VII. GAME TYPE 4: NETWORKING GAMES

     What 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.

Basic Structure 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 Network Games

     The 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.

     As 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.

     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 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):

     The 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.

     How 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.

Question One

(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 correct answer.

Let's try another question.

Question Two

(2) What is a complete list of the cities from which a traveler could not drive directly to city B?

(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 network diagrams.

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 German.
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.

     Now, 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:

     Now, 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 diagram.

     The 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.

     Do 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 created.

     Okay, 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:

      You 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.

Question One

(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

Solution
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 answer.

Let's try a second question.

Question Two

(2) Besides Ms. Win, who can converse with Ms. Farmer without an interpreter?

(A) Mr. Andrews and Ms. Petrucci
(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.

Question Three

(3) Which of the following pairs of people cannot converse without an interpreter?

(A) Mr. Andrews and Ms. Petrucci
(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 game.

Question Four

(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

Solution
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 like this.

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 conditions.

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.

Question One

(1) Which of the following is a complete list of the people to whom a rumor can be passed directly from Q?

(A) R
(B) S
(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 C.

Question Two

(2) To which person in the group can any other person pass a rumor directly?

(A) P
(B) Q
(C) R
(D) S
(E) T

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.

Question Three

(3) A rumor that begins with R and reaches T must have been told to which of the other friends:

(A) P
(B) Q
(C) S
(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.
  

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1: Ordering Games
2: Characteristic Games
3: Grouping Games
4: Network Games
5: Non Linear Spatial Games
6: Map Games

Continue to:

V. GAME TYPE 5: SPATIAL GAMES