yet know whether or not this is a complete list, so
let's first go through the other answer choices. From D, there
is a two-way road to city B, so choice B is not correct. Likewise,
we can eliminate choices C and E since they too contain city
D as one of the cities. If we check choice D, which is cities
C and E, we see that in fact both C and E are cities from which
a person could not travel directly to city B. Therefore choice
D is the correct answer.
Let's try another game, one that
is a little more complicated, to get more practice in creating
Sample Game Two
In celebration of a very profitable
year, the board of directors of an international corporation
decides to hold a special meeting in Basel, Switzerland to honor
the company's top six executives, Mr. Andrews, Mr. Das, Ms. Farmer,
Mr. Gaal, Ms. Petrucci, and Ms. Win.
Mr. Andrews speaks English and
Mr. Das speaks Hindi and French.
Ms. Farmer speaks German and Russian.
Mr. Gaal speaks Russian and English.
Ms. Petrucci speaks French and English.
Ms. Win speaks Russian and Hindi.
Okay, once we have read through
the game, we can see that the elements are the six executives.
What are the connections between the elements? These will be
languages, or the ability to communicate with each other. As
before, our first step is to create the roster. Since we are
given names, let's simply use the first letter for each name.
Roster = A, D, F, G, P, W
Now, let's arrange these six
elements into a circular pattern. Between the elements, in the
center of the circle, is where the connections will be placed.
how will we represent the connections? In this case, we can simply
use lines to connect people with a shared language. (We don't
need to use arrows, because by definition, sharing a language
means that each person can speak it to the other.) Let's now
translate the information given by the conditions into drawn
connections between our six elements. Where do we start? A good
strategy is just to start with one person, and go through each
of his languages, and see what other person (or persons) also
speaks those languages. Let's start with Mr. Andrews (A). He
speaks both English and German. Who else speaks English? Mr.
Gaal and Ms. Win (G and W) both speak English. So we can draw
connections from A to both G and W. What other connection can
we draw? Well, at this point, we could also draw in the connection
between G and W, since we've just noted that the two of them
speak English. At this point, just placing as many of the conditions
onto the diagram as possible should be your main strategy. After
you have gone through all the conditions and translated them
to your diagram, you can then go back over the conditions to
make sure you have not either missed or duplicated any of them.
Back to Mr. Andrews, we know that he also speaks German. Who
else speaks German? Ms. Farmer (F), so let's draw a line connecting
A and F.
The diagram now looks like this:
if you look at the diagram, can you tell who is speaking what
language? Do F and P speak the same language since they both
are connected to A? No, they don't, and it's important to remember
that just because two elements each connect to another element,
it does not mean that those two elements necessarily have any
connection at all between them. This would be an invalid assumption.
One way to avoid this confusion would be to label the connections
in some way that describes them, such as shown in the following
problem with doing this, however, is that by writing words on
our connections, it greatly increases our chances of making our
figure too complicated. Keep in mind that we have only placed
on our diagram the connections between one of our elements with
the others. We still have conditions describing the other five
elements! A better solution is just to keep in mind that you
must accurately translate each condition, and never assume any
other connections beyond that which is provided in the conditions.
we need to know who speaks which language? At this point (since
we haven't seen the questions), we don't know. In the interest
of simplicity, let's assume we don't need to know the languages
at this point. If a question comes up which requires knowing
this information, we can always add it to the diagram we have
now let's try to place the other connections onto our diagram,
working though each element in the order in which they appear
in the conditions. Once you have translated each condition, read
through the conditions one more time to be sure that you have
not missed any connections between elements and that all of the
connections you have drawn are accurate.
The diagram for this game should
now look like this:
may have noticed, as you work your way through the conditions,
you find that some you have already placed on the diagram as
a result of them being two-way connections. When this happens,
it just serves as another way for you to double-check the relationship
between the elements.
Now that we have the network
diagram, let's try some questions.
(1) If Mr. Das and
Mr. Gaal wish to converse, which of the following represents
a complete list of the people who could serve as an interpreter?
(A) Mr. Andrews only
(B) Ms. Petrucci only
(C) Ms. Win only
(D) Ms. Petrucci and Ms. Win only
(E) Mr. Andrews, Ms. Petrucci, and Ms. Win only
How do we solve this? We can see from our diagram that Mr. Das
and Mr. Gaal do not share a language, that is, we have drawn
no line connecting them. An interpreter would be someone who
can speak the same language as Mr. Das and also speak the same
language as Mr. Gaal. In the scheme of our network diagram, what
would this look like? Simply, we are looking for an element that
is connected to D and connected to G. Let's work through the
answer choices. Choice A is Mr. Andrews. According to our diagram,
A is connected to G, though not to D, so this cannot be the correct
answer. Choice B is Ms. Petrucci. In our diagram, P is connected
to both D and G, so this is definitely a correct answer. We are
asked for the complete list, however, so we must continue to
work through the answer choices. Choice C is Ms. Win. We can
eliminate this choice immediately on the basis of it not including
Ms. Petrucci. Choice D is Ms. Petrucci and Ms. Win. If we consult
our diagram, indeed Ms. Win is connected to both D and G. Choice
E adds Mr. Andrews to the list, but we have already eliminated
him as a possible interpreter. Therefore choice D is the correct
Let's try a second question.
(2) Besides Ms. Win, who
can converse with Ms. Farmer without an interpreter?
(A) Mr. Andrews and Ms.
(B) Mr. Gaal and Mr. Das
(C) Mr. Gaal and Mr. Andrews
(D) Mr. Gaal and Ms. Petrucci
(E) Mr. Das and Ms. Petrucci
Again, this question is relatively
easy with the use of our network diagram. We are asked who can
converse with Ms. Farmer without an interpreter. In the context
of our network diagram, this question is asking to which other
elements, other than W, is element F directly connected. By looking
at our diagram, we see that F is connected to A, W, and G. Therefore
the correct answer is choice C, Mr. Gaal and Mr. Andrews (G and
A on our diagram). If we look at the other choices, we can compare
them to our diagram and see that these are incorrect.
(3) Which of the following pairs
of people cannot converse without an interpreter?
(A) Mr. Andrews and Ms.
(B) Mr. Das and Ms. Win
(C) Mr. Gaal and Ms. Petrucci
(D) Ms. Farmer and Mr. Das
(E) Mr. Andrews and Ms. Farmer
Again, we only need to consult
our network diagram to answer this question. We are looking for
two elements that are not connected with a line. Starting with
choice A, we see that A and P are connected on our diagram, so
this cannot be the correct answer. Choice B, D and W are also
connected, so we can eliminate this choice. Choice C, also, G
and P are connected and thus not the correct answer. Choice D
seems to be the exception, as D and F are not connected on our
diagram. To be sure, let's check choice E. We see that A and
F are connected, so choice D represents the only pair of people
(of the choices given) that cannot converse without an interpreter.
Let's do one final question for
this game, which is structurally different from the ones we have
just solved, though is a reasonable question for this kind of
(4) Of the five languages spoken
by the six executives, which are the two most common?
(A) English and German
(B) English and Hindi
(C) French and Russian
(D) Russian and German
(E) Russian and English
As you remember, we have not included anything about which languages
are spoken by which people on our diagram. Can we solve this
question from our diagram? Unfortunately, no. We basically need
to determine how many people speak each of the languages. The
best way to do this is make a small table in which each column
represents a language, and underneath, we can list the people
that speak each of those languages. Then we can simply count
how many people are in each column. The table would look something
From this, it is easy to see
that English and Russian are the most commonly spoken languages.
Therefore, choice E is correct.
Let's try one final game to illustrate
the best way to solve network games. Unlike the previous games,
here we also show you the first question, like it would be on
the actual exam.
Sample Game Three
In a group of five friends- P,
Q, R, S, and T- rumors are passed according to the following
P passes rumors to R, S, and
T, but nobody from the group passes rumors to P.
Rumors can pass from R to S, but not from S to R.
Rumors can pass in either direction between S and T.
Rumors can pass in either direction between Q and R.
Rumors can pass from Q to S, but not from S to Q.
(1) Which of the following
is a complete list of the people to whom a rumor can be passed
directly from Q?
(C) R, S
(D) P, R, S
(E) R, S, T
Okay, let's start by creating
our network diagram. What are the elements? In this case, the
elements are the five people, P, Q, R, S and T. What are the
connections? The connections between the people are the way in
which rumors can be passed. (This falls into the flow of information
category of network problems.) What type of graphical representation
will we need for these connections, lines or arrows? For this
problem, there is directionality of connections (some are one-way,
some are two-way), so we must use arrows to illustrate the direction
of the flow of information.
Your network diagram might look
something like this:
Like the other diagrams, we have
arranged the elements circularly to illustrate the connections
between them in the middle of the circle. We have used arrows,
some pointing in two directions, others in only one direction,
to indicate the direction of the flow of information.
Now let's tackle the question.
We are asked for the complete list of the people to whom a rumor
can be passed directly from Q. By consulting our diagram, we
see that a rumor from Q can go directly only to R and to S, choice
(2) To which person in the group
can any other person pass a rumor directly?
Consulting our diagram, we need
to look for the element that is connected to all other elements
(with the arrows pointing at that same element). We can see that
the only element that satisfies this condition is S.
(3) A rumor that begins with
R and reaches T must have been told to which of the other friends:
(D) Q and S
(E) P and S
Again, let's consult our diagram.
If we start with R, the rumor cannot be passed directly to T,
but instead must be passed to either S or Q. If it is passed
to S, then it can be passed directly from there to T. If R first
passed to Q, then it must be passed to S, and then to T. In both
scenarios, S is required. Since we are asked which friend must
have been told, choice C is the correct answer. Now, if we were
asked which group members the rumor might have been passed to,
the answer would be S and Q.
Summary of Strategy
for Network Games
Hopefully these three sample
games have helped to illustrate the different kinds of network
games that you might see on the exam. The critical skill needed
to successfully work through these games is to be able to draw
an effective and accurate network diagram. Look over the examples
of network diagrams presented here, and use these as guidelines
in creating your own.
To review the basic strategy:
- As with other games, start with
a roster of the elements.
- Once the roster has been created,
create a diagram in which the elements are arranged circularly.
Next, read through the conditions and represent them symbolically
on this network diagram.
- Read through the premise and
conditions again to be sure you have translated all the information
onto your diagram.
- Read the question and solve
by consulting the network diagram.
1: Ordering Games
2: Characteristic Games
3: Grouping Games
4: Network Games
5: Non Linear Spatial Games
6: Map Games
GAME TYPE 5: SPATIAL GAMES